1128:') System.SNRdb(iter) System.BCHBER disp('Simulating AWGN channel with BCH & BPSK') %Value of SNR Eb=System.SNR=10^(System.SNRdb(iter)/10)*System.N0; %Signal strength, power. System.Eb= System.SNR*System.N0; Ec=Eb*36.0/63.0; %Source Message System.Mesg=mod(floor(randn(1,System.Length)),2); Rxword=; Chunks=length(System.Mesg)/36; for biter=1:Chunks bmesg=System.Mesg((biter-1)*36+1:biter*36); cwmesg=GF_product(gen_poly,GF_polarize(bmesg)); %convert from the 0,-1 mode to the 1,0 mode. cwmesgTx=cwmesg+1; System.BCHCoded((biter-1)*63+1:(biter)*63)=cwmesg; %Modulate BPSK bch1=bpskmod(cwmesgTx,20,10,sqrt(Ec)); %Add noise bch2=bch1+sqrt(System.N0)*boxmuller(length(bch1)); %Demodulate BPSK cwRx=bpskdemod(bch2,20,10,sqrt(Ec)); % This is in the +1,+0 mode, % convert to a 0,-1 mode cwrecv=cwRx-1; Rxword=; %error pos gives roots w.r.t index of 0. errpos=BCH_decode_berlekamp(cwrecv,63,36,5);%_berlekamp %so add +1, to convert error position locations form 0 index, to +1 index. errpos=errpos+1; if isempty(errpos) %disp('Codeword is correct') correctw=cwrecv; else %printf('Correcting %d errors\n',length(errpos)) %errpos correctw=cwrecv; for i=1:length(errpos) %flip the bits, correctw(errpos(i))=-(1+correctw(errpos(i))); end end %disp('Corrected CW') %correctw %fgets(stdin) CorrectError=length(errpos); System.BCHCorrected(iter)=System.BCHCorrected(iter)+CorrectError; System.BCHDeCoded((biter-1)*63+1:biter*63)=correctw; end System.BCHActualErrors(iter)=Difference(System.BCHCoded,Rxword); System.BCHError(iter)=Difference(System.BCHCoded,System.BCHDeCoded); System.BCHBER(iter)=System.BCHError(iter)/System.BCHLength; %BPSK part of simulation, System.Coded=System.Mesg; %Modulate System.Modulated=bpskmod(System.Coded,System.CarrierRate,System.DataRate,sqrt(System.Eb)); %Channel with AWGN System.Channel=System.Modulated + sqrt(System.N0)*randn(1,length(System.Modulated)); %DeModulate System.DeModulated=bpskdemod(System.Channel,System.CarrierRate,System.DataRate,sqrt(System.Eb)); %Channel Decode System.DeCoded=System.DeModulated; %Error Calculation System.Error(iter)=Difference(System.Coded,System.DeCoded); System.BER(iter)=System.Error(iter)/System.Length; end disp('System Error') System.Error disp('System BER') System.BER disp('System.BCHError') System.BCHError disp('System.BCHBER') System.BCHBER disp('System BCH Actual') System.BCHActualErrors disp('System BCH Corrected') System.BCHCorrected disp('System SNRdb') System.SNRdb semilogy(System.SNRdb,System.BER,";Simulated BPSK;",System.SNRdb,0.5*erfc(sqrt(10.^(System.SNRdb/10))),";Theoretical BPSK;",System.SNRdb,System.BCHBER,";Simulated BCH, ;") title "BPSK Uncoded, Vs BCH Coded systems" save bchchannel_sim System fgets(stdin) fgets(stdin)
1127:#!/usr/bin/octave -q #A simple communications channel, %BPSK uncoded. System.BER=; System.Error=; System.Length=36*1000; System.Mesg=zeros(1,System.Length); System.Coded=; %BCH coded. System.BCHError=; System.BCHBER=; System.BCHLength=System.Length*(63.0/36.0); System.BCHCoded=zeros(1,System.BCHLength); System.BCHDeCoded=; %BPSK parameters. System.Modulated=; System.DeModulated=; System.DeCoded=; System.CarrierRate=10; System.DataRate=2; System.SNRdb=0:0.5:10; gen_poly=BCH_poly(); Length=length(System.SNRdb); System.BCHActualErrors=zeros(1,Length); System.BCHCorrected=zeros(1,Length); %Additive White Gaussian Noise, power. System.N0=5; for iter=1:Length disp('=: -->
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improvement. Even better, set the jargon aside for the theory section, and explain what is being done and why. Is the term "Galois Field" meaningful in any way to someone learning about BCH on
Knowledge? Does the phrase "primitive narrow-sense BCH codes" help the reader understand why they are different from general BCH codes, why one might prefer either of them, or help the reader understand how to construct BCH codes? It is understandable that every article cannot be a hand-holding tutorial that gingerly leads the reader along to illumination, but surely there is some happy medium between that and strict mathematical formalism.
1011:?" - it is just an example of a primitive polynomial used as the basis for GF(2^4). All non-zero numbers for this field are powers of x (hex 2). "How to calculate syndromes ... error locator polynomial" - I'm assuming the article has been updated since the question was asked, since it now explains the process. "cycles of roots" - I don't see this in the article anymore. I updated the article to explain the minimum polynomials for powers of 2 modulo the field primitive polynomial. The least common multiples are simply the product of non-duplicate minimum polynomials.
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than a genuine attempt to be helpful or to understand the problem. There most certenly is a helpful, intuitive level of explination between "its a cyclic block code" and the highly technical and inaccessable treatment given in this article. This isn't a criticism that should be dismissed or ridiculed - it goes to the heart of the public service that
Knowledge is trying to provide. Hopefully someone who has both an understanding of the subject and a desire to help others understand will make the needed improvements.
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1984:'s recent edit summary: "single digit symbols and errors make this a poor example." Thanks for your input. I feel that the example is a good one, as it is simple and practical. Additional examples could be added to fully illustrate the complexity of the subject, but I don't know how much value that would add to the article. Thoughts?
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that such codes are cyclic provides any benefit. Note that is not a cyclic code unless the evaluated data points are powers of the field primitive. Outside of academics, the most common usage for Reed
Solomon original view encoding is some implementations of jerasure, an erasure only code used for data storage and typically a set of
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the roots of it are the reciprocals of the error locations (which in fact will be the error location themselves from this definiton). If this is a mistake i hope someone would fix it and if its not please explain to me why. Also excuse me for the lack of formatting or any spelling mistakes, i was typing this on my phone.
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calculation, so the Forney formula could be used easily afterwards. Peterson algorithm is good only for explanation purposes. Computing several determinants cannot be faster than gcd computation. Massey ... algorithm is probably comparable especially for binary case. Both favourite algorithms compute
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I just read that two early chess programs used BCH coding for their hash functions. What advantage does BCH coding add to hashing ( I am hoping it might eliminate collisions. Was/Is it not used more widely due to speed considerations - chess is always looking for maximum speed (accepting some degree
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I am not familiar to the wikipedia encoding or anything like that but i want to point out something which seems to me like a mistake. Close to the beginning of the forney algorithm explanation, the exponent in the multiplicative formula for lambda(x) is negative, which contradicts the definiton that
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I have created Czech version of the page (based on material from this one), now I am going to update the
English in simillar way. The explanation of the unreadable characters correction leading to simple reuse of Euclidian algorithm with Forney formula ... will be added in near future (no new source
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Some helpful information might be how to get the corrections for a given code. For example if the message is 512 bytes and the BCH length is 8, then the maximum correction is 4 (always?). Or can different GF(2) fields be chosen to alter the corrective power. I think that a section on fields of 2
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Although BCH code is a class of cyclic error correction code, the fact that it is a cyclic code only makes clear that the maximum size of a message for a field based on GF(2^n) is (2^n-1) elements There is nothing in the encoding or decoding process of common error correcting codes where the fact
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It sounds like the above comments are not complaining about the description of BCH as a cyclic block code, but rather that this information is not sufficient to understand what BCH is or how it is used. A layman's explanation of what the jargon means, particularly why it is relevant, would be an
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This article is exactly as critized above - OBTUSE and of LITTLE USE to anyone who does not already have a sophisticated matamatical background. The observation that a BCH is a "cyclic block code" is of little help, and even seems to be a bit of a put down or an attempt at "witty repartee" rather
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I tried to run your code on my octave, but I don't have the required other functions. Perhaps, it is better to consider the theoretical error rate. The (63,36) Binary BCH code with
Berlekamp-Massey decoding should correct all error patterns of weight <= 5 and no error patterns of weight :
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0)+0; xhat = bchdec(gf(ybit,1),n,k); sim_ber(i) = sum(sum(xhat~=0))/M/k; end semilogy(ebno,bpsk_ber,'r-',ebno,coded_raw_ber,'g-',ebno,coded_ber,'b-',ebno,sim_ber,'bo'); title('(63,36) BCH with bounded distance decoding'); legend('BPSK','Input to decoder','After decoder','Simulation',3);
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t)+0).*X; for i=1:length(ebno) msg_err = binopdf(0:k,k,coded_raw_ber(i)); par_err = binopdf(0:(n-k),n-k,coded_raw_ber(i))'; joint = msg_err(ones(1,n-k+1),:) .* par_err(:,ones(1,k+1)); coded_ber(i) = sum(sum(T.*joint))/k; end % Simulation (all zeros codeword) M = 500; sim_ber =
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n = 63; k = 36; t = 5; rate = k/n; ebno = 0:0.5:10; bpsk_ber = 0.5*erfc(sqrt(10.^(ebno/10))); coded_raw_ber = 0.5*erfc(sqrt(10.^(rate*ebno/10))); coded_ber = 0*coded_raw_ber; b = (0:n)/n; b(1:(t+1)) = 0; for i=1:length(ebno) coded_ber(i) = dot(binopdf(0:n,n,coded_raw_ber(i)),b); end
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would be helpful for the people in the binary world. It is mathematically interesting for other fields, but a lot of people accessing the page will just be trying to understand GF(2) only; I don't know if there are then short-cuts/simplification to some of the explanations?
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Is there a reason for using matrix style notation versus the simpler notation shown in the other
Knowledge articles about extended Euclidean algorithm or Reed Solomon implementation of the extended Euclidean algorithm? For decoding purposes, there is no need to calculate
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Really? Describing BCH as a "cyclic block code" is seen as "witty repartee"? I think the entry is perfect just the way it is... If you think otherwise, then take a crack at rewriting it... *THAT* is the heart of the public service that
Knowledge provides...
669:, is not accessible to people without an account on the relevant database. As such, it seems that a full citation would be more appropriate: A Simple Step-by-Step Decoding of Binary BCH Codes CHR et al. IEICE Trans Fundamentals.2005; E88-A: 2236-2239 --
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I dont know how this is going to be useful, but to the best of my knowledge this is the graph I obtained from running my simulation code. The main program is given here (ancillary functions omitted for want of space). You can get the full program from
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If interested, I have example code that matches this wiki article for 3 error correction, using
Sugiyama's extended Euclid decoder (note the target audience for this was a hardware group, so the decoder emulates a pair of shift registers).
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n = 63; k = 36; t = 5; rate = k/n; ebno = 0:0.5:10; ebno_lin = 10.^(ebno/10); bpsk_ber = 0.5*erfc(sqrt(ebno_lin)); coded_raw_ber = 0.5*erfc(sqrt(rate*ebno_lin)); coded_ber = 0*coded_raw_ber; % Theory = meshgrid(0:k,0:n-k); T = (((X+Y):
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I'm not sure what sort of explanation you're looking for, beyond the fact that BCH is a cyclic block code! The fact that it is useful is due to its particular mathematical properties, which can only really be explained by presented
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polynomials and appending the values to the message? All methods for calculating CRC, etc can be used such as table look ups and calculations in parallel. If it is this simple, I don't think an extra section is needed.
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are technically more difficult, so I would expect them to be a little more difficult to understand (require more reading). I think some worked binary examples would be helpful; along with the mathematical notation.
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I have generalised the BCH code definition in Czech version to cover Reed-Solomon, what was somehow inconsistently suggested on this page. I am not sure if I am approved to do the same here in english version.
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functions. AFAIK I think it is correct. The method maybe different, but I think it is statistically significant, as I ran the code for 36000 bits and got BER for 1e-4 (10x more bits than 1/BER). Also the
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is the raw error rate at the input of the decoder (due to the rate adjusted SNR). This assumes further that all decoder failures are detected, which is not a terrible assumption for this code.
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semilogy(ebno,bpsk_ber,'r-',ebno,coded_raw_ber,'g-',ebno,coded_ber,'b-'); title('(63,36) BCH with bounded distance decoding'); legend('BPSK','Input to decoder','After decoder',3); grid on
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Here is an updated version of my code which uses the Matlab
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have some similar subject matter. Not everyone will be familiar with algebraic fields. However, those criticizing could be constructive by saying what is better in those pages.
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way would be to perform any simulation for just about till 10 errors or 100 errors accumulate at each SNR value as I understand from books by Moore on ECC. Let me know, now that
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Classic case of nerds writing for nerds. Most of us are just trying to figure out how things work, get things done, and make a little money - and don't have time to be nerds.
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7085:. My confusion about this is due to textbooks and articles I've read. One of my textbooks, Error Control Coding by Shu Lin and Daniel J Costello Jr, has an example in
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The example didn't seem to be right so I have expanded it. The encoding section needs to be revised accordingly but I don't have time to do that right now.
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4747:{\displaystyle \{\alpha ,\alpha +1,\alpha ^{2},\alpha ^{2}+1,\alpha ^{3}+1,\alpha ^{3}+\alpha +1,\alpha ^{3}+\alpha ^{2}+1,\alpha ^{3}+\alpha ^{2}+\alpha \}}
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evaluated data points are {0, 1, ... , m-1} resulting in a non-cyclic code, and for GF(2^n), the maximum message length is 2^n (as opposed to 2^n - 1).
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4123:{\displaystyle \alpha +1,\alpha ^{2},\alpha ^{2}+1,\alpha ^{3}+1,\alpha ^{3}+\alpha +1,\alpha ^{3}+\alpha ^{2}+1,\alpha ^{3}+\alpha ^{2}+\alpha }
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This entry should explain BCH codes ... not the mathematics behind them. This page should be moved to a separate referenced "BCH theory" page.
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means the redundancy polynomial. How do I calculate the syndrome values? How do I calculate the error locator polynomials? Shall I continue?
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should be equal to your "simulated BPSK" error rate. Perhaps you could add these theoretical curves to your graph to see what is going on.
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Can anyone please provide us with a complete algorithm description, detailed enough so it can be replicated on paper or in source code?
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5125:. One way is the 16 bytes: 0000, 0001, ..., 1111. They form a variant of the 16 vectors: (0,0,0,0), (0,0,0,1), ..., (1,1,1,1) showing
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is one of the vectors. Multiplication, also for the vector representation, is defined as for polynomials, but modulo for instance
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The results of this script are plotted here. You will notice the theory and simulation now match because they are both correct.
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of error in the process).) I was looking for a simple explanation but it appears this is not a simple subject. Thanks
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I agree, for nonbinary codes or for decodding with unreadable characters I think it is the best method as it computes
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7389:, even though that is more mathematically correct. The books and articles target implementations rather than theory.
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We're running around in circles. Compare it with the complex numbers. Extend the reals with a root of the polynomial
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Explanation of \Lambda could be simplified or rather fully replaced by the explanation with unreadable characters.
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related articles on
Knowledge. If you would like to participate, please visit the project page, where you can join
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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Hmmm, I should reindex the syndroms to be indexed in all sections the same way. ... in the next edit probably
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functions are avaiable. I will try & check your claims as well. --பராசக்தி 00:38, 19 September 2006 (UTC)
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would have made this simpler to understand, with the goal of finding the minimum polynomial that satisfies
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refers to this BCH article, perhaps the Euclidean decoder could be mentioned here as well with a link to
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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zeros(1,length(ebno)); for i=1:2:15 y = -1 + randn(M,n)/sqrt(2*rate*ebno_lin(i)); ybit = (y: -->
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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http://web.archive.org/web/20120930164509/http://www.stanford.edu/class/ee387/handouts/notes7.pdf
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before doing mass systematic removals. This message is updated dynamically through the template
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Replace this entry in entirety with something more intuitive and tutorial ... (... PLEASE ...)
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To answer Damix's original concern, I would like to note the following mathematical fact. The
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Here is my Matlab code (apologies but my Octave installation has never quite worked for me).
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Let me know whats the problem, if there is any. --பராசக்தி 16:48, 16 September 2006 (UTC)
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Most (if not all) of my textbooks for error correcting codes define a primitive element
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This is a general comment on the BCH entry ... it is AWFUL, TERRIBLE, OBTUSE, BAD, ....
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If you found an error with any archives or the URLs themselves, you can fix them with
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Requested articles/Applied arts and sciences/Computer science, computing, and Internet
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or another irreducible polynomial of degree 4. A third way is, as I wrote, extending
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Thank for you explanation. However, being a mathematician, it makes no sense to me.
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is another popular algorithm. I don't know if this is true for BCH codes, but since
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Page 29 of Error Control Coding, by Lin and Costello, seems to support this case.
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The examples are not very elaborate. How does one create the generator polynomial
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2983:, ... . It might have been more clear if another variable was used. For example,
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1376:{\displaystyle P_{b}=\sum _{j=6}^{63}{\frac {j}{n}}{n \choose j}p^{j}(1-p)^{n-j}}
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I don't have Lin and Costello, but addition in GF(2) is just exclusive or. The 2
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I have given a link to the code tarball on my website. Please use that for the
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is redefined as a power of alpha, which is why I think switching to using
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after the link to keep me from modifying it. Alternatively, you can add
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However, an "applications" section would probably benefit the article.
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This is in the lead. Thanks to whoever has contributed to this page.
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denote the residue class that contains X", followed by a table where
7230:. In the textbook Error Correcting Codes by Peterson and Weldon, for
2487:
times, you get zero. Arun Kandasamy 12:34, 25 February 2012 (UTC)
3918:
Thanks again, I do understand this all, except for the statement:
2598:
to keep me off the page altogether. I made the following changes:
585:
3086:{\displaystyle m_{2}\left(z=x^{2}\right)=z^{4}+z+1=x^{8}+x^{2}+1}
3177:{\displaystyle x^{8}+x^{2}+1{\bmod {\left(x^{4}+x+1\right)}}=0.}
7792:
7461:
7431:
7417:
7398:
7044:
6944:
6808:
6513:{\displaystyle x^{13}=x^{4}+x^{3}+x^{2}+x=x^{3}+x^{2}+1(=1101)}
6396:{\displaystyle x^{12}=x^{4}+x^{3}+x^{2}=x^{3}+x^{2}+x+1(=1111)}
5075:
4569:
4531:
4250:
3911:
3293:
3192:
2836:
2721:
2701:
2560:
2534:
2335:
2303:
2103:
2028:
1993:
1944:
1511:
xlabel('E_n / N_0'); ylabel('BER'); grid on; set(gca,'YLim',);
1106:
1052:
1030:
1020:
966:
673:
655:
641:
621:
568:
547:
531:
514:
7519:, or using negative constant (versus variable) powers such as
5177:. Another, equivalent, representation is with the polynomials
284:
Find pictures for the biographies of computer scientists (see
15:
3810:
3245:
looks like a polynomial, but if it is meant to be the number
3130:
2770:
2525:) 00:47, 30 December 2012 (UTC) I hope this is corrected now
2613:
When you have finished reviewing my changes, please set the
6931:, should eliminate the confusing part of the article where
5080:
The point is you're confusing different representations of
5551:. In the representation with polynomials, the polynomial
2574:
I have just added archive links to one external link on
7634:{\displaystyle \alpha ^{4}+\alpha ^{-6}+\alpha ^{-1}=0}
4548:, I changed the terms for the reducing polynomial from
2608:
http://www.stanford.edu/class/ee387/handouts/notes7.pdf
2579:
1060:
Encoding should just be the process of calculating the
7177:. The pdf file I linked to above just lists powers of
7762:
7723:
7677:
7647:
7582:
7555:
7525:
7498:
7360:
7327:
7307:
7268:
7236:
7203:
7183:
7150:
7130:
7091:
7056:
7006:
6986:
6960:
6847:
6827:
6746:
6719:
6692:
6618:
6527:
6410:
6293:
6221:
6130:
6071:
5980:
5889:
5823:
5764:
5712:
5672:
5632:
5599:
5577:
5557:
5518:
5486:
5466:
5434:
5395:
5350:
5272:
5183:
5151:
5131:
5086:
5020:
4981:
4795:
4789:, so why not just define the 8 primitive elements as
4760:
4582:
4495:
4475:
4455:
4435:
4415:
4376:
4350:
4324:
4304:
4271:
4227:
4207:
4168:
4136:
3970:
3950:
3924:
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3732:
3706:
3534:
3486:
3448:
3409:
3370:
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3251:
3212:
3099:
2989:
2951:
2913:
2893:
2855:
2740:
2473:
2433:
2413:
2393:
2352:
2265:
2250:{\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}=a^{2}+b^{2}}
2160:
2119:
2080:
2059:
2039:
1883:
1824:
1781:
1740:
1636:
1580:
1537:
1458:
1391:
1268:
1145:
1066:
978:
935:
908:
881:
854:
827:
788:
761:
734:
689:
433:, a collaborative effort to improve the coverage of
82:, a collaborative effort to improve the coverage of
6954:In the end the confusion arises from the statement
6604:{\displaystyle x^{14}=x^{4}+x^{3}+x=x^{3}+1(=1001)}
6207:{\displaystyle x^{10}=x^{4}+x^{2}=x^{2}+x+1(=0111)}
4344:. Of course there is the sloppy way of saying: for
2650:using the archive tool instructions below. Editors
7775:
7748:
7709:
7663:
7633:
7568:
7541:
7511:
7381:
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7313:
7293:
7254:
7222:
7189:
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7077:
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6992:
6972:
6915:
6833:
6791:
6732:
6705:
6668:
6603:
6512:
6395:
6278:
6206:
6115:
6057:{\displaystyle x^{8}=x^{4}+x^{2}+x=x^{2}+1(=0101)}
6056:
5966:{\displaystyle x^{7}=x^{4}+x^{3}=x^{3}+x+1(=1011)}
5965:
5874:
5808:
5749:
5697:
5657:
5617:
5583:
5563:
5543:
5504:
5472:
5452:
5420:
5381:
5336:
5259:{\displaystyle a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}}
5258:
5169:
5137:
5117:
5058:
5002:
4959:
4781:
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3356:
3336:
3276:
3237:
3176:
3085:
2975:
2937:
2899:
2879:
2816:
2540:Possible error in the Forney algorithm explanation
2479:
2460:
2419:
2399:
2379:
2283:
2249:
2144:
2086:
2065:
2045:
1922:
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1003:
948:
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894:
867:
840:
813:
774:
747:
720:
610:could include technology that currently uses them.
467:This article has not yet received a rating on the
190:Computer science articles needing expert attention
116:This article has not yet received a rating on the
1329:
1316:
651:The last two links in the references are broken.
3726:is the simplest choice. In the following lines,
1957:http://www.wolframalpha.com/input/?i=GF%28q^m%29
2314:does not represent a field multiply of 2 times
7827:Unknown-importance Telecommunications articles
7467:Decoding based on extended Euclidean algorithm
3364:. In the example, the reducing polynomial for
3344:is defined by a reducing polynomial of degree
2636:This message was posted before February 2018.
1968:http://www.wolframalpha.com/input/?i=n%3Dq^m-1
821:? The way to determine the cycles of roots of
330:WikiProject Computer science/Unreferenced BLPs
8:
7812:Unknown-importance Computer science articles
7000:is a polynomial. Formally this should read:
6910:
6848:
4954:
4796:
4741:
4583:
6916:{\displaystyle \{x,x+1,...,x^{3}+x^{2}+x\}}
6279:{\displaystyle x^{11}=x^{3}+x^{2}+x(=1110)}
5571:is a primitive root of unity. It generates
5480:, a root of an irreducible polynomial over
5337:{\displaystyle a=(a_{0},a_{1},a_{2},a_{3})}
1442:{\displaystyle p=Q(2{\sqrt {E_{s}/N_{0}}})}
1137:5. Since the theoretical BER for BPSK is
1036:material, just simplifying the reasoning).
929:are not explained. It's not clarified that
247:Computer science articles without infoboxes
185:Computer science articles needing attention
2546:
382:
151:Here are some tasks awaiting attention:
125:
47:
7767:
7761:
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7722:
7710:{\displaystyle \alpha ^{7}+\alpha ^{7}=0}
7695:
7682:
7676:
7655:
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7554:
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4640:
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4608:
4581:
4522:will have the value 0. No more than hat.
4500:
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4381:
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4349:
4323:
4303:
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859:
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832:
826:
793:
787:
766:
760:
739:
733:
694:
688:
7444:Should the scope of the Galois field be
5875:{\displaystyle x^{6}=x^{3}+x^{2}(=1100)}
5059:{\displaystyle \alpha ^{1}=z=0010_{2}=2}
3964:is a primitive element. The other 7 are
447:Knowledge:WikiProject Telecommunications
7492:. There also the issue of always using
7482:Reed–Solomon_error_correction#Example_3
6713:is also a primitive root of unity. But
6669:{\displaystyle x^{15}=x^{4}+x=1(=0001)}
3864:, since it's the 1 bit coefficients of
1949:
450:Template:WikiProject Telecommunications
384:
49:
19:
7756:and used 4 bit binary numbers such as
7478:Extended_Euclidean_algorithm#Example_2
4449:are the same, but merely: If you give
2714:2601:181:8301:4510:54EA:A058:77AF:BF1B
96:Knowledge:WikiProject Computer science
7817:WikiProject Computer science articles
4576:Defining the 8 primitive elements as
4556:, following the example on page 6 of
3902:that define the minimum polynomials.
3277:{\displaystyle \alpha ^{4}+\alpha +1}
2625:to let others know (documentation at
2427:. That is, if you sum any element of
99:Template:WikiProject Computer science
7:
6792:{\displaystyle (x^{3})^{5}=x^{15}=1}
6116:{\displaystyle x^{9}=x^{3}+x(=1010)}
5809:{\displaystyle x^{5}=x^{2}+x(=0110)}
5145:as a 4-dimensional vectorspace over
4318:. As far as I know, no one will say
4162:with any one of these will generate
2461:{\displaystyle \mathrm {GF} (p^{m})}
2380:{\displaystyle \mathrm {GF} (p^{m})}
1529:Wolfram Alpha writes the function :
427:This article is within the scope of
76:This article is within the scope of
7822:C-Class Telecommunications articles
7480:which results in the simpler still
4544:- To avoid having two meanings for
3895:{\displaystyle m_{i}\left(z\right)}
3305:- for this article, a finite field
2976:{\displaystyle m_{2}\left(x\right)}
2938:{\displaystyle m_{1}\left(x\right)}
647:Another problem with the references
38:It is of interest to the following
2438:
2435:
2357:
2354:
2081:
2060:
2040:
1722:{\displaystyle GeneratingFunction}
1320:
606:has some historical use. I think
266:Timeline of computing 2020–present
14:
7807:C-Class Computer science articles
5010:, matches the table on page 6 of
3442:, with a primitive element value
2578:. Please take a moment to review
782:, ...? What is the importance of
292:Computing articles needing images
7347:{\displaystyle \alpha =0010_{2}}
7170:{\displaystyle \alpha =0010_{2}}
5750:{\displaystyle x^{4}=x+1(=0011)}
3521:{\displaystyle \phi (2^{4}-1)=8}
2346:of all the non-zero elements in
2094:without 'guessing' it's degree.
1866:{\displaystyle q=(1+n)^{m^{-1}}}
1517:
414:
404:
386:
142:
69:
51:
20:
7354:. None of these use the syntax
7743:
7730:
7370:
7364:
7249:
7243:
7111:
7098:
7066:
7060:
7016:
7010:
6841:as a polynomial in x, such as
6761:
6747:
6663:
6654:
6598:
6589:
6507:
6498:
6390:
6381:
6273:
6264:
6201:
6192:
6110:
6101:
6051:
6042:
5960:
5951:
5869:
5860:
5803:
5794:
5744:
5735:
5692:
5683:
5652:
5643:
5612:
5603:
5499:
5493:
5447:
5441:
5382:{\displaystyle a_{i}\in GF(2)}
5376:
5370:
5331:
5279:
5164:
5158:
5112:
5099:
4991:
4985:
4770:
4764:
4188:
4175:
4149:
4143:
3509:
3490:
3390:
3377:
3331:
3318:
2561:12:57, 16 September 2015 (UTC)
2455:
2442:
2374:
2361:
2278:
2272:
2174:
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2126:
1903:
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1401:
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1244:
1212:
1184:
1149:
1083:
1077:
656:09:53, 14 September 2007 (UTC)
430:WikiProject Telecommunications
1:
7223:{\displaystyle \alpha ^{1}=z}
7050:I made your suggested change
6935:previously had two meanings.
5591:. The successive powers are:
2104:18:56, 29 December 2012 (UTC)
2009:Reed–Solomon_error_correction
1923:{\displaystyle q=(1+n)^{1/m}}
1773:is written alternatively as
1107:16:54, 16 December 2013 (UTC)
1053:18:39, 29 December 2012 (UTC)
1021:19:33, 16 December 2020 (UTC)
642:19:24, 16 December 2020 (UTC)
622:17:11, 16 December 2013 (UTC)
548:14:58, 30 December 2011 (UTC)
441:and see a list of open tasks.
346:Tag all relevant articles in
90:and see a list of open tasks.
7664:{\displaystyle 2\alpha ^{7}}
7542:{\displaystyle \alpha ^{-7}}
7432:12:19, 18 October 2021 (UTC)
7418:08:18, 18 October 2021 (UTC)
7399:08:10, 18 October 2021 (UTC)
7382:{\displaystyle \alpha (z)=z}
7078:{\displaystyle \alpha (z)=z}
7045:07:27, 18 October 2021 (UTC)
7028:{\displaystyle \alpha (x)=x}
6945:20:35, 17 October 2021 (UTC)
6809:19:31, 17 October 2021 (UTC)
5698:{\displaystyle x^{3}(=1000)}
5658:{\displaystyle x^{2}(=0100)}
5512:with degree 4, for instance
5076:12:11, 17 October 2021 (UTC)
5003:{\displaystyle \alpha (z)=z}
4570:02:43, 17 October 2021 (UTC)
4532:20:02, 16 October 2021 (UTC)
4251:11:13, 16 October 2021 (UTC)
3912:23:20, 15 October 2021 (UTC)
3294:21:16, 15 October 2021 (UTC)
3193:20:49, 15 October 2021 (UTC)
2837:21:46, 14 October 2021 (UTC)
2336:20:11, 15 January 2012 (UTC)
2304:11:23, 14 January 2012 (UTC)
2259:But I think that is true in
2029:04:30, 3 November 2011 (UTC)
674:07:55, 11 January 2007 (UTC)
569:17:04, 22 October 2012 (UTC)
355:WikiProject Computer science
131:WikiProject Computer science
79:WikiProject Computer science
7569:{\displaystyle \alpha ^{8}}
7512:{\displaystyle \alpha ^{i}}
5118:{\displaystyle K=GF(2^{4})}
2907:on the following lines for
2535:20:51, 4 January 2013 (UTC)
1945:07:34, 29 August 2010 (UTC)
972:"What is the importance of
721:{\displaystyle m_{1,3,...}}
453:Telecommunications articles
286:List of computer scientists
7843:
7576:, and it's not clear that
7462:21:00, 23 April 2024 (UTC)
3528:primitive element values:
2702:16:24, 22 March 2016 (UTC)
2667:(last update: 5 June 2024)
2596:|deny=InternetArchiveBot}}
2571:Hello fellow Wikipedians,
469:project's importance scale
118:project's importance scale
7793:04:43, 11 July 2024 (UTC)
7749:{\displaystyle GF(2^{4})}
7294:{\displaystyle X^{4}+X+1}
7117:{\displaystyle GF(2^{4})}
6973:{\displaystyle \alpha =x}
5544:{\displaystyle x^{4}+x+1}
5460:with a primitive element
5421:{\displaystyle x^{4}+x+1}
4402:{\displaystyle x^{2}+1=0}
4194:{\displaystyle GF(2^{4})}
3937:{\displaystyle \alpha =x}
3719:{\displaystyle \alpha =x}
3435:{\displaystyle x^{4}+x+1}
3396:{\displaystyle GF(2^{4})}
3337:{\displaystyle GF(2^{n})}
3238:{\displaystyle x^{4}+x+1}
2731:In the example it reads:
2722:13:15, 24 June 2016 (UTC)
2145:{\displaystyle GF(2^{4})}
2113:The article says that in
1994:15:30, 28 July 2011 (UTC)
1807:{\displaystyle q^{m}=1+n}
1766:{\displaystyle n=q^{m}-1}
1563:{\displaystyle GF(q^{m})}
1031:06:50, 30 June 2007 (UTC)
1004:{\displaystyle x^{4}+x+1}
967:16:54, 24 June 2007 (UTC)
814:{\displaystyle x^{4}+x+1}
665:The second reference, to
466:
399:
348:Category:Computer science
124:
115:
102:Computer science articles
64:
46:
7776:{\displaystyle 0101_{2}}
5618:{\displaystyle x(=0010)}
2087:{\displaystyle \Lambda }
2066:{\displaystyle \Lambda }
1619:{\displaystyle G_{m}(z)}
1089:{\displaystyle m_{y}(x)}
532:11:14, 6 June 2010 (UTC)
515:05:25, 6 June 2010 (UTC)
422:Telecommunication portal
350:and sub-categories with
7314:{\displaystyle \alpha }
7190:{\displaystyle \alpha }
7137:{\displaystyle \alpha }
6993:{\displaystyle \alpha }
6834:{\displaystyle \alpha }
5473:{\displaystyle \alpha }
4515:{\displaystyle x^{2}+1}
4291:{\displaystyle x^{2}+1}
4234:{\displaystyle \alpha }
3957:{\displaystyle \alpha }
3480:. There are a total of
3473:{\displaystyle alpha=x}
2880:{\displaystyle alpha=x}
2567:External links modified
2046:{\displaystyle \Omega }
7777:
7750:
7711:
7665:
7635:
7570:
7543:
7513:
7422:Okay, I rest my case.
7383:
7348:
7315:
7295:
7256:
7224:
7191:
7171:
7138:
7118:
7079:
7029:
6994:
6974:
6917:
6835:
6793:
6734:
6707:
6670:
6605:
6514:
6397:
6280:
6208:
6117:
6058:
5967:
5876:
5810:
5751:
5699:
5659:
5619:
5585:
5565:
5545:
5506:
5474:
5454:
5422:
5383:
5338:
5260:
5171:
5139:
5119:
5060:
5004:
4961:
4783:
4782:{\displaystyle a(x)=x}
4748:
4516:
4483:
4463:
4443:
4423:
4403:
4364:
4338:
4312:
4292:
4235:
4215:
4195:
4156:
4124:
3958:
3938:
3896:
3858:
3760:
3740:
3720:
3694:
3522:
3474:
3436:
3397:
3358:
3338:
3278:
3239:
3178:
3087:
2977:
2939:
2901:
2881:
2827:Seems an error to me.
2818:
2481:
2462:
2421:
2401:
2381:
2285:
2251:
2146:
2109:Simplified BCH Example
2088:
2067:
2047:
1924:
1867:
1808:
1767:
1723:
1620:
1564:
1466:
1443:
1377:
1302:
1251:
1090:
1005:
950:
923:
896:
869:
842:
815:
776:
749:
722:
661:Problem with reference
311:Computer science stubs
28:This article is rated
7778:
7751:
7712:
7666:
7636:
7571:
7544:
7514:
7440:Scope of Galois field
7384:
7349:
7316:
7296:
7257:
7255:{\displaystyle GF(2)}
7225:
7192:
7172:
7139:
7119:
7080:
7030:
6995:
6975:
6918:
6836:
6794:
6735:
6733:{\displaystyle x^{3}}
6708:
6706:{\displaystyle x^{2}}
6671:
6606:
6515:
6398:
6281:
6209:
6118:
6059:
5968:
5877:
5811:
5752:
5700:
5660:
5620:
5586:
5566:
5546:
5507:
5505:{\displaystyle GF(2)}
5475:
5455:
5453:{\displaystyle GF(2)}
5423:
5384:
5339:
5261:
5172:
5170:{\displaystyle GF(2)}
5140:
5120:
5061:
5005:
4962:
4784:
4749:
4517:
4484:
4464:
4444:
4424:
4404:
4365:
4339:
4313:
4293:
4236:
4221:as being the same as
4216:
4196:
4157:
4155:{\displaystyle GF(2)}
4125:
3959:
3939:
3897:
3859:
3761:
3741:
3721:
3695:
3523:
3475:
3437:
3398:
3359:
3339:
3279:
3240:
3179:
3088:
2978:
2940:
2902:
2882:
2819:
2482:
2463:
2422:
2402:
2382:
2286:
2284:{\displaystyle GF(2)}
2252:
2147:
2089:
2068:
2048:
1925:
1868:
1809:
1768:
1724:
1621:
1565:
1467:
1444:
1378:
1282:
1252:
1091:
1006:
951:
949:{\displaystyle C_{r}}
924:
922:{\displaystyle m_{7}}
897:
895:{\displaystyle m_{5}}
870:
868:{\displaystyle m_{3}}
843:
841:{\displaystyle m_{1}}
816:
777:
775:{\displaystyle m_{3}}
750:
748:{\displaystyle m_{1}}
723:
7760:
7721:
7675:
7645:
7580:
7553:
7523:
7496:
7358:
7325:
7305:
7266:
7234:
7201:
7181:
7148:
7128:
7089:
7054:
7004:
6984:
6958:
6845:
6825:
6744:
6717:
6690:
6616:
6525:
6408:
6291:
6219:
6128:
6069:
5978:
5887:
5821:
5762:
5710:
5670:
5630:
5597:
5575:
5555:
5516:
5484:
5464:
5432:
5393:
5348:
5270:
5181:
5149:
5129:
5084:
5018:
4979:
4793:
4758:
4580:
4493:
4473:
4453:
4433:
4413:
4374:
4348:
4322:
4302:
4269:
4225:
4205:
4166:
4134:
3968:
3948:
3944:. As far as I know,
3922:
3868:
3770:
3750:
3730:
3704:
3532:
3484:
3446:
3407:
3368:
3348:
3309:
3249:
3210:
3097:
2987:
2949:
2911:
2891:
2853:
2738:
2648:regular verification
2582:. If necessary, add
2471:
2431:
2411:
2391:
2350:
2263:
2158:
2117:
2078:
2057:
2037:
1937:James Michael DuPont
1935:More to come... --
1881:
1822:
1779:
1738:
1634:
1578:
1535:
1456:
1389:
1266:
1143:
1064:
976:
933:
906:
879:
852:
825:
786:
759:
732:
687:
486:Terrible Explication
129:Things you can help
7783:where appropriate.
7144:as a binary value:
4409:. This doen't mean
4363:{\displaystyle x=i}
4337:{\displaystyle i=x}
2887:, but then reusing
2638:After February 2018
2617:parameter below to
1999:Euclidean algorithm
7773:
7746:
7707:
7661:
7631:
7566:
7539:
7509:
7379:
7344:
7311:
7291:
7252:
7220:
7197:in a table, where
7187:
7167:
7134:
7124:where they define
7114:
7075:
7025:
6990:
6970:
6913:
6831:
6789:
6730:
6703:
6666:
6601:
6510:
6393:
6276:
6204:
6113:
6054:
5963:
5872:
5806:
5747:
5695:
5655:
5615:
5581:
5561:
5541:
5502:
5470:
5450:
5418:
5379:
5334:
5256:
5167:
5135:
5115:
5056:
5000:
4957:
4779:
4744:
4512:
4479:
4459:
4439:
4419:
4399:
4360:
4334:
4308:
4288:
4231:
4211:
4191:
4152:
4120:
3954:
3934:
3892:
3854:
3756:
3736:
3716:
3690:
3518:
3470:
3432:
3393:
3354:
3334:
3284:, why not say so.
3274:
3235:
3174:
3083:
2973:
2935:
2897:
2877:
2814:
2643:InternetArchiveBot
2513:Syndroms numbering
2477:
2458:
2417:
2397:
2377:
2281:
2247:
2142:
2084:
2063:
2043:
2003:For RS codes, the
1920:
1863:
1804:
1763:
1719:
1616:
1560:
1525:Ideas from Wolfram
1462:
1439:
1373:
1247:
1086:
1001:
946:
919:
892:
865:
838:
811:
772:
745:
718:
611:
444:Telecommunications
435:Telecommunications
394:Telecommunications
34:content assessment
5584:{\displaystyle K}
5564:{\displaystyle x}
5138:{\displaystyle K}
4482:{\displaystyle i}
4462:{\displaystyle x}
4442:{\displaystyle i}
4422:{\displaystyle x}
4311:{\displaystyle i}
4298:. Call this root
4214:{\displaystyle x}
3759:{\displaystyle z}
3739:{\displaystyle x}
3357:{\displaystyle n}
2900:{\displaystyle x}
2700:
2668:
2563:
2551:comment added by
2508:
2494:comment added by
2480:{\displaystyle p}
2420:{\displaystyle p}
2400:{\displaystyle p}
2152:it is true that:
2017:Euclidean_decoder
2005:Euclidean_decoder
1465:{\displaystyle p}
1434:
1327:
1311:
1242:
1198:
1182:
679:Encoding/Decoding
602:
559:comment added by
505:comment added by
483:
482:
479:
478:
475:
474:
381:
380:
377:
376:
373:
372:
369:
368:
7834:
7782:
7780:
7779:
7774:
7772:
7771:
7755:
7753:
7752:
7747:
7742:
7741:
7716:
7714:
7713:
7708:
7700:
7699:
7687:
7686:
7670:
7668:
7667:
7662:
7660:
7659:
7640:
7638:
7637:
7632:
7624:
7623:
7608:
7607:
7592:
7591:
7575:
7573:
7572:
7567:
7565:
7564:
7548:
7546:
7545:
7540:
7538:
7537:
7518:
7516:
7515:
7510:
7508:
7507:
7451:
7447:
7388:
7386:
7385:
7380:
7353:
7351:
7350:
7345:
7343:
7342:
7320:
7318:
7317:
7312:
7300:
7298:
7297:
7292:
7278:
7277:
7261:
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7253:
7229:
7227:
7226:
7221:
7213:
7212:
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7188:
7176:
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7168:
7166:
7165:
7143:
7141:
7140:
7135:
7123:
7121:
7120:
7115:
7110:
7109:
7084:
7082:
7081:
7076:
7034:
7032:
7031:
7026:
6999:
6997:
6996:
6991:
6979:
6977:
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6890:
6889:
6840:
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6832:
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6796:
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6782:
6781:
6769:
6768:
6759:
6758:
6739:
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6731:
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6728:
6712:
6710:
6709:
6704:
6702:
6701:
6675:
6673:
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6667:
6641:
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6628:
6627:
6610:
6608:
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6602:
6582:
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6563:
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6549:
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6511:
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6433:
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6402:
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6399:
6394:
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6355:
6354:
6342:
6341:
6329:
6328:
6316:
6315:
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6302:
6285:
6283:
6282:
6277:
6257:
6256:
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6231:
6230:
6213:
6211:
6210:
6205:
6179:
6178:
6166:
6165:
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6119:
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6094:
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5990:
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5925:
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5912:
5911:
5899:
5898:
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5873:
5859:
5858:
5846:
5845:
5833:
5832:
5815:
5813:
5812:
5807:
5787:
5786:
5774:
5773:
5756:
5754:
5753:
5748:
5722:
5721:
5704:
5702:
5701:
5696:
5682:
5681:
5664:
5662:
5661:
5656:
5642:
5641:
5624:
5622:
5621:
5616:
5590:
5588:
5587:
5582:
5570:
5568:
5567:
5562:
5550:
5548:
5547:
5542:
5528:
5527:
5511:
5509:
5508:
5503:
5479:
5477:
5476:
5471:
5459:
5457:
5456:
5451:
5427:
5425:
5424:
5419:
5405:
5404:
5388:
5386:
5385:
5380:
5360:
5359:
5343:
5341:
5340:
5335:
5330:
5329:
5317:
5316:
5304:
5303:
5291:
5290:
5265:
5263:
5262:
5257:
5255:
5254:
5239:
5238:
5226:
5225:
5216:
5215:
5203:
5202:
5193:
5192:
5176:
5174:
5173:
5168:
5144:
5142:
5141:
5136:
5124:
5122:
5121:
5116:
5111:
5110:
5065:
5063:
5062:
5057:
5049:
5048:
5030:
5029:
5009:
5007:
5006:
5001:
4966:
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4947:
4946:
4934:
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4915:
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4902:
4901:
4877:
4876:
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4857:
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4826:
4825:
4788:
4786:
4785:
4780:
4753:
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4734:
4733:
4721:
4720:
4702:
4701:
4689:
4688:
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4663:
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4644:
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4543:
4521:
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4513:
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4504:
4488:
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4468:
4466:
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4408:
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4341:
4340:
4335:
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4281:
4280:
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4192:
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4005:
4004:
3992:
3991:
3963:
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3808:
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3613:
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3594:
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3562:
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3524:
3519:
3502:
3501:
3479:
3477:
3476:
3471:
3441:
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3438:
3433:
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3402:
3400:
3399:
3394:
3389:
3388:
3363:
3361:
3360:
3355:
3343:
3341:
3340:
3335:
3330:
3329:
3304:
3283:
3281:
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3275:
3261:
3260:
3244:
3242:
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3236:
3222:
3221:
3203:
3183:
3181:
3180:
3175:
3167:
3166:
3165:
3161:
3148:
3147:
3122:
3121:
3109:
3108:
3092:
3090:
3089:
3084:
3076:
3075:
3063:
3062:
3038:
3037:
3025:
3021:
3020:
3019:
2999:
2998:
2982:
2980:
2979:
2974:
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2961:
2960:
2944:
2942:
2941:
2936:
2934:
2923:
2922:
2906:
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2903:
2898:
2886:
2884:
2883:
2878:
2848:
2823:
2821:
2820:
2815:
2807:
2806:
2805:
2801:
2788:
2787:
2768:
2764:
2763:
2750:
2749:
2696:
2695:Talk to my owner
2691:
2666:
2665:
2644:
2632:
2597:
2589:
2507:
2488:
2486:
2484:
2483:
2478:
2467:
2465:
2464:
2459:
2454:
2453:
2441:
2426:
2424:
2423:
2418:
2406:
2404:
2403:
2398:
2386:
2384:
2383:
2378:
2373:
2372:
2360:
2290:
2288:
2287:
2282:
2256:
2254:
2253:
2248:
2246:
2245:
2233:
2232:
2220:
2219:
2195:
2194:
2182:
2181:
2151:
2149:
2148:
2143:
2138:
2137:
2093:
2091:
2090:
2085:
2072:
2070:
2069:
2064:
2052:
2050:
2049:
2044:
2013:Peterson_decoder
1976:Decoding example
1970:
1965:
1959:
1954:
1929:
1927:
1926:
1921:
1919:
1918:
1914:
1875:or more simply
1872:
1870:
1869:
1864:
1862:
1861:
1860:
1859:
1813:
1811:
1810:
1805:
1791:
1790:
1772:
1770:
1769:
1764:
1756:
1755:
1728:
1726:
1725:
1720:
1703:
1702:
1625:
1623:
1622:
1617:
1603:
1602:
1590:
1589:
1569:
1567:
1566:
1561:
1556:
1555:
1521:
1471:
1469:
1468:
1463:
1448:
1446:
1445:
1440:
1435:
1433:
1432:
1423:
1418:
1417:
1408:
1382:
1380:
1379:
1374:
1372:
1371:
1344:
1343:
1334:
1333:
1332:
1319:
1312:
1304:
1301:
1296:
1278:
1277:
1256:
1254:
1253:
1248:
1243:
1241:
1240:
1231:
1226:
1225:
1216:
1199:
1191:
1183:
1181:
1180:
1171:
1166:
1165:
1153:
1115:Dispute on Graph
1095:
1093:
1092:
1087:
1076:
1075:
1010:
1008:
1007:
1002:
988:
987:
955:
953:
952:
947:
945:
944:
928:
926:
925:
920:
918:
917:
901:
899:
898:
893:
891:
890:
874:
872:
871:
866:
864:
863:
847:
845:
844:
839:
837:
836:
820:
818:
817:
812:
798:
797:
781:
779:
778:
773:
771:
770:
754:
752:
751:
746:
744:
743:
727:
725:
724:
719:
717:
716:
571:
517:
455:
454:
451:
448:
445:
424:
419:
418:
417:
408:
401:
400:
390:
383:
359:
353:
228:Computer science
157:Article requests
146:
139:
138:
126:
104:
103:
100:
97:
94:
93:Computer science
84:Computer science
73:
66:
65:
59:Computer science
55:
48:
31:
25:
24:
16:
7842:
7841:
7837:
7836:
7835:
7833:
7832:
7831:
7797:
7796:
7763:
7758:
7757:
7733:
7719:
7718:
7691:
7678:
7673:
7672:
7651:
7643:
7642:
7612:
7596:
7583:
7578:
7577:
7556:
7551:
7550:
7526:
7521:
7520:
7499:
7494:
7493:
7469:
7449:
7445:
7442:
7356:
7355:
7334:
7323:
7322:
7303:
7302:
7269:
7264:
7263:
7232:
7231:
7204:
7199:
7198:
7179:
7178:
7157:
7146:
7145:
7126:
7125:
7101:
7087:
7086:
7052:
7051:
7002:
7001:
6982:
6981:
6956:
6955:
6894:
6881:
6843:
6842:
6823:
6822:
6773:
6760:
6750:
6742:
6741:
6720:
6715:
6714:
6693:
6688:
6687:
6632:
6619:
6614:
6613:
6573:
6554:
6541:
6528:
6523:
6522:
6482:
6469:
6450:
6437:
6424:
6411:
6406:
6405:
6359:
6346:
6333:
6320:
6307:
6294:
6289:
6288:
6248:
6235:
6222:
6217:
6216:
6170:
6157:
6144:
6131:
6126:
6125:
6085:
6072:
6067:
6066:
6026:
6007:
5994:
5981:
5976:
5975:
5929:
5916:
5903:
5890:
5885:
5884:
5850:
5837:
5824:
5819:
5818:
5778:
5765:
5760:
5759:
5713:
5708:
5707:
5673:
5668:
5667:
5633:
5628:
5627:
5595:
5594:
5573:
5572:
5553:
5552:
5519:
5514:
5513:
5482:
5481:
5462:
5461:
5430:
5429:
5396:
5391:
5390:
5351:
5346:
5345:
5321:
5308:
5295:
5282:
5268:
5267:
5246:
5230:
5217:
5207:
5194:
5184:
5179:
5178:
5147:
5146:
5127:
5126:
5102:
5082:
5081:
5040:
5021:
5016:
5015:
4977:
4976:
4938:
4925:
4906:
4893:
4868:
4849:
4830:
4817:
4791:
4790:
4756:
4755:
4725:
4712:
4693:
4680:
4655:
4636:
4617:
4604:
4578:
4577:
4537:
4496:
4491:
4490:
4489:the polynomial
4471:
4470:
4451:
4450:
4431:
4430:
4411:
4410:
4377:
4372:
4371:
4346:
4345:
4320:
4319:
4300:
4299:
4272:
4267:
4266:
4258:
4223:
4222:
4203:
4202:
4178:
4164:
4163:
4132:
4131:
4104:
4091:
4072:
4059:
4034:
4015:
3996:
3983:
3966:
3965:
3946:
3945:
3920:
3919:
3881:
3871:
3866:
3865:
3819:
3818:
3814:
3794:
3787:
3783:
3773:
3768:
3767:
3748:
3747:
3728:
3727:
3702:
3701:
3674:
3661:
3642:
3629:
3604:
3585:
3566:
3553:
3530:
3529:
3493:
3482:
3481:
3444:
3443:
3410:
3405:
3404:
3380:
3366:
3365:
3346:
3345:
3321:
3307:
3306:
3298:
3252:
3247:
3246:
3213:
3208:
3207:
3197:
3139:
3138:
3134:
3113:
3100:
3095:
3094:
3067:
3054:
3029:
3011:
3004:
3000:
2990:
2985:
2984:
2962:
2952:
2947:
2946:
2924:
2914:
2909:
2908:
2889:
2888:
2851:
2850:
2842:
2779:
2778:
2774:
2755:
2751:
2741:
2736:
2735:
2729:
2709:
2699:
2694:
2659:
2652:have permission
2642:
2626:
2591:
2583:
2569:
2542:
2515:
2489:
2469:
2468:
2445:
2429:
2428:
2409:
2408:
2389:
2388:
2364:
2348:
2347:
2261:
2260:
2237:
2224:
2211:
2186:
2173:
2156:
2155:
2129:
2115:
2114:
2111:
2076:
2075:
2055:
2054:
2035:
2034:
2001:
1978:
1973:
1966:
1962:
1955:
1951:
1930:
1902:
1879:
1878:
1873:
1848:
1843:
1820:
1819:
1814:
1782:
1777:
1776:
1747:
1736:
1735:
1729:
1694:
1632:
1631:
1626:
1594:
1581:
1576:
1575:
1570:
1547:
1533:
1532:
1527:
1512:
1480:
1454:
1453:
1424:
1409:
1387:
1386:
1357:
1335:
1314:
1269:
1264:
1263:
1232:
1217:
1172:
1157:
1141:
1140:
1129:
1117:
1067:
1062:
1061:
979:
974:
973:
936:
931:
930:
909:
904:
903:
882:
877:
876:
855:
850:
849:
828:
823:
822:
789:
784:
783:
762:
757:
756:
735:
730:
729:
690:
685:
684:
681:
663:
653:194.171.252.100
649:
554:
500:
488:
452:
449:
446:
443:
442:
420:
415:
413:
365:
362:
357:
351:
339:Project-related
334:
315:
296:
270:
251:
232:
213:
194:
170:
101:
98:
95:
92:
91:
32:on Knowledge's
29:
12:
11:
5:
7840:
7838:
7830:
7829:
7824:
7819:
7814:
7809:
7799:
7798:
7770:
7766:
7745:
7740:
7736:
7732:
7729:
7726:
7706:
7703:
7698:
7694:
7690:
7685:
7681:
7658:
7654:
7650:
7630:
7627:
7622:
7619:
7615:
7611:
7606:
7603:
7599:
7595:
7590:
7586:
7563:
7559:
7536:
7533:
7529:
7506:
7502:
7484:which renames
7468:
7465:
7441:
7438:
7437:
7436:
7435:
7434:
7401:
7378:
7375:
7372:
7369:
7366:
7363:
7341:
7337:
7333:
7330:
7310:
7290:
7287:
7284:
7281:
7276:
7272:
7251:
7248:
7245:
7242:
7239:
7219:
7216:
7211:
7207:
7186:
7164:
7160:
7156:
7153:
7133:
7113:
7108:
7104:
7100:
7097:
7094:
7074:
7071:
7068:
7065:
7062:
7059:
7024:
7021:
7018:
7015:
7012:
7009:
6989:
6969:
6966:
6963:
6952:
6951:
6950:
6949:
6948:
6947:
6912:
6909:
6906:
6901:
6897:
6893:
6888:
6884:
6880:
6877:
6874:
6871:
6868:
6865:
6862:
6859:
6856:
6853:
6850:
6830:
6814:
6813:
6812:
6811:
6788:
6785:
6780:
6776:
6772:
6767:
6763:
6757:
6753:
6749:
6727:
6723:
6700:
6696:
6681:
6680:
6679:
6678:
6677:
6676:
6665:
6662:
6659:
6656:
6653:
6650:
6647:
6644:
6639:
6635:
6631:
6626:
6622:
6611:
6600:
6597:
6594:
6591:
6588:
6585:
6580:
6576:
6572:
6569:
6566:
6561:
6557:
6553:
6548:
6544:
6540:
6535:
6531:
6520:
6509:
6506:
6503:
6500:
6497:
6494:
6489:
6485:
6481:
6476:
6472:
6468:
6465:
6462:
6457:
6453:
6449:
6444:
6440:
6436:
6431:
6427:
6423:
6418:
6414:
6403:
6392:
6389:
6386:
6383:
6380:
6377:
6374:
6371:
6366:
6362:
6358:
6353:
6349:
6345:
6340:
6336:
6332:
6327:
6323:
6319:
6314:
6310:
6306:
6301:
6297:
6286:
6275:
6272:
6269:
6266:
6263:
6260:
6255:
6251:
6247:
6242:
6238:
6234:
6229:
6225:
6214:
6203:
6200:
6197:
6194:
6191:
6188:
6185:
6182:
6177:
6173:
6169:
6164:
6160:
6156:
6151:
6147:
6143:
6138:
6134:
6123:
6112:
6109:
6106:
6103:
6100:
6097:
6092:
6088:
6084:
6079:
6075:
6064:
6053:
6050:
6047:
6044:
6041:
6038:
6033:
6029:
6025:
6022:
6019:
6014:
6010:
6006:
6001:
5997:
5993:
5988:
5984:
5973:
5962:
5959:
5956:
5953:
5950:
5947:
5944:
5941:
5936:
5932:
5928:
5923:
5919:
5915:
5910:
5906:
5902:
5897:
5893:
5882:
5871:
5868:
5865:
5862:
5857:
5853:
5849:
5844:
5840:
5836:
5831:
5827:
5816:
5805:
5802:
5799:
5796:
5793:
5790:
5785:
5781:
5777:
5772:
5768:
5757:
5746:
5743:
5740:
5737:
5734:
5731:
5728:
5725:
5720:
5716:
5705:
5694:
5691:
5688:
5685:
5680:
5676:
5665:
5654:
5651:
5648:
5645:
5640:
5636:
5625:
5614:
5611:
5608:
5605:
5602:
5580:
5560:
5540:
5537:
5534:
5531:
5526:
5522:
5501:
5498:
5495:
5492:
5489:
5469:
5449:
5446:
5443:
5440:
5437:
5417:
5414:
5411:
5408:
5403:
5399:
5378:
5375:
5372:
5369:
5366:
5363:
5358:
5354:
5333:
5328:
5324:
5320:
5315:
5311:
5307:
5302:
5298:
5294:
5289:
5285:
5281:
5278:
5275:
5253:
5249:
5245:
5242:
5237:
5233:
5229:
5224:
5220:
5214:
5210:
5206:
5201:
5197:
5191:
5187:
5166:
5163:
5160:
5157:
5154:
5134:
5114:
5109:
5105:
5101:
5098:
5095:
5092:
5089:
5055:
5052:
5047:
5043:
5039:
5036:
5033:
5028:
5024:
5014:, which lists
4999:
4996:
4993:
4990:
4987:
4984:
4956:
4953:
4950:
4945:
4941:
4937:
4932:
4928:
4924:
4921:
4918:
4913:
4909:
4905:
4900:
4896:
4892:
4889:
4886:
4883:
4880:
4875:
4871:
4867:
4864:
4861:
4856:
4852:
4848:
4845:
4842:
4837:
4833:
4829:
4824:
4820:
4816:
4813:
4810:
4807:
4804:
4801:
4798:
4778:
4775:
4772:
4769:
4766:
4763:
4743:
4740:
4737:
4732:
4728:
4724:
4719:
4715:
4711:
4708:
4705:
4700:
4696:
4692:
4687:
4683:
4679:
4676:
4673:
4670:
4667:
4662:
4658:
4654:
4651:
4648:
4643:
4639:
4635:
4632:
4629:
4624:
4620:
4616:
4611:
4607:
4603:
4600:
4597:
4594:
4591:
4588:
4585:
4573:
4572:
4511:
4508:
4503:
4499:
4478:
4458:
4438:
4418:
4398:
4395:
4392:
4389:
4384:
4380:
4370:it holds that
4359:
4356:
4353:
4333:
4330:
4327:
4307:
4287:
4284:
4279:
4275:
4256:
4255:
4254:
4253:
4230:
4210:
4190:
4185:
4181:
4177:
4174:
4171:
4151:
4148:
4145:
4142:
4139:
4119:
4116:
4111:
4107:
4103:
4098:
4094:
4090:
4087:
4084:
4079:
4075:
4071:
4066:
4062:
4058:
4055:
4052:
4049:
4046:
4041:
4037:
4033:
4030:
4027:
4022:
4018:
4014:
4011:
4008:
4003:
3999:
3995:
3990:
3986:
3982:
3979:
3976:
3973:
3953:
3933:
3930:
3927:
3916:
3915:
3914:
3890:
3887:
3884:
3878:
3874:
3853:
3850:
3844:
3840:
3837:
3834:
3831:
3826:
3822:
3817:
3812:
3807:
3801:
3797:
3793:
3790:
3786:
3780:
3776:
3755:
3735:
3715:
3712:
3709:
3689:
3686:
3681:
3677:
3673:
3668:
3664:
3660:
3657:
3654:
3649:
3645:
3641:
3636:
3632:
3628:
3625:
3622:
3619:
3616:
3611:
3607:
3603:
3600:
3597:
3592:
3588:
3584:
3581:
3578:
3573:
3569:
3565:
3560:
3556:
3552:
3549:
3546:
3543:
3540:
3537:
3517:
3514:
3511:
3508:
3505:
3500:
3496:
3492:
3489:
3469:
3466:
3463:
3460:
3457:
3454:
3451:
3431:
3428:
3425:
3422:
3417:
3413:
3392:
3387:
3383:
3379:
3376:
3373:
3353:
3333:
3328:
3324:
3320:
3317:
3314:
3273:
3270:
3267:
3264:
3259:
3255:
3234:
3231:
3228:
3225:
3220:
3216:
3204:
3173:
3170:
3164:
3160:
3157:
3154:
3151:
3146:
3142:
3137:
3132:
3128:
3125:
3120:
3116:
3112:
3107:
3103:
3082:
3079:
3074:
3070:
3066:
3061:
3057:
3053:
3050:
3047:
3044:
3041:
3036:
3032:
3028:
3024:
3018:
3014:
3010:
3007:
3003:
2997:
2993:
2971:
2968:
2965:
2959:
2955:
2933:
2930:
2927:
2921:
2917:
2896:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2825:
2824:
2813:
2810:
2804:
2800:
2797:
2794:
2791:
2786:
2782:
2777:
2772:
2767:
2762:
2758:
2754:
2748:
2744:
2728:
2725:
2708:
2705:
2692:
2686:
2685:
2678:
2611:
2610:
2602:Added archive
2568:
2565:
2541:
2538:
2514:
2511:
2510:
2509:
2476:
2457:
2452:
2448:
2444:
2440:
2437:
2416:
2396:
2376:
2371:
2367:
2363:
2359:
2356:
2344:characteristic
2339:
2338:
2280:
2277:
2274:
2271:
2268:
2244:
2240:
2236:
2231:
2227:
2223:
2218:
2214:
2210:
2207:
2204:
2201:
2198:
2193:
2189:
2185:
2180:
2176:
2172:
2169:
2166:
2163:
2141:
2136:
2132:
2128:
2125:
2122:
2110:
2107:
2083:
2062:
2042:
2000:
1997:
1977:
1974:
1972:
1971:
1960:
1948:
1917:
1913:
1909:
1905:
1901:
1898:
1895:
1892:
1889:
1886:
1877:
1858:
1855:
1851:
1846:
1842:
1839:
1836:
1833:
1830:
1827:
1818:
1803:
1800:
1797:
1794:
1789:
1785:
1775:
1762:
1759:
1754:
1750:
1746:
1743:
1718:
1715:
1712:
1709:
1706:
1701:
1697:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1630:
1615:
1612:
1609:
1606:
1601:
1597:
1593:
1588:
1584:
1574:
1559:
1554:
1550:
1546:
1543:
1540:
1531:
1526:
1523:
1505:
1483:
1481:
1477:
1461:
1438:
1431:
1427:
1422:
1416:
1412:
1406:
1403:
1400:
1397:
1394:
1370:
1367:
1364:
1360:
1356:
1353:
1350:
1347:
1342:
1338:
1331:
1326:
1323:
1318:
1310:
1307:
1300:
1295:
1292:
1289:
1285:
1281:
1276:
1272:
1246:
1239:
1235:
1230:
1224:
1220:
1214:
1211:
1208:
1205:
1202:
1197:
1194:
1189:
1186:
1179:
1175:
1170:
1164:
1160:
1156:
1151:
1148:
1133:
1126:
1116:
1113:
1112:
1111:
1110:
1109:
1085:
1082:
1079:
1074:
1070:
1024:
1023:
1000:
997:
994:
991:
986:
982:
943:
939:
916:
912:
889:
885:
862:
858:
835:
831:
810:
807:
804:
801:
796:
792:
769:
765:
742:
738:
715:
712:
709:
706:
703:
700:
697:
693:
680:
677:
662:
659:
648:
645:
625:
624:
582:
581:
535:
534:
524:
523:
507:75.107.115.229
487:
484:
481:
480:
477:
476:
473:
472:
465:
459:
458:
456:
439:the discussion
426:
425:
409:
397:
396:
391:
379:
378:
375:
374:
371:
370:
367:
366:
364:
363:
361:
360:
343:
335:
333:
332:
326:
316:
314:
313:
307:
297:
295:
294:
289:
281:
271:
269:
268:
262:
252:
250:
249:
243:
233:
231:
230:
224:
214:
212:
211:
205:
195:
193:
192:
187:
181:
171:
169:
168:
162:
150:
148:
147:
135:
134:
122:
121:
114:
108:
107:
105:
88:the discussion
74:
62:
61:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
7839:
7828:
7825:
7823:
7820:
7818:
7815:
7813:
7810:
7808:
7805:
7804:
7802:
7795:
7794:
7790:
7786:
7768:
7764:
7738:
7734:
7727:
7724:
7704:
7701:
7696:
7692:
7688:
7683:
7679:
7656:
7652:
7648:
7628:
7625:
7620:
7617:
7613:
7609:
7604:
7601:
7597:
7593:
7588:
7584:
7561:
7557:
7534:
7531:
7527:
7504:
7500:
7491:
7487:
7483:
7479:
7475:
7466:
7464:
7463:
7459:
7455:
7439:
7433:
7429:
7425:
7421:
7420:
7419:
7415:
7411:
7407:
7402:
7400:
7396:
7392:
7376:
7373:
7367:
7361:
7339:
7335:
7331:
7328:
7308:
7288:
7285:
7282:
7279:
7274:
7270:
7246:
7240:
7237:
7217:
7214:
7209:
7205:
7184:
7162:
7158:
7154:
7151:
7131:
7106:
7102:
7095:
7092:
7072:
7069:
7063:
7057:
7049:
7048:
7047:
7046:
7042:
7038:
7022:
7019:
7013:
7007:
6987:
6967:
6964:
6961:
6946:
6942:
6938:
6934:
6930:
6926:
6907:
6904:
6899:
6895:
6891:
6886:
6882:
6878:
6875:
6872:
6869:
6866:
6863:
6860:
6857:
6854:
6851:
6828:
6820:
6819:
6818:
6817:
6816:
6815:
6810:
6806:
6802:
6786:
6783:
6778:
6774:
6770:
6765:
6755:
6751:
6725:
6721:
6698:
6694:
6685:
6684:
6683:
6682:
6660:
6657:
6651:
6648:
6645:
6642:
6637:
6633:
6629:
6624:
6620:
6612:
6595:
6592:
6586:
6583:
6578:
6574:
6570:
6567:
6564:
6559:
6555:
6551:
6546:
6542:
6538:
6533:
6529:
6521:
6504:
6501:
6495:
6492:
6487:
6483:
6479:
6474:
6470:
6466:
6463:
6460:
6455:
6451:
6447:
6442:
6438:
6434:
6429:
6425:
6421:
6416:
6412:
6404:
6387:
6384:
6378:
6375:
6372:
6369:
6364:
6360:
6356:
6351:
6347:
6343:
6338:
6334:
6330:
6325:
6321:
6317:
6312:
6308:
6304:
6299:
6295:
6287:
6270:
6267:
6261:
6258:
6253:
6249:
6245:
6240:
6236:
6232:
6227:
6223:
6215:
6198:
6195:
6189:
6186:
6183:
6180:
6175:
6171:
6167:
6162:
6158:
6154:
6149:
6145:
6141:
6136:
6132:
6124:
6107:
6104:
6098:
6095:
6090:
6086:
6082:
6077:
6073:
6065:
6048:
6045:
6039:
6036:
6031:
6027:
6023:
6020:
6017:
6012:
6008:
6004:
5999:
5995:
5991:
5986:
5982:
5974:
5957:
5954:
5948:
5945:
5942:
5939:
5934:
5930:
5926:
5921:
5917:
5913:
5908:
5904:
5900:
5895:
5891:
5883:
5866:
5863:
5855:
5851:
5847:
5842:
5838:
5834:
5829:
5825:
5817:
5800:
5797:
5791:
5788:
5783:
5779:
5775:
5770:
5766:
5758:
5741:
5738:
5732:
5729:
5726:
5723:
5718:
5714:
5706:
5689:
5686:
5678:
5674:
5666:
5649:
5646:
5638:
5634:
5626:
5609:
5606:
5600:
5593:
5592:
5578:
5558:
5538:
5535:
5532:
5529:
5524:
5520:
5496:
5490:
5487:
5467:
5444:
5438:
5435:
5415:
5412:
5409:
5406:
5401:
5397:
5373:
5367:
5364:
5361:
5356:
5352:
5326:
5322:
5318:
5313:
5309:
5305:
5300:
5296:
5292:
5287:
5283:
5276:
5273:
5251:
5247:
5243:
5240:
5235:
5231:
5227:
5222:
5218:
5212:
5208:
5204:
5199:
5195:
5189:
5185:
5161:
5155:
5152:
5132:
5107:
5103:
5096:
5093:
5090:
5087:
5079:
5078:
5077:
5073:
5069:
5053:
5050:
5045:
5041:
5037:
5034:
5031:
5026:
5022:
5013:
4997:
4994:
4988:
4982:
4974:
4970:
4951:
4948:
4943:
4939:
4935:
4930:
4926:
4922:
4919:
4916:
4911:
4907:
4903:
4898:
4894:
4890:
4887:
4884:
4881:
4878:
4873:
4869:
4865:
4862:
4859:
4854:
4850:
4846:
4843:
4840:
4835:
4831:
4827:
4822:
4818:
4814:
4811:
4808:
4805:
4802:
4799:
4776:
4773:
4767:
4761:
4738:
4735:
4730:
4726:
4722:
4717:
4713:
4709:
4706:
4703:
4698:
4694:
4690:
4685:
4681:
4677:
4674:
4671:
4668:
4665:
4660:
4656:
4652:
4649:
4646:
4641:
4637:
4633:
4630:
4627:
4622:
4618:
4614:
4609:
4605:
4601:
4598:
4595:
4592:
4589:
4586:
4575:
4574:
4571:
4567:
4563:
4559:
4555:
4551:
4547:
4541:
4536:
4535:
4534:
4533:
4529:
4525:
4509:
4506:
4501:
4497:
4476:
4456:
4436:
4416:
4396:
4393:
4390:
4387:
4382:
4378:
4357:
4354:
4351:
4331:
4328:
4325:
4305:
4285:
4282:
4277:
4273:
4262:
4252:
4248:
4244:
4241:from thereon.
4228:
4208:
4183:
4179:
4172:
4169:
4146:
4140:
4137:
4117:
4114:
4109:
4105:
4101:
4096:
4092:
4088:
4085:
4082:
4077:
4073:
4069:
4064:
4060:
4056:
4053:
4050:
4047:
4044:
4039:
4035:
4031:
4028:
4025:
4020:
4016:
4012:
4009:
4006:
4001:
3997:
3993:
3988:
3984:
3980:
3977:
3974:
3971:
3951:
3931:
3928:
3925:
3917:
3913:
3909:
3905:
3888:
3885:
3882:
3876:
3872:
3851:
3848:
3842:
3838:
3835:
3832:
3829:
3824:
3820:
3815:
3805:
3799:
3795:
3791:
3788:
3784:
3778:
3774:
3753:
3733:
3713:
3710:
3707:
3687:
3684:
3679:
3675:
3671:
3666:
3662:
3658:
3655:
3652:
3647:
3643:
3639:
3634:
3630:
3626:
3623:
3620:
3617:
3614:
3609:
3605:
3601:
3598:
3595:
3590:
3586:
3582:
3579:
3576:
3571:
3567:
3563:
3558:
3554:
3550:
3547:
3544:
3541:
3538:
3535:
3515:
3512:
3506:
3503:
3498:
3494:
3487:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3429:
3426:
3423:
3420:
3415:
3411:
3385:
3381:
3374:
3371:
3351:
3326:
3322:
3315:
3312:
3302:
3297:
3296:
3295:
3291:
3287:
3271:
3268:
3265:
3262:
3257:
3253:
3232:
3229:
3226:
3223:
3218:
3214:
3205:
3201:
3196:
3195:
3194:
3190:
3186:
3171:
3168:
3162:
3158:
3155:
3152:
3149:
3144:
3140:
3135:
3126:
3123:
3118:
3114:
3110:
3105:
3101:
3080:
3077:
3072:
3068:
3064:
3059:
3055:
3051:
3048:
3045:
3042:
3039:
3034:
3030:
3026:
3022:
3016:
3012:
3008:
3005:
3001:
2995:
2991:
2969:
2966:
2963:
2957:
2953:
2931:
2928:
2925:
2919:
2915:
2894:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2846:
2841:
2840:
2839:
2838:
2834:
2830:
2811:
2808:
2802:
2798:
2795:
2792:
2789:
2784:
2780:
2775:
2765:
2760:
2756:
2752:
2746:
2742:
2734:
2733:
2732:
2726:
2724:
2723:
2719:
2715:
2707:Chess hashing
2706:
2704:
2703:
2697:
2690:
2683:
2679:
2676:
2672:
2671:
2670:
2663:
2657:
2653:
2649:
2645:
2639:
2634:
2630:
2624:
2620:
2616:
2609:
2605:
2601:
2600:
2599:
2595:
2587:
2581:
2577:
2572:
2566:
2564:
2562:
2558:
2554:
2553:37.26.148.151
2550:
2539:
2537:
2536:
2532:
2528:
2524:
2520:
2512:
2505:
2501:
2497:
2493:
2474:
2450:
2446:
2414:
2394:
2369:
2365:
2345:
2341:
2340:
2337:
2333:
2329:
2325:
2321:
2317:
2313:
2309:
2308:
2307:
2305:
2301:
2297:
2292:
2275:
2269:
2266:
2257:
2242:
2238:
2234:
2229:
2225:
2221:
2216:
2212:
2208:
2205:
2202:
2199:
2196:
2191:
2187:
2183:
2178:
2170:
2167:
2164:
2153:
2134:
2130:
2123:
2120:
2108:
2106:
2105:
2101:
2097:
2031:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
1998:
1996:
1995:
1991:
1987:
1983:
1975:
1969:
1964:
1961:
1958:
1953:
1950:
1947:
1946:
1942:
1938:
1933:
1915:
1911:
1907:
1899:
1896:
1893:
1887:
1884:
1876:
1856:
1853:
1849:
1840:
1837:
1834:
1828:
1825:
1817:
1801:
1798:
1795:
1792:
1787:
1783:
1774:
1760:
1757:
1752:
1748:
1744:
1741:
1732:
1713:
1710:
1707:
1704:
1699:
1695:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1629:
1610:
1599:
1595:
1586:
1582:
1573:
1572:like this :
1552:
1548:
1541:
1538:
1530:
1524:
1522:
1520:
1515:
1504:
1501:
1498:
1496:
1492:
1487:
1476:
1473:
1459:
1452:I think that
1450:
1429:
1425:
1420:
1414:
1410:
1404:
1398:
1395:
1392:
1383:
1368:
1365:
1362:
1354:
1351:
1348:
1340:
1336:
1324:
1321:
1308:
1305:
1298:
1293:
1290:
1287:
1283:
1279:
1274:
1270:
1261:
1260:we can write
1258:
1237:
1233:
1228:
1222:
1218:
1209:
1206:
1203:
1200:
1195:
1192:
1187:
1177:
1173:
1168:
1162:
1158:
1154:
1146:
1138:
1132:
1125:
1123:
1114:
1108:
1104:
1100:
1099:71.19.161.130
1080:
1072:
1068:
1059:
1058:
1057:
1056:
1055:
1054:
1050:
1046:
1040:
1037:
1033:
1032:
1029:
1028:Richard Pinch
1022:
1018:
1014:
998:
995:
992:
989:
984:
980:
971:
970:
969:
968:
965:
960:
957:
941:
937:
914:
910:
887:
883:
860:
856:
833:
829:
808:
805:
802:
799:
794:
790:
767:
763:
740:
736:
713:
710:
707:
704:
701:
698:
695:
691:
678:
676:
675:
672:
668:
660:
658:
657:
654:
646:
644:
643:
639:
635:
631:
623:
619:
615:
614:71.19.161.130
609:
605:
599:
595:
591:
587:
584:
583:
578:
577:
576:
572:
570:
566:
562:
558:
550:
549:
545:
541:
533:
530:
526:
525:
520:
519:
518:
516:
512:
508:
504:
497:
494:
491:
485:
470:
464:
461:
460:
457:
440:
436:
432:
431:
423:
412:
410:
407:
403:
402:
398:
395:
392:
389:
385:
356:
349:
345:
344:
342:
340:
336:
331:
328:
327:
325:
323:
322:
317:
312:
309:
308:
306:
304:
303:
298:
293:
290:
287:
283:
282:
280:
278:
277:
272:
267:
264:
263:
261:
259:
258:
253:
248:
245:
244:
242:
240:
239:
234:
229:
226:
225:
223:
221:
220:
215:
210:
207:
206:
204:
202:
201:
196:
191:
188:
186:
183:
182:
180:
178:
177:
172:
167:
164:
163:
161:
159:
158:
153:
152:
149:
145:
141:
140:
137:
136:
132:
128:
127:
123:
119:
113:
110:
109:
106:
89:
85:
81:
80:
75:
72:
68:
67:
63:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
7489:
7485:
7473:
7470:
7448:rather than
7443:
6953:
6932:
6928:
6924:
5012:bch_code.pdf
4972:
4968:
4558:bch_code.pdf
4553:
4549:
4545:
4257:
4130:. Extending
3700:, but using
2826:
2730:
2710:
2687:
2662:source check
2641:
2635:
2622:
2618:
2614:
2612:
2573:
2570:
2547:— Preceding
2543:
2516:
2490:— Preceding
2323:
2319:
2315:
2311:
2296:82.57.60.212
2293:
2258:
2154:
2112:
2032:
2002:
1979:
1963:
1952:
1934:
1931:
1874:
1815:
1733:
1730:
1627:
1571:
1528:
1516:
1513:
1502:
1499:
1494:
1490:
1485:
1482:
1474:
1451:
1384:
1262:
1259:
1139:
1134:
1130:
1118:
1041:
1038:
1034:
1025:
961:
958:
682:
664:
650:
629:
626:
598:Reed-Solomon
590:Reed-Solomon
573:
561:173.73.54.71
555:— Preceding
551:
536:
498:
495:
492:
489:
428:
338:
337:
321:Unreferenced
319:
318:
300:
299:
274:
273:
255:
254:
236:
235:
217:
216:
198:
197:
174:
173:
155:
154:
77:
40:WikiProjects
7549:instead of
7476:as seen in
7454:RossBoswell
2629:Sourcecheck
2387:(for prime
2318:but rather
2053:during the
604:Golay codes
540:Alan.A.Mick
501:—Preceding
7801:Categories
4469:the value
667:| this PDF
7301:... "let
2682:this tool
2675:this tool
529:Oli Filth
209:Computing
7641:or that
6980:, where
6740:is not:
4754:assumes
2688:Cheers.—
2586:cbignore
2576:BCH code
2549:unsigned
2527:Hippo.69
2519:Hippo.69
2504:contribs
2492:unsigned
2096:Hippo.69
2011:section
1045:Hippo.69
557:unsigned
503:unsigned
257:Maintain
200:Copyedit
7406:bch15.c
7262:modulo
4975:, then
2727:Example
2698::Online
2615:checked
2580:my edit
2496:Arun ks
1986:Bobmath
238:Infobox
176:Cleanup
30:C-class
7785:Rcgldr
7671:means
7424:Madyno
7410:Rcgldr
7391:Rcgldr
7037:Madyno
6937:Rcgldr
6801:Madyno
6686:Hence
5266:where
5068:Rcgldr
4562:Rcgldr
4540:Madyno
4524:Madyno
4243:Madyno
3904:Rcgldr
3301:Madyno
3286:Madyno
3200:Rcgldr
3185:Rcgldr
3093:, and
2845:Madyno
2829:Madyno
2623:failed
2594:nobots
2306:Damix
2021:Rcgldr
1385:where
1122:| here
1013:Rcgldr
671:Dyfrgi
634:Rcgldr
219:Expand
36:scale.
5344:with
2407:) is
1980:From
1932:see
1731:see
1628:and
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1491:right
1486:other
964:Zom-B
728:from
586:CRC32
522:them.
302:Stubs
276:Photo
133:with:
7789:talk
7765:0101
7458:talk
7428:talk
7414:talk
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2025:talk
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2656:RfC
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1507:-->
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608:BCH
594:BCH
463:???
112:???
7803::
7791:)
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7680:α
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1213:(
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1196:2
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1188:=
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1101:(
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341::
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279::
260::
241::
222::
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179::
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42::
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