115:
972:
Not much can be said about these codes at this level of generality, but if the incidence structure has some "regularity" the codes produced this way can be analyzed and information about the codes and the incidence structures can be gleaned from each other. When the incidence structure is a finite
326:
on the lines of the plane. Each of the three non-Desarguesian planes of order nine have collineation groups having two orbits on the lines, producing two non-isomorphic affine planes of order nine, depending on which orbit the line to be removed is selected from.
314:, the removal of different lines could result in non-isomorphic affine planes. For instance, there are exactly four projective planes of order nine, and seven affine planes of order nine. There is only one affine plane corresponding to the
1141:
can be defined in an analogous manner to the construction of affine planes from projective planes. It is also possible to provide a system of axioms for the higher-dimensional affine spaces which does not refer to the corresponding
379:. While in general it is often easier to work with projective planes, in this context the affine planes are preferred and several authors simply use the term translation plane to mean affine translation plane.
952:
191:
All known finite affine planes have orders that are prime or prime power integers. The smallest affine plane (of order 2) is obtained by removing a line and the three points on that line from the
215:
exists (however, the definition of order in these two cases is not the same). Thus, there is no affine plane of order 6 or order 10 since there are no projective planes of those orders. The
1189:
75:
Since no concepts other than those involving the relationship between points and lines are involved in the axioms, an affine plane is an object of study belonging to
981:
of the field does not divide the order of the plane, the code generated is the full space and does not carry any information. On the other hand,
1350:
216:
299:
by removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane by adding a
1539:
1521:
1499:
1481:
1454:
1407:
195:. A similar construction, starting from the projective plane of order 3, produces the affine plane of order 3 sometimes called the
278:
41:
Given any line and any point not on that line there is a unique line which contains the point and does not meet the given line. (
903:
1389:
799:
Given a translation net, it is not always possible to add parallel classes to the net to form an affine plane. However, if
1359:
1198:
1557:
1562:
355:
323:
977:. How much information the code carries about the affine plane depends in part on the choice of field. If the
80:
1368:, QMW Maths Notes, vol. 13, London: Queen Mary and Westfield College School of Mathematical Sciences,
1099:
and the minimum weight vectors are precisely the scalar multiples of the (incidence vectors of) lines of
1047:
and all the minimum weight vectors are constant multiples of vectors whose entries are either zero or one.
978:
558:
311:
99:
219:
provides further limitations on the order of a projective plane, and thus, the order of an affine plane.
255:
of lines. The lines in any parallel class form a partition the points of the affine plane. Each of the
64:
Given a point and a line, there is a unique line which contains the point and is parallel to the line.
69:
842:
832:
505:
400:
358:
196:
42:
1220:
819:
members can be extended and the translation net can be completed to an affine translation plane.
315:
76:
1535:
1517:
1495:
1477:
1450:
1403:
1385:
1346:
1212:
793:
304:
126:
If the number of points in an affine plane is finite, then if one line of the plane contains
1202:
1143:
1127:
828:
300:
296:
290:
206:
1426:
1373:
1369:
87:
626:
Parallelism (as defined in affine planes) is an equivalence relation on the set of lines.
1510:
48:
There exist four points such that no three are collinear (points not on a single line).
887:. Another related code that contains information about the incidence structure is the
1551:
251:
lines apiece under the equivalence relation of parallelism. These classes are called
103:
95:
57:
1138:
396:
319:
114:
91:
20:
90:
is an affine plane. There are many finite and infinite affine planes. As well as
864:
102:, not derived from coordinates in a division ring, satisfying these axioms. The
382:
An alternate view of affine translation planes can be obtained as follows: Let
192:
1216:
375:
and the affine plane obtained by removing the translation line is called an
262:
lines that pass through a single point lies in a different parallel class.
27:
1363:
1187:
Moulton, Forest Ray (1902), "A Simple Non-Desarguesian Plane
Geometry",
514:
and whose lines are the cosets of components, that is, sets of the form
1224:
580:
is an automorphism group acting regularly on the points of this plane.
1207:
595:
An incidence structure more general than a finite affine plane is a
60:. Using this definition, Playfair's axiom above can be replaced by:
34:
is a system of points and lines that satisfy the following axioms:
796:, any subset of the parallel classes will form a translation net.
113:
641:
lines (so each parallel class of lines partitions the point set).
281:. Only the incidence relations are needed for this construction.
973:
affine plane, the codes belong to a class of codes known as
947:{\displaystyle \operatorname {Hull} (C)=C\cap C^{\perp },}
371:. A projective plane with a translation line is called a
265:
The parallel class structure of an affine plane of order
361:
on the points of the affine plane obtained by removing
650:
parallel classes of lines. Each point lies on exactly
906:
1490:
Lindner, Charles C.; Rodger, Christopher A. (1997),
307:
where an equivalence class of parallel lines meets.
19:
For the algebraic definition of the same space, see
741:, any two of which intersect only in {0} (called a
1509:
946:
1190:Transactions of the American Mathematical Society
38:Any two distinct points lie on a unique line.
8:
1534:, San Francisco: W.H. Freeman and Company,
1248:
1162:
735:-dimensional subspaces of the vector space
705:mutually orthogonal Latin squares of order
1174:
1308:
1296:
1284:
1272:
1260:
1236:
1206:
935:
905:
805:is an infinite field, any partial spread
295:An affine plane can be obtained from any
52:In an affine plane, two lines are called
1324:
1155:
436:that partition the non-zero vectors of
16:Axiomatically defined geometrical space
1021:, then the minimum weight of the code
674:is precisely an affine plane of order
635:points, and every parallel class has
7:
1447:Projective Geometry: An Introduction
1341:Assmus, E.F. Jr.; Key, J.D. (1992),
1320:
1120:the geometric code generated is the
656:lines, one from each parallel class.
557:is an affine plane and the group of
1449:, Oxford: Oxford University Press,
1420:, Berlin: Deutscher Verlag d. Wiss.
349:if the group of elations with axis
1418:Grundlagen der Elementarmathematik
271:may be used to construct a set of
232:lines of an affine plane of order
14:
1474:Introduction to Finite Geometries
279:mutually orthogonal latin squares
72:on the lines of an affine plane.
1530:Stevenson, Frederick W. (1972),
508:whose points are the vectors of
303:, each of whose points is that
285:Relation with projective planes
106:is an example of one of these.
1398:Hughes, D.; Piper, F. (1973),
1345:, Cambridge University Press,
919:
913:
322:of that projective plane acts
1:
1199:American Mathematical Society
1073:is a field of characteristic
1003:is a field of characteristic
542:is a component of the spread
474:are distinct components then
83:satisfying Playfair's axiom.
1476:, Amsterdam: North-Holland,
1323:, p. 138, but see also
1061:is an affine plane of order
991:is an affine plane of order
1516:, Berlin: Springer Verlag,
1382:Geometry: Euclid and Beyond
1365:Projective and Polar Spaces
698:is equivalent to a set of
310:If the projective plane is
199:. An affine plane of order
145:each point is contained in
1579:
963:is the orthogonal code to
792:. Starting with an affine
430:-dimensional subspaces of
288:
217:Bruck–Ryser–Chowla theorem
79:. They are non-degenerate
18:
1467:, Berlin: Springer Verlag
1463:Dembowski, Peter (1968),
715:Example: translation nets
331:Affine translation planes
1508:Lüneburg, Heinz (1980),
1425:Moorhouse, Eric (2007),
377:affine translation plane
318:of order nine since the
205:exists if and only if a
119:Affine plane of order 3
1380:Hartshorne, R. (2000),
1343:Designs and their Codes
1249:Hughes & Piper 1973
1197:(2), Providence, R.I.:
1163:Hughes & Piper 1973
827:Given the "line/point"
719:For an arbitrary field
629:Every line has exactly
245:equivalence classes of
100:non-Desarguesian planes
98:), there are also many
948:
867:that we can denote by
755:, form the lines of a
749:, and their cosets in
341:in a projective plane
123:
1472:Kárteszi, F. (1976),
1309:Assmus & Key 1992
1297:Assmus & Key 1992
1285:Assmus & Key 1992
949:
897:which is defined as:
452:of the spread and if
188:of the affine plane.
117:
56:if they are equal or
904:
164:there is a total of
110:Finite affine planes
70:equivalence relation
1445:Casse, Rey (2006),
1402:, Springer-Verlag,
833:incidence structure
610:. This consists of
506:incidence structure
197:Hesse configuration
136:each line contains
1558:Incidence geometry
1512:Translation Planes
1428:Incidence Geometry
944:
745:). The members of
320:collineation group
316:Desarguesian plane
161:points in all, and
124:
121:9 points, 12 lines
77:incidence geometry
68:Parallelism is an
1563:Planes (geometry)
1532:Projective Planes
1465:Finite Geometries
1416:Lenz, H. (1961),
1400:Projective Planes
1360:Cameron, Peter J.
1352:978-0-521-41361-9
851:the row space of
794:translation plane
759:on the points of
622:lines such that:
442:. The members of
373:translation plane
305:point at infinity
94:over fields (and
1570:
1544:
1526:
1515:
1504:
1486:
1468:
1459:
1434:
1433:
1421:
1412:
1394:
1376:
1355:
1328:
1318:
1312:
1306:
1300:
1294:
1288:
1282:
1276:
1275:, pp. 21–22
1270:
1264:
1258:
1252:
1246:
1240:
1234:
1228:
1227:
1210:
1184:
1178:
1172:
1166:
1160:
1144:projective space
1128:Reed-Muller Code
1125:
1119:
1104:
1098:
1078:
1072:
1066:
1060:
1046:
1040:
1020:
1014:
1008:
1002:
996:
990:
968:
962:
953:
951:
950:
945:
940:
939:
896:
886:
862:
856:
850:
840:
829:incidence matrix
818:
816:
809:with fewer than
808:
804:
791:
789:
781:
775:
770:
764:
754:
748:
740:
734:
728:
724:
710:
704:
697:
688:
679:
673:
667:
655:
649:
640:
634:
621:
615:
609:
600:
590:
585:Generalization:
579:
573:
556:
547:
541:
535:
529:
523:
513:
503:
497:
473:
462:
447:
441:
435:
429:
423:
417:
407:
394:
387:
370:
366:
354:
347:translation line
344:
340:
312:non-Desarguesian
301:line at infinity
297:projective plane
291:Projective plane
277:
270:
261:
253:parallel classes
250:
244:
237:
231:
214:
207:projective plane
204:
183:
173:
160:
151:
141:
131:
43:Playfair's axiom
1578:
1577:
1573:
1572:
1571:
1569:
1568:
1567:
1548:
1547:
1542:
1529:
1524:
1507:
1502:
1489:
1484:
1471:
1462:
1457:
1444:
1441:
1439:Further reading
1431:
1424:
1415:
1410:
1397:
1392:
1379:
1358:
1353:
1340:
1337:
1332:
1331:
1319:
1315:
1307:
1303:
1295:
1291:
1283:
1279:
1271:
1267:
1259:
1255:
1247:
1243:
1235:
1231:
1208:10.2307/1986419
1186:
1185:
1181:
1175:Hartshorne 2000
1173:
1169:
1161:
1157:
1152:
1136:
1121:
1110:
1100:
1092:
1080:
1074:
1068:
1062:
1056:
1042:
1034:
1022:
1016:
1010:
1004:
998:
992:
986:
975:geometric codes
964:
958:
931:
902:
901:
892:
880:
868:
858:
852:
846:
836:
825:
823:Geometric codes
812:
810:
806:
800:
785:
783:
777:
768:
766:
760:
757:translation net
750:
746:
736:
730:
726:
720:
717:
706:
699:
693:
684:
675:
669:
661:
651:
645:
636:
630:
617:
611:
605:
596:
593:
586:
575:
561:
552:
543:
537:
531:
530:is a vector of
525:
515:
509:
499:
492:
483:
475:
472:
464:
461:
453:
448:are called the
443:
437:
431:
425:
419:
413:
403:
389:
383:
368:
367:from the plane
362:
350:
342:
336:
333:
293:
287:
272:
266:
256:
246:
239:
233:
223:
210:
200:
179:
165:
156:
146:
137:
127:
122:
120:
112:
88:Euclidean plane
24:
17:
12:
11:
5:
1576:
1574:
1566:
1565:
1560:
1550:
1549:
1546:
1545:
1540:
1527:
1522:
1505:
1500:
1487:
1482:
1469:
1460:
1455:
1440:
1437:
1436:
1435:
1422:
1413:
1408:
1395:
1390:
1377:
1356:
1351:
1336:
1333:
1330:
1329:
1313:
1301:
1289:
1277:
1273:Moorhouse 2007
1265:
1261:Moorhouse 2007
1253:
1241:
1237:Moorhouse 2007
1229:
1179:
1167:
1154:
1153:
1151:
1148:
1135:
1132:
1107:
1106:
1088:
1051:Furthermore,
1049:
1048:
1030:
979:characteristic
955:
954:
943:
938:
934:
930:
927:
924:
921:
918:
915:
912:
909:
876:
831:of any finite
824:
821:
782:-net of order
743:partial spread
716:
713:
668:-net of order
658:
657:
642:
627:
592:
583:
582:
581:
488:
479:
468:
457:
332:
329:
289:Main article:
286:
283:
184:is called the
176:
175:
162:
153:
143:
118:
111:
108:
96:division rings
66:
65:
50:
49:
46:
39:
15:
13:
10:
9:
6:
4:
3:
2:
1575:
1564:
1561:
1559:
1556:
1555:
1553:
1543:
1541:0-7167-0443-9
1537:
1533:
1528:
1525:
1523:0-387-09614-0
1519:
1514:
1513:
1506:
1503:
1501:0-8493-3986-3
1497:
1494:, CRC Press,
1493:
1492:Design Theory
1488:
1485:
1483:0-7204-2832-7
1479:
1475:
1470:
1466:
1461:
1458:
1456:0-19-929886-6
1452:
1448:
1443:
1442:
1438:
1430:
1429:
1423:
1419:
1414:
1411:
1409:0-387-90044-6
1405:
1401:
1396:
1393:
1387:
1383:
1378:
1375:
1371:
1367:
1366:
1361:
1357:
1354:
1348:
1344:
1339:
1338:
1334:
1326:
1322:
1317:
1314:
1311:, p. 211
1310:
1305:
1302:
1299:, p. 208
1298:
1293:
1290:
1286:
1281:
1278:
1274:
1269:
1266:
1262:
1257:
1254:
1251:, p. 100
1250:
1245:
1242:
1238:
1233:
1230:
1226:
1222:
1218:
1214:
1209:
1204:
1200:
1196:
1192:
1191:
1183:
1180:
1176:
1171:
1168:
1164:
1159:
1156:
1149:
1147:
1145:
1140:
1139:Affine spaces
1134:Affine spaces
1133:
1131:
1129:
1124:
1117:
1113:
1103:
1096:
1091:
1087:
1083:
1077:
1071:
1065:
1059:
1054:
1053:
1052:
1045:
1038:
1033:
1029:
1025:
1019:
1013:
1007:
1001:
995:
989:
984:
983:
982:
980:
976:
970:
967:
961:
941:
936:
932:
928:
925:
922:
916:
910:
907:
900:
899:
898:
895:
890:
884:
879:
875:
871:
866:
861:
855:
849:
844:
839:
834:
830:
822:
820:
815:
803:
797:
795:
788:
780:
774:
763:
758:
753:
744:
739:
733:
723:
714:
712:
709:
702:
696:
692:
687:
681:
678:
672:
665:
654:
648:
643:
639:
633:
628:
625:
624:
623:
620:
614:
608:
604:
599:
589:
584:
578:
574:for a vector
572:
568:
564:
560:
555:
551:
550:
549:
546:
540:
534:
528:
522:
518:
512:
507:
502:
496:
491:
487:
482:
478:
471:
467:
460:
456:
451:
446:
440:
434:
428:
422:
416:
411:
406:
402:
398:
395:-dimensional
393:
386:
380:
378:
374:
365:
360:
357:
353:
348:
339:
330:
328:
325:
321:
317:
313:
308:
306:
302:
298:
292:
284:
282:
280:
275:
269:
263:
259:
254:
249:
242:
236:
230:
226:
220:
218:
213:
208:
203:
198:
194:
189:
187:
182:
172:
168:
163:
159:
154:
149:
144:
140:
135:
134:
133:
132:points then:
130:
116:
109:
107:
105:
104:Moulton plane
101:
97:
93:
92:affine planes
89:
86:The familiar
84:
82:
81:linear spaces
78:
73:
71:
63:
62:
61:
59:
55:
47:
44:
40:
37:
36:
35:
33:
29:
22:
1531:
1511:
1491:
1473:
1464:
1446:
1427:
1417:
1399:
1384:, Springer,
1381:
1364:
1342:
1325:Cameron 1991
1316:
1304:
1292:
1287:, p. 43
1280:
1268:
1263:, p. 13
1256:
1244:
1239:, p. 11
1232:
1194:
1188:
1182:
1177:, p. 71
1170:
1165:, p. 82
1158:
1137:
1122:
1115:
1111:
1108:
1101:
1094:
1089:
1085:
1081:
1075:
1069:
1063:
1057:
1050:
1043:
1036:
1031:
1027:
1023:
1017:
1011:
1005:
999:
993:
987:
974:
971:
965:
959:
956:
893:
888:
882:
877:
873:
869:
859:
853:
847:
837:
826:
813:
801:
798:
786:
778:
772:
761:
756:
751:
742:
737:
731:
729:be a set of
721:
718:
707:
700:
694:
691:net of order
690:
685:
682:
676:
670:
663:
659:
652:
646:
637:
631:
618:
612:
606:
603:net of order
602:
597:
594:
587:
576:
570:
566:
562:
559:translations
553:
544:
538:
532:
526:
520:
516:
510:
500:
494:
489:
485:
480:
476:
469:
465:
458:
454:
449:
444:
438:
432:
426:
420:
414:
409:
404:
397:vector space
391:
384:
381:
376:
372:
363:
359:transitively
351:
346:
337:
334:
324:transitively
309:
294:
273:
267:
264:
257:
252:
247:
240:
234:
228:
224:
221:
211:
201:
190:
185:
180:
177:
170:
166:
157:
147:
138:
128:
125:
85:
74:
67:
53:
51:
32:affine plane
31:
25:
21:Affine space
1327:, chapter 3
1201:: 192–195,
865:linear code
616:points and
178:The number
1552:Categories
1391:0387986502
1335:References
841:, and any
776:this is a
644:There are
450:components
238:fall into
193:Fano plane
155:there are
1321:Lenz 1961
1217:0002-9947
937:⊥
929:∩
911:
771:| =
418:is a set
209:of order
1362:(1991),
1114:= AG(2,
1015:divides
1009:, where
548:. Then:
58:disjoint
54:parallel
28:geometry
1374:1153019
1225:1986419
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1079:, then
1026:= Hull(
504:be the
399:over a
335:A line
142:points,
1538:
1520:
1498:
1480:
1453:
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1388:
1372:
1349:
1223:
1215:
957:where
817:|
811:|
790:|
784:|
767:|
725:, let
524:where
498:. Let
410:spread
174:lines.
152:lines,
1432:(PDF)
1221:JSTOR
1150:Notes
1126:-ary
1109:When
863:is a
857:over
843:field
765:. If
591:-nets
401:field
388:be a
345:is a
186:order
30:, an
1536:ISBN
1518:ISBN
1496:ISBN
1478:ISBN
1451:ISBN
1404:ISBN
1386:ISBN
1347:ISBN
1213:ISSN
1067:and
997:and
908:Hull
889:Hull
666:+ 1)
536:and
463:and
408:. A
356:acts
222:The
1203:doi
1055:If
1041:is
985:If
891:of
703:− 2
660:An
424:of
412:of
276:− 1
260:+ 1
243:+ 1
150:+ 1
26:In
1554::
1370:MR
1219:,
1211:,
1193:,
1146:.
1130:.
1097:))
1039:))
969:.
872:=
845:,
835:,
711:.
683:A
680:.
619:nk
569:+
565:→
519:+
493:=
484:⊕
227:+
169:+
1205::
1195:3
1123:q
1118:)
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1105:.
1102:π
1095:π
1093:(
1090:F
1086:C
1082:C
1076:p
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1064:p
1058:π
1044:n
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1035:(
1032:F
1028:C
1024:B
1018:n
1012:p
1006:p
1000:F
994:n
988:π
966:C
960:C
942:,
933:C
926:C
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920:)
917:C
914:(
894:C
885:)
883:M
881:(
878:F
874:C
870:C
860:F
854:M
848:F
838:M
814:F
807:Σ
802:F
787:F
779:k
773:k
769:Σ
762:F
752:F
747:Σ
738:F
732:n
727:Σ
722:F
708:n
701:k
695:n
689:-
686:k
677:n
671:n
664:n
662:(
653:k
647:k
638:n
632:n
613:n
607:n
601:-
598:k
588:k
577:w
571:w
567:x
563:x
554:A
545:S
539:U
533:V
527:v
521:U
517:v
511:V
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495:V
490:j
486:V
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477:V
470:j
466:V
459:i
455:V
445:S
439:V
433:V
427:n
421:S
415:V
405:F
392:n
390:2
385:V
369:Π
364:l
352:l
343:Π
338:l
274:n
268:n
258:n
248:n
241:n
235:n
229:n
225:n
212:n
202:n
181:n
171:n
167:n
158:n
148:n
139:n
129:n
45:)
23:.
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