Best Known (68, 68+∞, s)-Nets in Base 9
(68, 68+∞, 165)-Net over F9 — Constructive and digital
Digital (68, m, 165)-net over F9 for arbitrarily large m, using
- net from sequence [i] based on digital (68, 164)-sequence over F9, using
- t-expansion [i] based on digital (64, 164)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 64 and N(F) ≥ 165, using
- T4 from the second tower of function fields by GarcÃa and Stichtenoth over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 64 and N(F) ≥ 165, using
- t-expansion [i] based on digital (64, 164)-sequence over F9, using
(68, 68+∞, 192)-Net over F9 — Digital
Digital (68, m, 192)-net over F9 for arbitrarily large m, using
- net from sequence [i] based on digital (68, 191)-sequence over F9, using
- t-expansion [i] based on digital (61, 191)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 61 and N(F) ≥ 192, using
- t-expansion [i] based on digital (61, 191)-sequence over F9, using
(68, 68+∞, 570)-Net in Base 9 — Upper bound on s
There is no (68, m, 571)-net in base 9 for arbitrarily large m, because
- m-reduction [i] would yield (68, 1709, 571)-net in base 9, but
- extracting embedded OOA [i] would yield OOA(91709, 571, S9, 3, 1641), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 57 413300 671233 147501 654482 014177 417240 620296 013666 059252 285538 591108 901679 720701 056935 595621 712926 773556 211376 643049 983006 275845 346283 352983 876074 597635 960020 248749 431529 532430 509221 960643 810953 488212 482112 256005 063263 996823 098985 049624 436683 187448 684818 183035 374492 957063 556919 651510 584424 422668 839534 683543 197857 816240 044402 739228 497903 810159 028939 105821 972492 805680 412377 516268 339743 554868 203546 558607 805690 847329 912397 128702 026962 055975 045372 604772 535992 040749 956320 264311 697880 570050 669253 170514 631973 902159 036562 193226 298688 951473 503269 418844 988299 175815 646026 932515 153867 702568 299729 566601 877891 268422 032575 870684 769454 452429 232148 339745 150930 294119 950053 604211 206397 735155 303446 939903 927277 981330 218244 057794 075276 598920 941318 513848 346169 579407 625263 286057 384272 638394 652639 641780 752689 288426 519757 703644 484743 720423 041365 982366 262553 357330 721694 151934 120477 492859 354144 971224 358712 628978 018502 653896 205631 798275 702093 860867 749528 106836 049718 399946 527190 347104 235465 771797 649070 986004 801714 170999 881853 138206 695193 454617 616057 575487 319711 126324 291090 526817 935993 518237 630989 175502 790624 030924 266391 921288 860501 567785 485856 169169 940641 221732 754963 904904 863341 923377 944082 225671 520333 422074 849996 617770 875316 833968 505744 923480 020387 378962 994465 779102 157030 030442 290370 550042 790226 668991 889151 182956 791512 416099 331748 046914 210715 112771 449353 058393 983692 654267 828800 670333 969639 545067 759089 015469 035037 573668 879303 251219 679385 512568 001360 573295 052030 224080 653729 517885 981208 415739 156258 521845 812381 755673 886586 299092 493702 699225 815139 035472 308382 030809 735276 215589 943631 511698 326883 748275 532878 549561 178458 172829 679776 180770 972501 / 821 > 91709 [i]
- extracting embedded OOA [i] would yield OOA(91709, 571, S9, 3, 1641), but