Best Known (102, 102+∞, s)-Nets in Base 7
(102, 102+∞, 83)-Net over F7 — Constructive and digital
Digital (102, m, 83)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (102, 82)-sequence over F7, using
- Niederreiter–Xing sequence construction III based on function field F2/F7 with g(F2) = 21, N(F2) ≥ 2, and N(F2) + N2(F2) ≥ 83 from GarcÃa–Stichtenoth tower as constant field extension [i]
(102, 102+∞, 105)-Net over F7 — Digital
Digital (102, m, 105)-net over F7 for arbitrarily large m, using
- net from sequence [i] based on digital (102, 104)-sequence over F7, using
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 43 and N(F) ≥ 105, using
- t-expansion [i] based on digital (43, 104)-sequence over F7, using
(102, 102+∞, 636)-Net in Base 7 — Upper bound on s
There is no (102, m, 637)-net in base 7 for arbitrarily large m, because
- m-reduction [i] would yield (102, 1907, 637)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(71907, 637, S7, 3, 1805), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 195955 919452 794526 847012 836441 097234 745674 826998 568695 217300 675944 733153 144832 017969 284872 827556 625359 998582 041404 972966 147110 844430 861139 778064 653706 523846 737160 458288 779485 069089 132493 752058 666721 059284 643418 495002 449091 898185 812037 688255 279760 094319 427366 875703 917619 787467 433940 122022 685387 650406 029141 458896 060231 064425 337160 947412 681271 024839 034638 196032 782786 141551 578294 266276 385527 903549 983431 973083 657906 989299 960765 189932 766959 112530 429525 607859 900103 585749 393640 885345 544510 800408 996320 540618 459964 161013 912523 375940 968914 312485 184421 382692 152496 525653 858354 853585 018107 077018 550567 663133 648745 479952 727782 884493 952467 869645 804034 570331 969598 215145 259587 229143 364386 957328 754034 122270 434019 785799 154845 284384 514208 482161 402479 890960 041603 581022 833072 289967 436903 094437 242226 651849 517042 407004 877932 428344 663661 768736 183446 670678 363849 724129 029837 777097 786888 845490 371668 587858 540942 249062 356406 556355 332818 010145 781356 891652 271704 101414 012904 992545 613118 743235 511489 383568 113790 883853 119373 855033 013160 628644 233392 819113 798018 175108 721068 620735 786269 136216 821086 507164 690686 333596 928738 011515 808008 037715 723277 168123 574824 946434 622273 080811 085156 230919 054866 970562 132625 873592 672489 084306 625712 156720 611859 149088 525885 498484 629749 658265 557144 053187 188729 522397 830050 362515 442523 311723 623507 687644 015267 065605 454067 445492 991249 783851 860947 476533 129652 932255 048306 629197 193176 862762 476960 194840 550153 183910 497794 876938 126216 399992 922086 305252 624420 098882 447961 955985 399203 578525 093556 834587 470034 689993 586438 201991 784176 031821 020803 309815 167326 511723 316297 824602 331757 544923 079131 054604 183805 178976 333607 / 43 > 71907 [i]
- extracting embedded OOA [i] would yield OOA(71907, 637, S7, 3, 1805), but