Best Known (211, 211+∞, s)-Nets in Base 3
(211, 211+∞, 112)-Net over F3 — Constructive and digital
Digital (211, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (211, 111)-sequence over F3, using
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on function field F4/F3 with g(F4) = 79, N(F4) ≥ 2, and N(F4) + N2(F4) ≥ 112 from GarcÃa–Stichtenoth tower as constant field extension [i]
- t-expansion [i] based on digital (189, 111)-sequence over F3, using
(211, 211+∞, 163)-Net over F3 — Digital
Digital (211, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (211, 162)-sequence over F3, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 151 and N(F) ≥ 163, using
- t-expansion [i] based on digital (151, 162)-sequence over F3, using
(211, 211+∞, 436)-Net in Base 3 — Upper bound on s
There is no (211, m, 437)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (211, 2615, 437)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32615, 437, S3, 6, 2404), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 246 740474 354684 808190 801691 342583 698330 319314 327033 241319 762308 357756 813012 207612 649163 274979 376171 922106 943344 958316 622393 261698 701126 896550 725552 361093 055940 160085 372887 492309 379689 340555 817984 644286 596885 627005 634734 809975 458857 373513 228004 810038 637420 043111 161671 883856 278361 080942 409981 756991 062540 497184 494383 238394 648135 301676 036080 308945 205591 501269 143090 818166 635784 382387 555496 621704 157119 422178 073882 633546 652799 237657 949029 586172 469699 037959 813661 807032 107812 828921 773591 503152 700539 547452 102764 157336 966044 829656 809354 752623 462160 780684 777875 986066 368415 575620 913491 912070 501296 819084 588680 604949 507967 301377 740940 715857 292251 108212 986349 485519 573543 284105 913893 975795 390589 361126 026575 632860 691547 311273 137544 180578 241842 015413 111309 216894 788784 045810 336236 360198 145258 647635 751629 653091 668501 848642 527878 368278 234782 120146 630778 094953 690647 918592 436173 162422 986390 484851 374932 645320 624707 598138 659022 479867 761413 211585 593320 234363 940731 794423 554123 400673 228670 912432 547452 637088 458100 400224 218483 849928 736158 557001 513922 040055 240967 491162 699387 766172 127904 729203 637771 005375 349326 859401 812740 060316 272379 438061 784440 061619 498295 488418 677376 837613 570099 896830 091316 011991 888040 873164 579372 453085 926117 803389 117014 330875 890672 320930 276175 / 481 > 32615 [i]
- extracting embedded OOA [i] would yield OOA(32615, 437, S3, 6, 2404), but