Best Known (199, 199+∞, s)-Nets in Base 3
(199, 199+∞, 112)-Net over F3 — Constructive and digital
Digital (199, m, 112)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (199, 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
(199, 199+∞, 163)-Net over F3 — Digital
Digital (199, m, 163)-net over F3 for arbitrarily large m, using
- net from sequence [i] based on digital (199, 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
(199, 199+∞, 412)-Net in Base 3 — Upper bound on s
There is no (199, m, 413)-net in base 3 for arbitrarily large m, because
- m-reduction [i] would yield (199, 2471, 413)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(32471, 413, S3, 6, 2272), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 2 464139 479267 488887 882560 110282 715607 404231 870997 969514 477227 387640 277558 039386 997546 497836 905989 856849 707585 047950 389142 768078 946986 757138 492845 900558 979709 045212 231226 309541 521027 013279 705044 258253 829528 506646 863297 420424 223317 874653 835744 860346 791478 130680 558357 473881 010352 783023 463896 576855 781289 339678 878165 639381 869280 556088 298031 191316 854927 357301 857285 466746 920112 830457 461192 794188 576559 219250 667280 213583 971145 690692 804407 976733 212681 738346 633219 545012 719660 389097 409537 021240 722844 909955 733764 528487 955467 909743 963951 055817 340110 314576 368262 284089 745914 208622 997381 079199 119917 279172 713054 188745 631936 127152 356200 823345 829155 382233 424330 120090 605561 108545 310063 181170 201121 209219 257234 208588 759237 408431 453916 414103 594919 466063 074872 895182 704444 621576 409203 496057 783533 932404 936718 919487 072064 782164 708530 066575 666388 013970 088826 304640 058897 750517 649036 375613 340344 241186 941073 428073 152726 190481 706625 267246 270979 829034 398632 597052 053345 108392 506557 148894 651037 219452 254477 351264 338606 634351 325273 657179 324251 099831 900701 636469 518767 990738 867539 533770 410743 505831 961795 239394 178151 779209 233282 332572 744146 754551 658294 533515 227584 898222 663251 882621 244345 684415 297506 708567 / 2273 > 32471 [i]
- extracting embedded OOA [i] would yield OOA(32471, 413, S3, 6, 2272), but