MinT - Best Known (159, 159+∞, s)-Nets in Base 3

Best Known (159, 159+∞, s)-Nets in Base 3

Digital (159, m, 82)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (159, 81)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ III based on function field F₄/F₃ with g(F₄)Â =Â 79, N(F₄)Â â‰¥Â 2, and N(F₄)Â +Â N₂(F₄)Â â‰¥Â 82 from GarcÃaâ€“Stichtenoth tower as constant field extension [i]

Digital (159, m, 163)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (159, 162)-sequence over F₃, using
- t-expansion [i] based on digital (151, 162)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 151 and N(F) ≥ 163, using
    - Drinfeld modules of rank 1 [i]

There is no (159, m, 333)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (159, 1658, 333)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁶⁵⁸, 333, S₃, 5, 1499), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 34 657463 499225 153043 523162 617638 176055 709495 533436 634855 624432 242872 209084 681127 263661 819695 161461 939438 216224 448793 958719 640436 615461 620290 390633 150203 169456 190768 399156 334421 792437 323518 213402 682055 878012 982750 250681 794275 485407 715561 396669 226103 005583 335472 361938 880101 097405 863979 721952 095677 848809 421731 614125 502169 001350 230239 663225 307521 600949 495341 138519 873470 801775 380405 807473 164198 788467 512232 212361 156701 792077 686295 338547 616559 129208 396451 030262 942841 236873 420109 954138 638950 276361 992221 367264 354094 364080 544247 091403 942091 735282 479117 814288 139212 448255 022000 261691 681972 731796 164677 056715 861326 490899 611095 425885 121706 830767 060889 894282 557459 916963 716850 456423 505144 150972 672696 692143 184864 772746 969622 976425 434740 571373 420724 194116 958424 689136 802572 387872 766045 285788 657179 111633 / 250 > 3¹⁶⁵⁸ [i]