Best Known (38, 38+∞, s)-Nets in Base 4
(38, 38+∞, 56)-Net over F4 — Constructive and digital
Digital (38, m, 56)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (38, 55)-sequence over F4, using
- t-expansion [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- t-expansion [i] based on digital (33, 55)-sequence over F4, using
(38, 38+∞, 66)-Net over F4 — Digital
Digital (38, m, 66)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (38, 65)-sequence over F4, using
- t-expansion [i] based on digital (37, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 37 and N(F) ≥ 66, using
- t-expansion [i] based on digital (37, 65)-sequence over F4, using
(38, 38+∞, 127)-Net in Base 4 — Upper bound on s
There is no (38, m, 128)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (38, 507, 128)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4507, 128, S4, 4, 469), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 44 942328 371557 897693 232629 769725 618340 449424 473557 664318 357520 289433 168951 375240 783177 119330 601884 005280 028469 967848 339414 697442 203604 155623 211857 659868 531094 441973 356216 371319 075554 900311 523529 863270 738021 251442 209537 670585 615720 368478 277635 206809 290837 627671 146574 559986 811484 619929 076208 839082 406056 034304 / 235 > 4507 [i]
- extracting embedded OOA [i] would yield OOA(4507, 128, S4, 4, 469), but