Best Known (40, 40+∞, s)-Nets in Base 4
(40, 40+∞, 56)-Net over F4 — Constructive and digital
Digital (40, m, 56)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (40, 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
(40, 40+∞, 75)-Net over F4 — Digital
Digital (40, m, 75)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 40 and N(F) ≥ 75, using
(40, 40+∞, 133)-Net in Base 4 — Upper bound on s
There is no (40, m, 134)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (40, 531, 134)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4531, 134, S4, 4, 491), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 2075 413730 201915 520216 016738 817023 167988 617990 745597 893308 070565 710955 630335 889787 466390 653188 564264 843226 872632 154809 638274 405998 584796 393655 899120 486088 236499 934526 273385 541334 865753 214823 019124 040586 546839 671113 184647 046835 227866 689536 361636 889982 010259 368534 928789 354981 302206 953395 105426 209805 452212 660870 050079 571968 / 41 > 4531 [i]
- extracting embedded OOA [i] would yield OOA(4531, 134, S4, 4, 491), but