Best Known (47, ∞, s)-Nets in Base 4
(47, ∞, 56)-Net over F4 — Constructive and digital
Digital (47, m, 56)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (47, 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
(47, ∞, 81)-Net over F4 — Digital
Digital (47, m, 81)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (47, 80)-sequence over F4, using
- t-expansion [i] based on digital (46, 80)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 46 and N(F) ≥ 81, using
- t-expansion [i] based on digital (46, 80)-sequence over F4, using
(47, ∞, 155)-Net in Base 4 — Upper bound on s
There is no (47, m, 156)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (47, 619, 156)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4619, 156, S4, 4, 572), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1287 370449 494445 233669 314515 756482 827102 150318 413143 262226 548613 084460 790665 231798 384106 299704 941274 411402 755047 476800 482061 812553 885821 423597 942428 725323 792384 807905 616011 309598 573403 085515 895203 864130 230145 578320 653239 727168 141258 432024 358884 158027 354350 859607 000413 853915 829862 528664 520692 998064 228010 708758 373056 868650 437200 801412 898438 040363 012003 839795 778273 766715 424768 / 191 > 4619 [i]
- extracting embedded OOA [i] would yield OOA(4619, 156, S4, 4, 572), but