Best Known (59, 59+∞, s)-Nets in Base 4
(59, 59+∞, 66)-Net over F4 — Constructive and digital
Digital (59, m, 66)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (59, 65)-sequence over F4, using
- t-expansion [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second 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) = 49 and N(F) ≥ 66, using
- t-expansion [i] based on digital (49, 65)-sequence over F4, using
(59, 59+∞, 91)-Net over F4 — Digital
Digital (59, m, 91)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (59, 90)-sequence over F4, using
- t-expansion [i] based on digital (50, 90)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 50 and N(F) ≥ 91, using
- t-expansion [i] based on digital (50, 90)-sequence over F4, using
(59, 59+∞, 191)-Net in Base 4 — Upper bound on s
There is no (59, m, 192)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (59, 763, 192)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4763, 192, S4, 4, 704), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 602578 106730 258147 145029 151507 078528 228023 311986 422237 839918 759766 314347 897950 800167 271256 026836 512415 862191 570222 001196 965604 244698 739581 242403 246972 693662 863803 334845 406306 192695 519479 420374 919211 385784 346955 014399 336464 476149 777365 346780 524876 991249 453910 335575 169407 368338 820404 357102 948540 991946 467592 592242 277305 397985 763558 002890 689612 520144 896962 725248 428723 070869 161657 795378 765951 218343 795565 126561 748209 474676 492753 131460 831100 308246 714251 190084 829184 / 235 > 4763 [i]
- extracting embedded OOA [i] would yield OOA(4763, 192, S4, 4, 704), but