Best Known (53, ∞, s)-Nets in Base 4
(53, ∞, 66)-Net over F4 — Constructive and digital
Digital (53, m, 66)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (53, 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
(53, ∞, 91)-Net over F4 — Digital
Digital (53, m, 91)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (53, 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
(53, ∞, 173)-Net in Base 4 — Upper bound on s
There is no (53, m, 174)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (53, 691, 174)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4691, 174, S4, 4, 638), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 9288 309531 565531 055035 747194 080313 886357 243394 884141 026377 950641 329674 237294 720699 224931 011169 188670 578716 212580 138923 112388 336154 259176 002714 322875 853864 937534 540168 870085 192377 501482 456621 190278 259334 886398 844258 928126 296021 052323 787642 518303 073805 666285 047041 187506 967047 516962 726360 280637 157314 088991 125416 953233 900193 744130 302058 950456 585745 306540 077087 926219 017930 333982 227198 998203 964159 324405 624829 149500 669952 / 71 > 4691 [i]
- extracting embedded OOA [i] would yield OOA(4691, 174, S4, 4, 638), but