Best Known (256−182, 256, s)-Nets in Base 4
(256−182, 256, 104)-Net over F4 — Constructive and digital
Digital (74, 256, 104)-net over F4, using
- t-expansion [i] based on digital (73, 256, 104)-net over F4, using
- net from sequence [i] based on digital (73, 103)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 73 and N(F) ≥ 104, using
- F6 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) = 73 and N(F) ≥ 104, using
- net from sequence [i] based on digital (73, 103)-sequence over F4, using
(256−182, 256, 112)-Net over F4 — Digital
Digital (74, 256, 112)-net over F4, using
- t-expansion [i] based on digital (73, 256, 112)-net over F4, using
- net from sequence [i] based on digital (73, 111)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 73 and N(F) ≥ 112, using
- net from sequence [i] based on digital (73, 111)-sequence over F4, using
(256−182, 256, 466)-Net over F4 — Upper bound on s (digital)
There is no digital (74, 256, 467)-net over F4, because
- 2 times m-reduction [i] would yield digital (74, 254, 467)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4254, 467, F4, 180) (dual of [467, 213, 181]-code), but
- residual code [i] would yield OA(474, 286, S4, 45), but
- the linear programming bound shows that M ≥ 22 769120 707181 828041 954378 161822 084052 954335 244294 016427 586097 147930 001332 191588 780049 100806 553600 / 63648 956571 718794 371382 905612 240945 674126 728108 353777 > 474 [i]
- residual code [i] would yield OA(474, 286, S4, 45), but
- extracting embedded orthogonal array [i] would yield linear OA(4254, 467, F4, 180) (dual of [467, 213, 181]-code), but
(256−182, 256, 499)-Net in Base 4 — Upper bound on s
There is no (74, 256, 500)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 15692 362447 398367 800924 362399 493906 185389 785815 178538 728571 185092 475961 357233 380612 726482 825869 175380 108317 672981 825933 832988 594880 152800 806781 909391 588756 > 4256 [i]