Best Known (58−36, 58, s)-Nets in Base 128
(58−36, 58, 342)-Net over F128 — Constructive and digital
Digital (22, 58, 342)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 19, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- digital (3, 39, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- digital (1, 19, 150)-net over F128, using
(58−36, 58, 390)-Net over F128 — Digital
Digital (22, 58, 390)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12858, 390, F128, 3, 36) (dual of [(390, 3), 1112, 37]-NRT-code), using
- construction X applied to AG(3;F,1118P) ⊂ AG(3;F,1126P) [i] based on
- linear OOA(12851, 385, F128, 3, 36) (dual of [(385, 3), 1104, 37]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1118P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386, using
- linear OOA(12843, 385, F128, 3, 28) (dual of [(385, 3), 1112, 29]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1126P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386 (see above)
- linear OOA(1287, 5, F128, 3, 7) (dual of [(5, 3), 8, 8]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1287, 128, F128, 3, 7) (dual of [(128, 3), 377, 8]-NRT-code), using
- Reed–Solomon NRT-code RS(3;377,128) [i]
- discarding factors / shortening the dual code based on linear OOA(1287, 128, F128, 3, 7) (dual of [(128, 3), 377, 8]-NRT-code), using
- construction X applied to AG(3;F,1118P) ⊂ AG(3;F,1126P) [i] based on
(58−36, 58, 513)-Net in Base 128
(22, 58, 513)-net in base 128, using
- t-expansion [i] based on (17, 58, 513)-net in base 128, using
- 14 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
- base change [i] based on digital (8, 63, 513)-net over F256, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- K1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F256 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- base change [i] based on digital (8, 63, 513)-net over F256, using
- 14 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
(58−36, 58, 366618)-Net in Base 128 — Upper bound on s
There is no (22, 58, 366619)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 165 264529 515502 001692 465109 789365 906468 970670 337709 459980 082141 603181 155483 813081 422251 615797 312807 609029 820793 119095 906595 > 12858 [i]