Best Known (56−34, 56, s)-Nets in Base 128
(56−34, 56, 345)-Net over F128 — Constructive and digital
Digital (22, 56, 345)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 17, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- digital (5, 39, 216)-net over F128, using
- net from sequence [i] based on digital (5, 215)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 5 and N(F) ≥ 216, using
- net from sequence [i] based on digital (5, 215)-sequence over F128, using
- digital (0, 17, 129)-net over F128, using
(56−34, 56, 386)-Net in Base 128 — Constructive
(22, 56, 386)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (0, 17, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- (5, 39, 257)-net in base 128, using
- 1 times m-reduction [i] based on (5, 40, 257)-net in base 128, using
- base change [i] based on digital (0, 35, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 35, 257)-net over F256, using
- 1 times m-reduction [i] based on (5, 40, 257)-net in base 128, using
- digital (0, 17, 129)-net over F128, using
(56−34, 56, 408)-Net over F128 — Digital
Digital (22, 56, 408)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12856, 408, F128, 34) (dual of [408, 352, 35]-code), using
- 131 step Varšamov–Edel lengthening with (ri) = (9, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 12 times 0, 1, 21 times 0, 1, 33 times 0, 1, 47 times 0) [i] based on linear OA(12841, 262, F128, 34) (dual of [262, 221, 35]-code), using
- extended algebraic-geometric code AGe(F,227P) [i] based on function field F/F128 with g(F) = 7 and N(F) ≥ 262, using
- 131 step Varšamov–Edel lengthening with (ri) = (9, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 12 times 0, 1, 21 times 0, 1, 33 times 0, 1, 47 times 0) [i] based on linear OA(12841, 262, F128, 34) (dual of [262, 221, 35]-code), using
(56−34, 56, 513)-Net in Base 128
(22, 56, 513)-net in base 128, using
- t-expansion [i] based on (17, 56, 513)-net in base 128, using
- 16 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
- 16 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
(56−34, 56, 493778)-Net in Base 128 — Upper bound on s
There is no (22, 56, 493779)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 10087 204520 985152 237604 647058 161599 165628 794116 350175 417900 922102 608732 853874 216767 288974 416391 332506 998701 235931 596134 > 12856 [i]