Best Known (50−30, 50, s)-Nets in Base 128
(50−30, 50, 345)-Net over F128 — Constructive and digital
Digital (20, 50, 345)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 15, 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, 35, 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, 15, 129)-net over F128, using
(50−30, 50, 386)-Net in Base 128 — Constructive
(20, 50, 386)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (0, 15, 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, 35, 257)-net in base 128, using
- 5 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
- 5 times m-reduction [i] based on (5, 40, 257)-net in base 128, using
- digital (0, 15, 129)-net over F128, using
(50−30, 50, 422)-Net over F128 — Digital
Digital (20, 50, 422)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12850, 422, F128, 30) (dual of [422, 372, 31]-code), using
- 31 step Varšamov–Edel lengthening with (ri) = (4, 5 times 0, 1, 24 times 0) [i] based on linear OA(12845, 386, F128, 30) (dual of [386, 341, 31]-code), using
- extended algebraic-geometric code AGe(F,355P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386, using
- 31 step Varšamov–Edel lengthening with (ri) = (4, 5 times 0, 1, 24 times 0) [i] based on linear OA(12845, 386, F128, 30) (dual of [386, 341, 31]-code), using
(50−30, 50, 513)-Net in Base 128
(20, 50, 513)-net in base 128, using
- t-expansion [i] based on (17, 50, 513)-net in base 128, using
- 22 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
- 22 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
(50−30, 50, 534552)-Net in Base 128 — Upper bound on s
There is no (20, 50, 534553)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 2293 530483 479129 519112 122089 547930 691865 344812 857181 227042 071189 637202 456557 004951 237751 374404 083335 518352 > 12850 [i]