Best Known (12, 30, s)-Nets in Base 128
(12, 30, 321)-Net over F128 — Constructive and digital
Digital (12, 30, 321)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 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 (3, 21, 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 (0, 9, 129)-net over F128, using
(12, 30, 347)-Net over F128 — Digital
Digital (12, 30, 347)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12830, 347, F128, 18) (dual of [347, 317, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12830, 381, F128, 18) (dual of [381, 351, 19]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(12830, 381, F128, 18) (dual of [381, 351, 19]-code), using
(12, 30, 386)-Net in Base 128 — Constructive
(12, 30, 386)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 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
- (3, 21, 257)-net in base 128, using
- 3 times m-reduction [i] based on (3, 24, 257)-net in base 128, using
- base change [i] based on digital (0, 21, 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, 21, 257)-net over F256, using
- 3 times m-reduction [i] based on (3, 24, 257)-net in base 128, using
- digital (0, 9, 129)-net over F128, using
(12, 30, 513)-Net in Base 128
(12, 30, 513)-net in base 128, using
- 2 times m-reduction [i] based on (12, 32, 513)-net in base 128, using
- base change [i] based on digital (8, 28, 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, 28, 513)-net over F256, using
(12, 30, 345124)-Net in Base 128 — Upper bound on s
There is no (12, 30, 345125)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 1645 524436 279789 081042 426942 489338 739321 248596 057512 692931 156176 > 12830 [i]