Best Known (35−21, 35, s)-Nets in Base 128
(35−21, 35, 342)-Net over F128 — Constructive and digital
Digital (14, 35, 342)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 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, 24, 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, 11, 150)-net over F128, using
(35−21, 35, 360)-Net over F128 — Digital
Digital (14, 35, 360)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12835, 360, F128, 21) (dual of [360, 325, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12835, 381, F128, 21) (dual of [381, 346, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(12835, 381, F128, 21) (dual of [381, 346, 22]-code), using
(35−21, 35, 407)-Net in Base 128 — Constructive
(14, 35, 407)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 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
- (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
- digital (1, 11, 150)-net over F128, using
(35−21, 35, 513)-Net in Base 128
(14, 35, 513)-net in base 128, using
- 13 times m-reduction [i] based on (14, 48, 513)-net in base 128, using
- base change [i] based on digital (8, 42, 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, 42, 513)-net over F256, using
(35−21, 35, 520814)-Net in Base 128 — Upper bound on s
There is no (14, 35, 520815)-net in base 128, because
- 1 times m-reduction [i] would yield (14, 34, 520815)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 441720 134789 443196 977755 359195 705148 480555 696991 045550 720562 639690 511917 > 12834 [i]