Best Known (15, 15+22, s)-Nets in Base 128
(15, 15+22, 342)-Net over F128 — Constructive and digital
Digital (15, 37, 342)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 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, 25, 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, 12, 150)-net over F128, using
(15, 15+22, 386)-Net in Base 128 — Constructive
(15, 37, 386)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 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
- (4, 26, 257)-net in base 128, using
- 6 times m-reduction [i] based on (4, 32, 257)-net in base 128, using
- base change [i] based on digital (0, 28, 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, 28, 257)-net over F256, using
- 6 times m-reduction [i] based on (4, 32, 257)-net in base 128, using
- digital (0, 11, 129)-net over F128, using
(15, 15+22, 388)-Net over F128 — Digital
Digital (15, 37, 388)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12837, 388, F128, 22) (dual of [388, 351, 23]-code), using
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(12836, 383, F128, 22) (dual of [383, 347, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(12836, 382, F128, 22) (dual of [382, 346, 23]-code), using an extension Ce(21) of the narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(12835, 382, F128, 21) (dual of [382, 347, 22]-code), using an extension Ce(20) of the narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1280, 1, F128, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(12836, 383, F128, 22) (dual of [383, 347, 23]-code), using
(15, 15+22, 513)-Net in Base 128
(15, 37, 513)-net in base 128, using
- 19 times m-reduction [i] based on (15, 56, 513)-net in base 128, using
- base change [i] based on digital (8, 49, 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, 49, 513)-net over F256, using
(15, 15+22, 473253)-Net in Base 128 — Upper bound on s
There is no (15, 37, 473254)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 926347 183985 489159 557962 946278 744025 216742 748551 958003 405276 993892 865531 090056 > 12837 [i]