Best Known (45, 45+19, s)-Nets in Base 128
(45, 45+19, 233146)-Net over F128 — Constructive and digital
Digital (45, 64, 233146)-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 (36, 55, 233017)-net over F128, using
- net defined by OOA [i] based on linear OOA(12855, 233017, F128, 19, 19) (dual of [(233017, 19), 4427268, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12855, 2097154, F128, 19) (dual of [2097154, 2097099, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12855, 2097155, F128, 19) (dual of [2097155, 2097100, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 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(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(12855, 2097155, F128, 19) (dual of [2097155, 2097100, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12855, 2097154, F128, 19) (dual of [2097154, 2097099, 20]-code), using
- net defined by OOA [i] based on linear OOA(12855, 233017, F128, 19, 19) (dual of [(233017, 19), 4427268, 20]-NRT-code), using
- digital (0, 9, 129)-net over F128, using
(45, 45+19, 932066)-Net in Base 128 — Constructive
(45, 64, 932066)-net in base 128, using
- base change [i] based on digital (37, 56, 932066)-net over F256, using
- 2561 times duplication [i] based on digital (36, 55, 932066)-net over F256, using
- net defined by OOA [i] based on linear OOA(25655, 932066, F256, 19, 19) (dual of [(932066, 19), 17709199, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25655, 8388595, F256, 19) (dual of [8388595, 8388540, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25655, 8388595, F256, 19) (dual of [8388595, 8388540, 20]-code), using
- net defined by OOA [i] based on linear OOA(25655, 932066, F256, 19, 19) (dual of [(932066, 19), 17709199, 20]-NRT-code), using
- 2561 times duplication [i] based on digital (36, 55, 932066)-net over F256, using
(45, 45+19, 2097284)-Net over F128 — Digital
Digital (45, 64, 2097284)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12864, 2097284, F128, 19) (dual of [2097284, 2097220, 20]-code), using
- (u, u+v)-construction [i] based on
- linear OA(1289, 129, F128, 9) (dual of [129, 120, 10]-code or 129-arc in PG(8,128)), using
- extended Reed–Solomon code RSe(120,128) [i]
- the expurgated narrow-sense BCH-code C(I) with length 129 | 1282−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(12855, 2097155, F128, 19) (dual of [2097155, 2097100, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 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(18) ⊂ Ce(17) [i] based on
- linear OA(1289, 129, F128, 9) (dual of [129, 120, 10]-code or 129-arc in PG(8,128)), using
- (u, u+v)-construction [i] based on
(45, 45+19, 3579378)-Net in Base 128
(45, 64, 3579378)-net in base 128, using
- base change [i] based on digital (37, 56, 3579378)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25656, 3579378, F256, 2, 19) (dual of [(3579378, 2), 7158700, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25656, 4194301, F256, 2, 19) (dual of [(4194301, 2), 8388546, 20]-NRT-code), using
- 2561 times duplication [i] based on linear OOA(25655, 4194301, F256, 2, 19) (dual of [(4194301, 2), 8388547, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25655, 8388602, F256, 19) (dual of [8388602, 8388547, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- OOA 2-folding [i] based on linear OA(25655, 8388602, F256, 19) (dual of [8388602, 8388547, 20]-code), using
- 2561 times duplication [i] based on linear OOA(25655, 4194301, F256, 2, 19) (dual of [(4194301, 2), 8388547, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25656, 4194301, F256, 2, 19) (dual of [(4194301, 2), 8388546, 20]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25656, 3579378, F256, 2, 19) (dual of [(3579378, 2), 7158700, 20]-NRT-code), using
(45, 45+19, large)-Net in Base 128 — Upper bound on s
There is no (45, 64, large)-net in base 128, because
- 17 times m-reduction [i] would yield (45, 47, large)-net in base 128, but