Best Known (27, 27+15, s)-Nets in Base 128
(27, 27+15, 4681)-Net over F128 — Constructive and digital
Digital (27, 42, 4681)-net over F128, using
- net defined by OOA [i] based on linear OOA(12842, 4681, F128, 15, 15) (dual of [(4681, 15), 70173, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12842, 32768, F128, 15) (dual of [32768, 32726, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12842, 32770, F128, 15) (dual of [32770, 32728, 16]-code), using
- (u, u+v)-construction [i] based on
- linear OA(12813, 16385, F128, 7) (dual of [16385, 16372, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12813, 16385, F128, 7) (dual of [16385, 16372, 8]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(12842, 32770, F128, 15) (dual of [32770, 32728, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12842, 32768, F128, 15) (dual of [32768, 32726, 16]-code), using
(27, 27+15, 9619)-Net in Base 128 — Constructive
(27, 42, 9619)-net in base 128, using
- (u, u+v)-construction [i] based on
- (1, 8, 257)-net in base 128, using
- base change [i] based on digital (0, 7, 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, 7, 257)-net over F256, using
- (19, 34, 9362)-net in base 128, using
- net defined by OOA [i] based on OOA(12834, 9362, S128, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12834, 65535, S128, 15), using
- discarding factors based on OA(12834, 65538, S128, 15), using
- discarding parts of the base [i] based on linear OA(25629, 65538, F256, 15) (dual of [65538, 65509, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(25629, 65536, F256, 15) (dual of [65536, 65507, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(25629, 65538, F256, 15) (dual of [65538, 65509, 16]-code), using
- discarding factors based on OA(12834, 65538, S128, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12834, 65535, S128, 15), using
- net defined by OOA [i] based on OOA(12834, 9362, S128, 15, 15), using
- (1, 8, 257)-net in base 128, using
(27, 27+15, 99842)-Net over F128 — Digital
Digital (27, 42, 99842)-net over F128, using
(27, 27+15, large)-Net in Base 128 — Upper bound on s
There is no (27, 42, large)-net in base 128, because
- 13 times m-reduction [i] would yield (27, 29, large)-net in base 128, but