Best Known (26, 26+18, s)-Nets in Base 128
(26, 26+18, 1949)-Net over F128 — Constructive and digital
Digital (26, 44, 1949)-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 (17, 35, 1820)-net over F128, using
- net defined by OOA [i] based on linear OOA(12835, 1820, F128, 18, 18) (dual of [(1820, 18), 32725, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(12835, 16380, F128, 18) (dual of [16380, 16345, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(12835, 16380, F128, 18) (dual of [16380, 16345, 19]-code), using
- net defined by OOA [i] based on linear OOA(12835, 1820, F128, 18, 18) (dual of [(1820, 18), 32725, 19]-NRT-code), using
- digital (0, 9, 129)-net over F128, using
(26, 26+18, 7283)-Net in Base 128 — Constructive
(26, 44, 7283)-net in base 128, using
- net defined by OOA [i] based on OOA(12844, 7283, S128, 18, 18), using
- OA 9-folding and stacking [i] based on OA(12844, 65547, S128, 18), using
- discarding parts of the base [i] based on linear OA(25638, 65547, F256, 18) (dual of [65547, 65509, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(25638, 65547, F256, 18) (dual of [65547, 65509, 19]-code), using
- OA 9-folding and stacking [i] based on OA(12844, 65547, S128, 18), using
(26, 26+18, 16515)-Net over F128 — Digital
Digital (26, 44, 16515)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12844, 16515, F128, 18) (dual of [16515, 16471, 19]-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(12835, 16386, F128, 18) (dual of [16386, 16351, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(12833, 16384, F128, 17) (dual of [16384, 16351, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 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(17) ⊂ Ce(16) [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
(26, 26+18, large)-Net in Base 128 — Upper bound on s
There is no (26, 44, large)-net in base 128, because
- 16 times m-reduction [i] would yield (26, 28, large)-net in base 128, but