Best Known (33−13, 33, s)-Nets in Base 128
(33−13, 33, 2881)-Net over F128 — Constructive and digital
Digital (20, 33, 2881)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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 (13, 26, 2731)-net over F128, using
- net defined by OOA [i] based on linear OOA(12826, 2731, F128, 13, 13) (dual of [(2731, 13), 35477, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12826, 16387, F128, 13) (dual of [16387, 16361, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12826, 16390, F128, 13) (dual of [16390, 16364, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(12821, 16385, F128, 11) (dual of [16385, 16364, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12826, 16390, F128, 13) (dual of [16390, 16364, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12826, 16387, F128, 13) (dual of [16387, 16361, 14]-code), using
- net defined by OOA [i] based on linear OOA(12826, 2731, F128, 13, 13) (dual of [(2731, 13), 35477, 14]-NRT-code), using
- digital (1, 7, 150)-net over F128, using
(33−13, 33, 10924)-Net in Base 128 — Constructive
(20, 33, 10924)-net in base 128, using
- 1281 times duplication [i] based on (19, 32, 10924)-net in base 128, using
- base change [i] based on digital (15, 28, 10924)-net over F256, using
- net defined by OOA [i] based on linear OOA(25628, 10924, F256, 13, 13) (dual of [(10924, 13), 141984, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(25628, 65545, F256, 13) (dual of [65545, 65517, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(25628, 65548, F256, 13) (dual of [65548, 65520, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(25617, 65537, F256, 9) (dual of [65537, 65520, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [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 C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25628, 65548, F256, 13) (dual of [65548, 65520, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(25628, 65545, F256, 13) (dual of [65545, 65517, 14]-code), using
- net defined by OOA [i] based on linear OOA(25628, 10924, F256, 13, 13) (dual of [(10924, 13), 141984, 14]-NRT-code), using
- base change [i] based on digital (15, 28, 10924)-net over F256, using
(33−13, 33, 25971)-Net over F128 — Digital
Digital (20, 33, 25971)-net over F128, using
(33−13, 33, 32419)-Net in Base 128
(20, 33, 32419)-net in base 128, using
- 1281 times duplication [i] based on (19, 32, 32419)-net in base 128, using
- base change [i] based on digital (15, 28, 32419)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25628, 32419, F256, 2, 13) (dual of [(32419, 2), 64810, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25628, 32774, F256, 2, 13) (dual of [(32774, 2), 65520, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25628, 65548, F256, 13) (dual of [65548, 65520, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(25617, 65537, F256, 9) (dual of [65537, 65520, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [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 C([0,6]) ⊂ C([0,4]) [i] based on
- OOA 2-folding [i] based on linear OA(25628, 65548, F256, 13) (dual of [65548, 65520, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(25628, 32774, F256, 2, 13) (dual of [(32774, 2), 65520, 14]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25628, 32419, F256, 2, 13) (dual of [(32419, 2), 64810, 14]-NRT-code), using
- base change [i] based on digital (15, 28, 32419)-net over F256, using
(33−13, 33, large)-Net in Base 128 — Upper bound on s
There is no (20, 33, large)-net in base 128, because
- 11 times m-reduction [i] would yield (20, 22, large)-net in base 128, but