Best Known (29, 29+19, s)-Nets in Base 128
(29, 29+19, 1971)-Net over F128 — Constructive and digital
Digital (29, 48, 1971)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 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 (19, 38, 1821)-net over F128, using
- net defined by OOA [i] based on linear OOA(12838, 1821, F128, 19, 19) (dual of [(1821, 19), 34561, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12838, 16390, F128, 19) (dual of [16390, 16352, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(12837, 16385, F128, 19) (dual of [16385, 16348, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,9]) ⊂ C([0,8]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(12838, 16390, F128, 19) (dual of [16390, 16352, 20]-code), using
- net defined by OOA [i] based on linear OOA(12838, 1821, F128, 19, 19) (dual of [(1821, 19), 34561, 20]-NRT-code), using
- digital (1, 10, 150)-net over F128, using
(29, 29+19, 7283)-Net in Base 128 — Constructive
(29, 48, 7283)-net in base 128, using
- base change [i] based on digital (23, 42, 7283)-net over F256, using
- 2562 times duplication [i] based on digital (21, 40, 7283)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 7283, F256, 19, 19) (dual of [(7283, 19), 138337, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(25629, 65537, F256, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,9]) ⊂ C([0,7]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
- net defined by OOA [i] based on linear OOA(25640, 7283, F256, 19, 19) (dual of [(7283, 19), 138337, 20]-NRT-code), using
- 2562 times duplication [i] based on digital (21, 40, 7283)-net over F256, using
(29, 29+19, 24758)-Net over F128 — Digital
Digital (29, 48, 24758)-net over F128, using
(29, 29+19, 27957)-Net in Base 128
(29, 48, 27957)-net in base 128, using
- base change [i] based on digital (23, 42, 27957)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25642, 27957, F256, 2, 19) (dual of [(27957, 2), 55872, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25642, 32777, F256, 2, 19) (dual of [(32777, 2), 65512, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25642, 65554, F256, 19) (dual of [65554, 65512, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 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(2565, 17, F256, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,256)), using
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- Reed–Solomon code RS(251,256) [i]
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- OOA 2-folding [i] based on linear OA(25642, 65554, F256, 19) (dual of [65554, 65512, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(25642, 32777, F256, 2, 19) (dual of [(32777, 2), 65512, 20]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25642, 27957, F256, 2, 19) (dual of [(27957, 2), 55872, 20]-NRT-code), using
(29, 29+19, large)-Net in Base 128 — Upper bound on s
There is no (29, 48, large)-net in base 128, because
- 17 times m-reduction [i] would yield (29, 31, large)-net in base 128, but