Best Known (45−19, 45, s)-Nets in Base 128
(45−19, 45, 1823)-Net over F128 — Constructive and digital
Digital (26, 45, 1823)-net over F128, using
- 1281 times duplication [i] based on digital (25, 44, 1823)-net over F128, using
- net defined by OOA [i] based on linear OOA(12844, 1823, F128, 19, 19) (dual of [(1823, 19), 34593, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12844, 16408, F128, 19) (dual of [16408, 16364, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,5]) [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(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(1287, 23, F128, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to C([0,9]) ⊂ C([0,5]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(12844, 16408, F128, 19) (dual of [16408, 16364, 20]-code), using
- net defined by OOA [i] based on linear OOA(12844, 1823, F128, 19, 19) (dual of [(1823, 19), 34593, 20]-NRT-code), using
(45−19, 45, 7282)-Net in Base 128 — Constructive
(26, 45, 7282)-net in base 128, using
- 1281 times duplication [i] based on (25, 44, 7282)-net in base 128, using
- net defined by OOA [i] based on OOA(12844, 7282, S128, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12844, 65539, S128, 19), using
- discarding factors based on OA(12844, 65542, S128, 19), using
- discarding parts of the base [i] based on linear OA(25638, 65542, F256, 19) (dual of [65542, 65504, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [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(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 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
- discarding parts of the base [i] based on linear OA(25638, 65542, F256, 19) (dual of [65542, 65504, 20]-code), using
- discarding factors based on OA(12844, 65542, S128, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12844, 65539, S128, 19), using
- net defined by OOA [i] based on OOA(12844, 7282, S128, 19, 19), using
(45−19, 45, 16065)-Net over F128 — Digital
Digital (26, 45, 16065)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12845, 16065, F128, 19) (dual of [16065, 16020, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12845, 16394, F128, 19) (dual of [16394, 16349, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([1,9]) [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(12836, 16385, F128, 10) (dual of [16385, 16349, 11]-code), using the narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [1,9], and minimum distance d ≥ |{−9,−7,−5,…,9}|+1 = 11 (BCH-bound) [i]
- linear OA(1288, 9, F128, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,128)), using
- dual of repetition code with length 9 [i]
- construction X applied to C([0,9]) ⊂ C([1,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12845, 16394, F128, 19) (dual of [16394, 16349, 20]-code), using
(45−19, 45, large)-Net in Base 128 — Upper bound on s
There is no (26, 45, large)-net in base 128, because
- 17 times m-reduction [i] would yield (26, 28, large)-net in base 128, but