Best Known (38−13, 38, s)-Nets in Base 128
(38−13, 38, 349526)-Net over F128 — Constructive and digital
Digital (25, 38, 349526)-net over F128, using
- net defined by OOA [i] based on linear OOA(12838, 349526, F128, 13, 13) (dual of [(349526, 13), 4543800, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12838, 2097157, F128, 13) (dual of [2097157, 2097119, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12838, 2097160, F128, 13) (dual of [2097160, 2097122, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 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(12838, 2097160, F128, 13) (dual of [2097160, 2097122, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12838, 2097157, F128, 13) (dual of [2097157, 2097119, 14]-code), using
(38−13, 38, 1048580)-Net over F128 — Digital
Digital (25, 38, 1048580)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12838, 1048580, F128, 2, 13) (dual of [(1048580, 2), 2097122, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12838, 2097160, F128, 13) (dual of [2097160, 2097122, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 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
- OOA 2-folding [i] based on linear OA(12838, 2097160, F128, 13) (dual of [2097160, 2097122, 14]-code), using
(38−13, 38, large)-Net in Base 128 — Upper bound on s
There is no (25, 38, large)-net in base 128, because
- 11 times m-reduction [i] would yield (25, 27, large)-net in base 128, but