Best Known (32−11, 32, s)-Nets in Base 128
(32−11, 32, 419431)-Net over F128 — Constructive and digital
Digital (21, 32, 419431)-net over F128, using
- net defined by OOA [i] based on linear OOA(12832, 419431, F128, 11, 11) (dual of [(419431, 11), 4613709, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(12832, 2097156, F128, 11) (dual of [2097156, 2097124, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(12832, 2097160, F128, 11) (dual of [2097160, 2097128, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 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(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12832, 2097160, F128, 11) (dual of [2097160, 2097128, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(12832, 2097156, F128, 11) (dual of [2097156, 2097124, 12]-code), using
(32−11, 32, 1048580)-Net over F128 — Digital
Digital (21, 32, 1048580)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12832, 1048580, F128, 2, 11) (dual of [(1048580, 2), 2097128, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12832, 2097160, F128, 11) (dual of [2097160, 2097128, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 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(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,5]) ⊂ C([0,4]) [i] based on
- OOA 2-folding [i] based on linear OA(12832, 2097160, F128, 11) (dual of [2097160, 2097128, 12]-code), using
(32−11, 32, large)-Net in Base 128 — Upper bound on s
There is no (21, 32, large)-net in base 128, because
- 9 times m-reduction [i] would yield (21, 23, large)-net in base 128, but