Best Known (26−9, 26, s)-Nets in Base 128
(26−9, 26, 524289)-Net over F128 — Constructive and digital
Digital (17, 26, 524289)-net over F128, using
- net defined by OOA [i] based on linear OOA(12826, 524289, F128, 9, 9) (dual of [(524289, 9), 4718575, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12826, 2097157, F128, 9) (dual of [2097157, 2097131, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(12826, 2097160, F128, 9) (dual of [2097160, 2097134, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- 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(12819, 2097153, F128, 7) (dual of [2097153, 2097134, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12826, 2097160, F128, 9) (dual of [2097160, 2097134, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12826, 2097157, F128, 9) (dual of [2097157, 2097131, 10]-code), using
(26−9, 26, 1048580)-Net over F128 — Digital
Digital (17, 26, 1048580)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12826, 1048580, F128, 2, 9) (dual of [(1048580, 2), 2097134, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12826, 2097160, F128, 9) (dual of [2097160, 2097134, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- 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(12819, 2097153, F128, 7) (dual of [2097153, 2097134, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,4]) ⊂ C([0,3]) [i] based on
- OOA 2-folding [i] based on linear OA(12826, 2097160, F128, 9) (dual of [2097160, 2097134, 10]-code), using
(26−9, 26, large)-Net in Base 128 — Upper bound on s
There is no (17, 26, large)-net in base 128, because
- 7 times m-reduction [i] would yield (17, 19, large)-net in base 128, but