Best Known (9, 18, s)-Nets in Base 128
(9, 18, 4097)-Net over F128 — Constructive and digital
Digital (9, 18, 4097)-net over F128, using
- net defined by OOA [i] based on linear OOA(12818, 4097, F128, 9, 9) (dual of [(4097, 9), 36855, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12818, 16389, F128, 9) (dual of [16389, 16371, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(12818, 16390, F128, 9) (dual of [16390, 16372, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(12817, 16385, F128, 9) (dual of [16385, 16368, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(12813, 16385, F128, 7) (dual of [16385, 16372, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12818, 16390, F128, 9) (dual of [16390, 16372, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12818, 16389, F128, 9) (dual of [16389, 16371, 10]-code), using
(9, 18, 8195)-Net over F128 — Digital
Digital (9, 18, 8195)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12818, 8195, F128, 2, 9) (dual of [(8195, 2), 16372, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12818, 16390, F128, 9) (dual of [16390, 16372, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(12817, 16385, F128, 9) (dual of [16385, 16368, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(12813, 16385, F128, 7) (dual of [16385, 16372, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,4]) ⊂ C([0,3]) [i] based on
- OOA 2-folding [i] based on linear OA(12818, 16390, F128, 9) (dual of [16390, 16372, 10]-code), using
(9, 18, large)-Net in Base 128 — Upper bound on s
There is no (9, 18, large)-net in base 128, because
- 7 times m-reduction [i] would yield (9, 11, large)-net in base 128, but