Best Known (20, 20+18, s)-Nets in Base 128
(20, 20+18, 1821)-Net over F128 — Constructive and digital
Digital (20, 38, 1821)-net over F128, using
- t-expansion [i] based on digital (19, 38, 1821)-net over F128, using
- net defined by OOA [i] based on linear OOA(12838, 1821, F128, 19, 19) (dual of [(1821, 19), 34561, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12838, 16390, F128, 19) (dual of [16390, 16352, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [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(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,9]) ⊂ C([0,8]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(12838, 16390, F128, 19) (dual of [16390, 16352, 20]-code), using
- net defined by OOA [i] based on linear OOA(12838, 1821, F128, 19, 19) (dual of [(1821, 19), 34561, 20]-NRT-code), using
(20, 20+18, 5765)-Net over F128 — Digital
Digital (20, 38, 5765)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12838, 5765, F128, 2, 18) (dual of [(5765, 2), 11492, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12838, 8197, F128, 2, 18) (dual of [(8197, 2), 16356, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12838, 16394, F128, 18) (dual of [16394, 16356, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12838, 16395, F128, 18) (dual of [16395, 16357, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(12827, 16384, F128, 14) (dual of [16384, 16357, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(12838, 16395, F128, 18) (dual of [16395, 16357, 19]-code), using
- OOA 2-folding [i] based on linear OA(12838, 16394, F128, 18) (dual of [16394, 16356, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(12838, 8197, F128, 2, 18) (dual of [(8197, 2), 16356, 19]-NRT-code), using
(20, 20+18, large)-Net in Base 128 — Upper bound on s
There is no (20, 38, large)-net in base 128, because
- 16 times m-reduction [i] would yield (20, 22, large)-net in base 128, but