Best Known (19, 19+9, s)-Nets in Base 128
(19, 19+9, 524291)-Net over F128 — Constructive and digital
Digital (19, 28, 524291)-net over F128, using
- net defined by OOA [i] based on linear OOA(12828, 524291, F128, 9, 9) (dual of [(524291, 9), 4718591, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12828, 2097165, F128, 9) (dual of [2097165, 2097137, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(12828, 2097168, F128, 9) (dual of [2097168, 2097140, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [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(12813, 2097153, F128, 5) (dual of [2097153, 2097140, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-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 C([0,4]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12828, 2097168, F128, 9) (dual of [2097168, 2097140, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12828, 2097165, F128, 9) (dual of [2097165, 2097137, 10]-code), using
(19, 19+9, 2097168)-Net over F128 — Digital
Digital (19, 28, 2097168)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12828, 2097168, F128, 9) (dual of [2097168, 2097140, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [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(12813, 2097153, F128, 5) (dual of [2097153, 2097140, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-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 C([0,4]) ⊂ C([0,2]) [i] based on
(19, 19+9, large)-Net in Base 128 — Upper bound on s
There is no (19, 28, large)-net in base 128, because
- 7 times m-reduction [i] would yield (19, 21, large)-net in base 128, but