Best Known (41, 41+19, s)-Nets in Base 128
(41, 41+19, 233019)-Net over F128 — Constructive and digital
Digital (41, 60, 233019)-net over F128, using
- net defined by OOA [i] based on linear OOA(12860, 233019, F128, 19, 19) (dual of [(233019, 19), 4427301, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12860, 2097172, F128, 19) (dual of [2097172, 2097112, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12860, 2097176, F128, 19) (dual of [2097176, 2097116, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(12855, 2097153, F128, 19) (dual of [2097153, 2097098, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1285, 23, F128, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12860, 2097176, F128, 19) (dual of [2097176, 2097116, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12860, 2097172, F128, 19) (dual of [2097172, 2097112, 20]-code), using
(41, 41+19, 1162510)-Net over F128 — Digital
Digital (41, 60, 1162510)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12860, 1162510, F128, 19) (dual of [1162510, 1162450, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12860, 2097176, F128, 19) (dual of [2097176, 2097116, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(12855, 2097153, F128, 19) (dual of [2097153, 2097098, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1285, 23, F128, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12860, 2097176, F128, 19) (dual of [2097176, 2097116, 20]-code), using
(41, 41+19, large)-Net in Base 128 — Upper bound on s
There is no (41, 60, large)-net in base 128, because
- 17 times m-reduction [i] would yield (41, 43, large)-net in base 128, but