Best Known (56, 56+19, s)-Nets in Base 64
(56, 56+19, 932066)-Net over F64 — Constructive and digital
Digital (56, 75, 932066)-net over F64, using
- 642 times duplication [i] based on digital (54, 73, 932066)-net over F64, using
- net defined by OOA [i] based on linear OOA(6473, 932066, F64, 19, 19) (dual of [(932066, 19), 17709181, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6473, 8388595, F64, 19) (dual of [8388595, 8388522, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6473, 8388595, F64, 19) (dual of [8388595, 8388522, 20]-code), using
- net defined by OOA [i] based on linear OOA(6473, 932066, F64, 19, 19) (dual of [(932066, 19), 17709181, 20]-NRT-code), using
(56, 56+19, 8294676)-Net over F64 — Digital
Digital (56, 75, 8294676)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6475, 8294676, F64, 19) (dual of [8294676, 8294601, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(6475, large, F64, 19) (dual of [large, large−75, 20]-code), using
- 2 times code embedding in larger space [i] based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 2 times code embedding in larger space [i] based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(6475, large, F64, 19) (dual of [large, large−75, 20]-code), using
(56, 56+19, large)-Net in Base 64 — Upper bound on s
There is no (56, 75, large)-net in base 64, because
- 17 times m-reduction [i] would yield (56, 58, large)-net in base 64, but