Best Known (54−13, 54, s)-Nets in Base 64
(54−13, 54, 1398100)-Net over F64 — Constructive and digital
Digital (41, 54, 1398100)-net over F64, using
- 645 times duplication [i] based on digital (36, 49, 1398100)-net over F64, using
- net defined by OOA [i] based on linear OOA(6449, 1398100, F64, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6449, 8388601, F64, 13) (dual of [8388601, 8388552, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6449, 8388601, F64, 13) (dual of [8388601, 8388552, 14]-code), using
- net defined by OOA [i] based on linear OOA(6449, 1398100, F64, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
(54−13, 54, large)-Net over F64 — Digital
Digital (41, 54, large)-net over F64, using
- 1 times m-reduction [i] based on digital (41, 55, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6455, large, F64, 14) (dual of [large, large−55, 15]-code), using
- 2 times code embedding in larger space [i] based on linear OA(6453, large, F64, 14) (dual of [large, large−53, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 2 times code embedding in larger space [i] based on linear OA(6453, large, F64, 14) (dual of [large, large−53, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6455, large, F64, 14) (dual of [large, large−55, 15]-code), using
(54−13, 54, large)-Net in Base 64 — Upper bound on s
There is no (41, 54, large)-net in base 64, because
- 11 times m-reduction [i] would yield (41, 43, large)-net in base 64, but