Best Known (58, 58+18, s)-Nets in Base 64
(58, 58+18, 932067)-Net over F64 — Constructive and digital
Digital (58, 76, 932067)-net over F64, using
- t-expansion [i] based on digital (57, 76, 932067)-net over F64, using
- net defined by OOA [i] based on linear OOA(6476, 932067, F64, 21, 19) (dual of [(932067, 21), 19573331, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(6476, 2796202, F64, 3, 19) (dual of [(2796202, 3), 8388530, 20]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(6473, 2796201, F64, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
- OOA 3-folding [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]
- OOA 3-folding [i] based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(6473, 2796201, F64, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(6476, 2796202, F64, 3, 19) (dual of [(2796202, 3), 8388530, 20]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6476, 932067, F64, 21, 19) (dual of [(932067, 21), 19573331, 20]-NRT-code), using
(58, 58+18, large)-Net over F64 — Digital
Digital (58, 76, large)-net over F64, using
- t-expansion [i] based on digital (57, 76, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6476, large, F64, 19) (dual of [large, large−76, 20]-code), using
- 3 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]
- 3 times code embedding in larger space [i] based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6476, large, F64, 19) (dual of [large, large−76, 20]-code), using
(58, 58+18, large)-Net in Base 64 — Upper bound on s
There is no (58, 76, large)-net in base 64, because
- 16 times m-reduction [i] would yield (58, 60, large)-net in base 64, but