Best Known (39, 47, s)-Nets in Base 64
(39, 47, 6291450)-Net over F64 — Constructive and digital
Digital (39, 47, 6291450)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (3, 5, 2097150)-net over F64, using
- s-reduction based on digital (3, 5, large)-net over F64, using
- digital (9, 13, 2097150)-net over F64, using
- s-reduction based on digital (9, 13, 4194301)-net over F64, using
- net defined by OOA [i] based on linear OOA(6413, 4194301, F64, 4, 4) (dual of [(4194301, 4), 16777191, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(6413, 4194301, F64, 3, 4) (dual of [(4194301, 3), 12582890, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(6413, 8388602, F64, 4) (dual of [8388602, 8388589, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(6413, large, F64, 4) (dual of [large, large−13, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(6413, large, F64, 4) (dual of [large, large−13, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(6413, 8388602, F64, 4) (dual of [8388602, 8388589, 5]-code), using
- appending kth column [i] based on linear OOA(6413, 4194301, F64, 3, 4) (dual of [(4194301, 3), 12582890, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6413, 4194301, F64, 4, 4) (dual of [(4194301, 4), 16777191, 5]-NRT-code), using
- s-reduction based on digital (9, 13, 4194301)-net over F64, using
- digital (21, 29, 2097150)-net over F64, using
- net defined by OOA [i] based on linear OOA(6429, 2097150, F64, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(6429, 8388600, F64, 8) (dual of [8388600, 8388571, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(6429, 8388600, F64, 8) (dual of [8388600, 8388571, 9]-code), using
- net defined by OOA [i] based on linear OOA(6429, 2097150, F64, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
- digital (3, 5, 2097150)-net over F64, using
(39, 47, large)-Net over F64 — Digital
Digital (39, 47, large)-net over F64, using
- t-expansion [i] based on digital (37, 47, large)-net over F64, using
- 3 times m-reduction [i] based on digital (37, 50, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6450, large, F64, 13) (dual of [large, large−50, 14]-code), using
- 1 times code embedding in larger space [i] 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]
- 1 times code embedding in larger space [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6450, large, F64, 13) (dual of [large, large−50, 14]-code), using
- 3 times m-reduction [i] based on digital (37, 50, large)-net over F64, using
(39, 47, large)-Net in Base 64 — Upper bound on s
There is no (39, 47, large)-net in base 64, because
- 6 times m-reduction [i] would yield (39, 41, large)-net in base 64, but