Best Known (52, 63, s)-Nets in Base 64
(52, 63, 3363705)-Net over F64 — Constructive and digital
Digital (52, 63, 3363705)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 5, 8265)-net over F64, using
- net defined by OOA [i] based on linear OOA(645, 8265, F64, 3, 3) (dual of [(8265, 3), 24790, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(645, 8265, F64, 2, 3) (dual of [(8265, 2), 16525, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(645, 8265, F64, 3, 3) (dual of [(8265, 3), 24790, 4]-NRT-code), using
- digital (12, 17, 1677720)-net over F64, using
- s-reduction based on digital (12, 17, 4194301)-net over F64, using
- net defined by OOA [i] based on linear OOA(6417, 4194301, F64, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(6417, large, F64, 5) (dual of [large, large−17, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(6417, large, F64, 5) (dual of [large, large−17, 6]-code), using
- net defined by OOA [i] based on linear OOA(6417, 4194301, F64, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- s-reduction based on digital (12, 17, 4194301)-net over F64, using
- digital (30, 41, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- digital (2, 5, 8265)-net over F64, using
(52, 63, large)-Net over F64 — Digital
Digital (52, 63, large)-net over F64, using
- t-expansion [i] based on digital (50, 63, large)-net over F64, using
- 4 times m-reduction [i] based on digital (50, 67, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6467, large, F64, 17) (dual of [large, large−67, 18]-code), using
- 2 times code embedding in larger space [i] based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 2 times code embedding in larger space [i] based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6467, large, F64, 17) (dual of [large, large−67, 18]-code), using
- 4 times m-reduction [i] based on digital (50, 67, large)-net over F64, using
(52, 63, large)-Net in Base 64 — Upper bound on s
There is no (52, 63, large)-net in base 64, because
- 9 times m-reduction [i] would yield (52, 54, large)-net in base 64, but