Best Known (31, 31+8, s)-Nets in Base 64
(31, 31+8, 2228223)-Net over F64 — Constructive and digital
Digital (31, 39, 2228223)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (6, 10, 131073)-net over F64, using
- net defined by OOA [i] based on linear OOA(6410, 131073, F64, 4, 4) (dual of [(131073, 4), 524282, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(6410, 131073, F64, 3, 4) (dual of [(131073, 3), 393209, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(6410, 262146, F64, 4) (dual of [262146, 262136, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(6410, 262147, F64, 4) (dual of [262147, 262137, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(6410, 262144, F64, 4) (dual of [262144, 262134, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(647, 262144, F64, 3) (dual of [262144, 262137, 4]-code or 262144-cap in PG(6,64)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(6410, 262147, F64, 4) (dual of [262147, 262137, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(6410, 262146, F64, 4) (dual of [262146, 262136, 5]-code), using
- appending kth column [i] based on linear OOA(6410, 131073, F64, 3, 4) (dual of [(131073, 3), 393209, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6410, 131073, F64, 4, 4) (dual of [(131073, 4), 524282, 5]-NRT-code), 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 (6, 10, 131073)-net over F64, using
(31, 31+8, large)-Net over F64 — Digital
Digital (31, 39, large)-net over F64, using
- 3 times m-reduction [i] based on digital (31, 42, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6442, large, F64, 11) (dual of [large, large−42, 12]-code), using
- 1 times code embedding in larger space [i] 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]
- 1 times code embedding in larger space [i] based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6442, large, F64, 11) (dual of [large, large−42, 12]-code), using
(31, 31+8, large)-Net in Base 64 — Upper bound on s
There is no (31, 39, large)-net in base 64, because
- 6 times m-reduction [i] would yield (31, 33, large)-net in base 64, but