Best Known (26−12, 26, s)-Nets in Base 64
(26−12, 26, 684)-Net over F64 — Constructive and digital
Digital (14, 26, 684)-net over F64, using
- 641 times duplication [i] based on digital (13, 25, 684)-net over F64, using
- net defined by OOA [i] based on linear OOA(6425, 684, F64, 12, 12) (dual of [(684, 12), 8183, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6425, 4104, F64, 12) (dual of [4104, 4079, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(6417, 4096, F64, 9) (dual of [4096, 4079, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- OA 6-folding and stacking [i] based on linear OA(6425, 4104, F64, 12) (dual of [4104, 4079, 13]-code), using
- net defined by OOA [i] based on linear OOA(6425, 684, F64, 12, 12) (dual of [(684, 12), 8183, 13]-NRT-code), using
(26−12, 26, 2351)-Net over F64 — Digital
Digital (14, 26, 2351)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6426, 2351, F64, 12) (dual of [2351, 2325, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6426, 4107, F64, 12) (dual of [4107, 4081, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(6426, 4107, F64, 12) (dual of [4107, 4081, 13]-code), using
(26−12, 26, 3189048)-Net in Base 64 — Upper bound on s
There is no (14, 26, 3189049)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 91344 010259 571709 742783 200211 597617 312347 311188 > 6426 [i]