Best Known (13, 27, s)-Nets in Base 64
(13, 27, 585)-Net over F64 — Constructive and digital
Digital (13, 27, 585)-net over F64, using
- net defined by OOA [i] based on linear OOA(6427, 585, F64, 14, 14) (dual of [(585, 14), 8163, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6427, 4095, F64, 14) (dual of [4095, 4068, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(6427, 4095, F64, 14) (dual of [4095, 4068, 15]-code), using
(13, 27, 1366)-Net over F64 — Digital
Digital (13, 27, 1366)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6427, 1366, F64, 3, 14) (dual of [(1366, 3), 4071, 15]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6427, 4098, F64, 14) (dual of [4098, 4071, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6425, 4096, F64, 13) (dual of [4096, 4071, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 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(13) ⊂ Ce(12) [i] based on
- OOA 3-folding [i] based on linear OA(6427, 4098, F64, 14) (dual of [4098, 4071, 15]-code), using
(13, 27, 496900)-Net in Base 64 — Upper bound on s
There is no (13, 27, 496901)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 5 846035 372647 188156 849530 851069 428761 107750 425344 > 6427 [i]