Best Known (40−14, 40, s)-Nets in Base 64
(40−14, 40, 37449)-Net over F64 — Constructive and digital
Digital (26, 40, 37449)-net over F64, using
- net defined by OOA [i] based on linear OOA(6440, 37449, F64, 14, 14) (dual of [(37449, 14), 524246, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6440, 262143, F64, 14) (dual of [262143, 262103, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6440, 262144, F64, 14) (dual of [262144, 262104, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(6440, 262144, F64, 14) (dual of [262144, 262104, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(6440, 262143, F64, 14) (dual of [262143, 262103, 15]-code), using
(40−14, 40, 131073)-Net over F64 — Digital
Digital (26, 40, 131073)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6440, 131073, F64, 2, 14) (dual of [(131073, 2), 262106, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6440, 262146, F64, 14) (dual of [262146, 262106, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6440, 262147, F64, 14) (dual of [262147, 262107, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(6440, 262144, F64, 14) (dual of [262144, 262104, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(6440, 262147, F64, 14) (dual of [262147, 262107, 15]-code), using
- OOA 2-folding [i] based on linear OA(6440, 262146, F64, 14) (dual of [262146, 262106, 15]-code), using
(40−14, 40, large)-Net in Base 64 — Upper bound on s
There is no (26, 40, large)-net in base 64, because
- 12 times m-reduction [i] would yield (26, 28, large)-net in base 64, but