Best Known (56, 56+29, s)-Nets in Base 64
(56, 56+29, 18724)-Net over F64 — Constructive and digital
Digital (56, 85, 18724)-net over F64, using
- net defined by OOA [i] based on linear OOA(6485, 18724, F64, 29, 29) (dual of [(18724, 29), 542911, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(6485, 262137, F64, 29) (dual of [262137, 262052, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(6485, 262144, F64, 29) (dual of [262144, 262059, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(6485, 262144, F64, 29) (dual of [262144, 262059, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(6485, 262137, F64, 29) (dual of [262137, 262052, 30]-code), using
(56, 56+29, 97674)-Net over F64 — Digital
Digital (56, 85, 97674)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6485, 97674, F64, 2, 29) (dual of [(97674, 2), 195263, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6485, 131073, F64, 2, 29) (dual of [(131073, 2), 262061, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6485, 262146, F64, 29) (dual of [262146, 262061, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(6485, 262147, F64, 29) (dual of [262147, 262062, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(6485, 262144, F64, 29) (dual of [262144, 262059, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(6482, 262144, F64, 28) (dual of [262144, 262062, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(6485, 262147, F64, 29) (dual of [262147, 262062, 30]-code), using
- OOA 2-folding [i] based on linear OA(6485, 262146, F64, 29) (dual of [262146, 262061, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(6485, 131073, F64, 2, 29) (dual of [(131073, 2), 262061, 30]-NRT-code), using
(56, 56+29, large)-Net in Base 64 — Upper bound on s
There is no (56, 85, large)-net in base 64, because
- 27 times m-reduction [i] would yield (56, 58, large)-net in base 64, but