Best Known (29−14, 29, s)-Nets in Base 64
(29−14, 29, 586)-Net over F64 — Constructive and digital
Digital (15, 29, 586)-net over F64, using
- net defined by OOA [i] based on linear OOA(6429, 586, F64, 14, 14) (dual of [(586, 14), 8175, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6429, 4102, F64, 14) (dual of [4102, 4073, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6429, 4104, F64, 14) (dual of [4104, 4075, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [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(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(6429, 4104, F64, 14) (dual of [4104, 4075, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(6429, 4102, F64, 14) (dual of [4102, 4073, 15]-code), using
(29−14, 29, 2052)-Net over F64 — Digital
Digital (15, 29, 2052)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6429, 2052, F64, 2, 14) (dual of [(2052, 2), 4075, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6429, 4104, F64, 14) (dual of [4104, 4075, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [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(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(13) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(6429, 4104, F64, 14) (dual of [4104, 4075, 15]-code), using
(29−14, 29, 1630507)-Net in Base 64 — Upper bound on s
There is no (15, 29, 1630508)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 23945 265031 716989 142721 865725 193106 153877 273435 149646 > 6429 [i]