Best Known (13, 13+12, s)-Nets in Base 64
(13, 13+12, 684)-Net over F64 — Constructive and digital
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
(13, 13+12, 2052)-Net over F64 — Digital
Digital (13, 25, 2052)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6425, 2052, F64, 2, 12) (dual of [(2052, 2), 4079, 13]-NRT-code), using
- OOA 2-folding [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
- OOA 2-folding [i] based on linear OA(6425, 4104, F64, 12) (dual of [4104, 4079, 13]-code), using
(13, 13+12, 1594522)-Net in Base 64 — Upper bound on s
There is no (13, 25, 1594523)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 1427 249073 402239 931645 173456 520574 971811 130230 > 6425 [i]