Best Known (63−20, 63, s)-Nets in Base 64
(63−20, 63, 26216)-Net over F64 — Constructive and digital
Digital (43, 63, 26216)-net over F64, using
- 641 times duplication [i] based on digital (42, 62, 26216)-net over F64, using
- net defined by OOA [i] based on linear OOA(6462, 26216, F64, 20, 20) (dual of [(26216, 20), 524258, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(6462, 262160, F64, 20) (dual of [262160, 262098, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(6462, 262163, F64, 20) (dual of [262163, 262101, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(6443, 262144, F64, 15) (dual of [262144, 262101, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(644, 19, F64, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(6462, 262163, F64, 20) (dual of [262163, 262101, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(6462, 262160, F64, 20) (dual of [262160, 262098, 21]-code), using
- net defined by OOA [i] based on linear OOA(6462, 26216, F64, 20, 20) (dual of [(26216, 20), 524258, 21]-NRT-code), using
(63−20, 63, 199553)-Net over F64 — Digital
Digital (43, 63, 199553)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6463, 199553, F64, 20) (dual of [199553, 199490, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(6463, 262167, F64, 20) (dual of [262167, 262104, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- 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(645, 23, F64, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(6463, 262167, F64, 20) (dual of [262167, 262104, 21]-code), using
(63−20, 63, large)-Net in Base 64 — Upper bound on s
There is no (43, 63, large)-net in base 64, because
- 18 times m-reduction [i] would yield (43, 45, large)-net in base 64, but