Best Known (63−19, 63, s)-Nets in Base 64
(63−19, 63, 29130)-Net over F64 — Constructive and digital
Digital (44, 63, 29130)-net over F64, using
- 642 times duplication [i] based on digital (42, 61, 29130)-net over F64, using
- net defined by OOA [i] based on linear OOA(6461, 29130, F64, 19, 19) (dual of [(29130, 19), 553409, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6461, 262171, F64, 19) (dual of [262171, 262110, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(11) [i] based on
- linear OA(6455, 262144, F64, 19) (dual of [262144, 262089, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(6434, 262144, F64, 12) (dual of [262144, 262110, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(646, 27, F64, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,64)), using
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- Reed–Solomon code RS(58,64) [i]
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- construction X applied to Ce(18) ⊂ Ce(11) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(6461, 262171, F64, 19) (dual of [262171, 262110, 20]-code), using
- net defined by OOA [i] based on linear OOA(6461, 29130, F64, 19, 19) (dual of [(29130, 19), 553409, 20]-NRT-code), using
(63−19, 63, 262179)-Net over F64 — Digital
Digital (44, 63, 262179)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6463, 262179, F64, 19) (dual of [262179, 262116, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(9) [i] based on
- linear OA(6455, 262144, F64, 19) (dual of [262144, 262089, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(648, 35, F64, 8) (dual of [35, 27, 9]-code or 35-arc in PG(7,64)), using
- discarding factors / shortening the dual code based on linear OA(648, 64, F64, 8) (dual of [64, 56, 9]-code or 64-arc in PG(7,64)), using
- Reed–Solomon code RS(56,64) [i]
- discarding factors / shortening the dual code based on linear OA(648, 64, F64, 8) (dual of [64, 56, 9]-code or 64-arc in PG(7,64)), using
- construction X applied to Ce(18) ⊂ Ce(9) [i] based on
(63−19, 63, large)-Net in Base 64 — Upper bound on s
There is no (44, 63, large)-net in base 64, because
- 17 times m-reduction [i] would yield (44, 46, large)-net in base 64, but