Best Known (26, 26+12, s)-Nets in Base 64
(26, 26+12, 43693)-Net over F64 — Constructive and digital
Digital (26, 38, 43693)-net over F64, using
- 641 times duplication [i] based on digital (25, 37, 43693)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 43693, F64, 12, 12) (dual of [(43693, 12), 524279, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6437, 262158, F64, 12) (dual of [262158, 262121, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6437, 262159, F64, 12) (dual of [262159, 262122, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- 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(6422, 262144, F64, 8) (dual of [262144, 262122, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(643, 15, F64, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,64) or 15-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(6437, 262159, F64, 12) (dual of [262159, 262122, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(6437, 262158, F64, 12) (dual of [262158, 262121, 13]-code), using
- net defined by OOA [i] based on linear OOA(6437, 43693, F64, 12, 12) (dual of [(43693, 12), 524279, 13]-NRT-code), using
(26, 26+12, 262163)-Net over F64 — Digital
Digital (26, 38, 262163)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6438, 262163, F64, 12) (dual of [262163, 262125, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- 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(6419, 262144, F64, 7) (dual of [262144, 262125, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(11) ⊂ Ce(6) [i] based on
(26, 26+12, large)-Net in Base 64 — Upper bound on s
There is no (26, 38, large)-net in base 64, because
- 10 times m-reduction [i] would yield (26, 28, large)-net in base 64, but