Best Known (42−14, 42, s)-Nets in Base 64
(42−14, 42, 37450)-Net over F64 — Constructive and digital
Digital (28, 42, 37450)-net over F64, using
- 641 times duplication [i] based on digital (27, 41, 37450)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 37450, F64, 14, 14) (dual of [(37450, 14), 524259, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6441, 262150, F64, 14) (dual of [262150, 262109, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6441, 262151, F64, 14) (dual of [262151, 262110, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- 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(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(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(6441, 262151, F64, 14) (dual of [262151, 262110, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(6441, 262150, F64, 14) (dual of [262150, 262109, 15]-code), using
- net defined by OOA [i] based on linear OOA(6441, 37450, F64, 14, 14) (dual of [(37450, 14), 524259, 15]-NRT-code), using
(42−14, 42, 131077)-Net over F64 — Digital
Digital (28, 42, 131077)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6442, 131077, F64, 2, 14) (dual of [(131077, 2), 262112, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6442, 262154, F64, 14) (dual of [262154, 262112, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6442, 262155, F64, 14) (dual of [262155, 262113, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- 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(6431, 262144, F64, 11) (dual of [262144, 262113, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(642, 11, F64, 2) (dual of [11, 9, 3]-code or 11-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(6442, 262155, F64, 14) (dual of [262155, 262113, 15]-code), using
- OOA 2-folding [i] based on linear OA(6442, 262154, F64, 14) (dual of [262154, 262112, 15]-code), using
(42−14, 42, large)-Net in Base 64 — Upper bound on s
There is no (28, 42, large)-net in base 64, because
- 12 times m-reduction [i] would yield (28, 30, large)-net in base 64, but