Best Known (40, 60, s)-Nets in Base 64
(40, 60, 26215)-Net over F64 — Constructive and digital
Digital (40, 60, 26215)-net over F64, using
- 641 times duplication [i] based on digital (39, 59, 26215)-net over F64, using
- net defined by OOA [i] based on linear OOA(6459, 26215, F64, 20, 20) (dual of [(26215, 20), 524241, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(6459, 262150, F64, 20) (dual of [262150, 262091, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(6459, 262151, F64, 20) (dual of [262151, 262092, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [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(6452, 262144, F64, 18) (dual of [262144, 262092, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(19) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(6459, 262151, F64, 20) (dual of [262151, 262092, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(6459, 262150, F64, 20) (dual of [262150, 262091, 21]-code), using
- net defined by OOA [i] based on linear OOA(6459, 26215, F64, 20, 20) (dual of [(26215, 20), 524241, 21]-NRT-code), using
(40, 60, 131077)-Net over F64 — Digital
Digital (40, 60, 131077)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6460, 131077, F64, 2, 20) (dual of [(131077, 2), 262094, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6460, 262154, F64, 20) (dual of [262154, 262094, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(6460, 262155, F64, 20) (dual of [262155, 262095, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [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(6449, 262144, F64, 17) (dual of [262144, 262095, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(6460, 262155, F64, 20) (dual of [262155, 262095, 21]-code), using
- OOA 2-folding [i] based on linear OA(6460, 262154, F64, 20) (dual of [262154, 262094, 21]-code), using
(40, 60, large)-Net in Base 64 — Upper bound on s
There is no (40, 60, large)-net in base 64, because
- 18 times m-reduction [i] would yield (40, 42, large)-net in base 64, but