Best Known (56, 56+28, s)-Nets in Base 64
(56, 56+28, 18725)-Net over F64 — Constructive and digital
Digital (56, 84, 18725)-net over F64, using
- 641 times duplication [i] based on digital (55, 83, 18725)-net over F64, using
- net defined by OOA [i] based on linear OOA(6483, 18725, F64, 28, 28) (dual of [(18725, 28), 524217, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(6483, 262150, F64, 28) (dual of [262150, 262067, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(6483, 262151, F64, 28) (dual of [262151, 262068, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(6482, 262144, F64, 28) (dual of [262144, 262062, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(6476, 262144, F64, 26) (dual of [262144, 262068, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(6483, 262151, F64, 28) (dual of [262151, 262068, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(6483, 262150, F64, 28) (dual of [262150, 262067, 29]-code), using
- net defined by OOA [i] based on linear OOA(6483, 18725, F64, 28, 28) (dual of [(18725, 28), 524217, 29]-NRT-code), using
(56, 56+28, 131077)-Net over F64 — Digital
Digital (56, 84, 131077)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6484, 131077, F64, 2, 28) (dual of [(131077, 2), 262070, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6484, 262154, F64, 28) (dual of [262154, 262070, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(6484, 262155, F64, 28) (dual of [262155, 262071, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(24) [i] based on
- linear OA(6482, 262144, F64, 28) (dual of [262144, 262062, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(6473, 262144, F64, 25) (dual of [262144, 262071, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(27) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(6484, 262155, F64, 28) (dual of [262155, 262071, 29]-code), using
- OOA 2-folding [i] based on linear OA(6484, 262154, F64, 28) (dual of [262154, 262070, 29]-code), using
(56, 56+28, large)-Net in Base 64 — Upper bound on s
There is no (56, 84, large)-net in base 64, because
- 26 times m-reduction [i] would yield (56, 58, large)-net in base 64, but