Best Known (52, 52+24, s)-Nets in Base 64
(52, 52+24, 21847)-Net over F64 — Constructive and digital
Digital (52, 76, 21847)-net over F64, using
- 641 times duplication [i] based on digital (51, 75, 21847)-net over F64, using
- net defined by OOA [i] based on linear OOA(6475, 21847, F64, 24, 24) (dual of [(21847, 24), 524253, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(6475, 262164, F64, 24) (dual of [262164, 262089, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(6475, 262167, F64, 24) (dual of [262167, 262092, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(17) [i] based on
- linear OA(6470, 262144, F64, 24) (dual of [262144, 262074, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(645, 23, F64, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to Ce(23) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(6475, 262167, F64, 24) (dual of [262167, 262092, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(6475, 262164, F64, 24) (dual of [262164, 262089, 25]-code), using
- net defined by OOA [i] based on linear OOA(6475, 21847, F64, 24, 24) (dual of [(21847, 24), 524253, 25]-NRT-code), using
(52, 52+24, 206501)-Net over F64 — Digital
Digital (52, 76, 206501)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6476, 206501, F64, 24) (dual of [206501, 206425, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(6476, 262171, F64, 24) (dual of [262171, 262095, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(16) [i] based on
- linear OA(6470, 262144, F64, 24) (dual of [262144, 262074, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(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(23) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(6476, 262171, F64, 24) (dual of [262171, 262095, 25]-code), using
(52, 52+24, large)-Net in Base 64 — Upper bound on s
There is no (52, 76, large)-net in base 64, because
- 22 times m-reduction [i] would yield (52, 54, large)-net in base 64, but