Best Known (52, 78, s)-Nets in Base 64
(52, 78, 20165)-Net over F64 — Constructive and digital
Digital (52, 78, 20165)-net over F64, using
- 1 times m-reduction [i] based on digital (52, 79, 20165)-net over F64, using
- net defined by OOA [i] based on linear OOA(6479, 20165, F64, 27, 27) (dual of [(20165, 27), 544376, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(6479, 262146, F64, 27) (dual of [262146, 262067, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(6479, 262147, F64, 27) (dual of [262147, 262068, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(6479, 262144, F64, 27) (dual of [262144, 262065, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(6479, 262147, F64, 27) (dual of [262147, 262068, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(6479, 262146, F64, 27) (dual of [262146, 262067, 28]-code), using
- net defined by OOA [i] based on linear OOA(6479, 20165, F64, 27, 27) (dual of [(20165, 27), 544376, 28]-NRT-code), using
(52, 78, 131077)-Net over F64 — Digital
Digital (52, 78, 131077)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6478, 131077, F64, 2, 26) (dual of [(131077, 2), 262076, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6478, 262154, F64, 26) (dual of [262154, 262076, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(6478, 262155, F64, 26) (dual of [262155, 262077, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- 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(6467, 262144, F64, 23) (dual of [262144, 262077, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(25) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(6478, 262155, F64, 26) (dual of [262155, 262077, 27]-code), using
- OOA 2-folding [i] based on linear OA(6478, 262154, F64, 26) (dual of [262154, 262076, 27]-code), using
(52, 78, large)-Net in Base 64 — Upper bound on s
There is no (52, 78, large)-net in base 64, because
- 24 times m-reduction [i] would yield (52, 54, large)-net in base 64, but