Best Known (39, 57, s)-Nets in Base 64
(39, 57, 29129)-Net over F64 — Constructive and digital
Digital (39, 57, 29129)-net over F64, using
- 641 times duplication [i] based on digital (38, 56, 29129)-net over F64, using
- net defined by OOA [i] based on linear OOA(6456, 29129, F64, 18, 18) (dual of [(29129, 18), 524266, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6456, 262161, F64, 18) (dual of [262161, 262105, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(6456, 262163, F64, 18) (dual of [262163, 262107, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- 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(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(644, 19, F64, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(6456, 262163, F64, 18) (dual of [262163, 262107, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(6456, 262161, F64, 18) (dual of [262161, 262105, 19]-code), using
- net defined by OOA [i] based on linear OOA(6456, 29129, F64, 18, 18) (dual of [(29129, 18), 524266, 19]-NRT-code), using
(39, 57, 226368)-Net over F64 — Digital
Digital (39, 57, 226368)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6457, 226368, F64, 18) (dual of [226368, 226311, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(6457, 262167, F64, 18) (dual of [262167, 262110, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- 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(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(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(17) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(6457, 262167, F64, 18) (dual of [262167, 262110, 19]-code), using
(39, 57, large)-Net in Base 64 — Upper bound on s
There is no (39, 57, large)-net in base 64, because
- 16 times m-reduction [i] would yield (39, 41, large)-net in base 64, but