Best Known (37, 55, s)-Nets in Base 64
(37, 55, 29128)-Net over F64 — Constructive and digital
Digital (37, 55, 29128)-net over F64, using
- 641 times duplication [i] based on digital (36, 54, 29128)-net over F64, using
- net defined by OOA [i] based on linear OOA(6454, 29128, F64, 18, 18) (dual of [(29128, 18), 524250, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6454, 262152, F64, 18) (dual of [262152, 262098, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(6454, 262155, F64, 18) (dual of [262155, 262101, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [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(6443, 262144, F64, 15) (dual of [262144, 262101, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(6454, 262155, F64, 18) (dual of [262155, 262101, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(6454, 262152, F64, 18) (dual of [262152, 262098, 19]-code), using
- net defined by OOA [i] based on linear OOA(6454, 29128, F64, 18, 18) (dual of [(29128, 18), 524250, 19]-NRT-code), using
(37, 55, 134596)-Net over F64 — Digital
Digital (37, 55, 134596)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6455, 134596, F64, 18) (dual of [134596, 134541, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(6455, 262159, F64, 18) (dual of [262159, 262104, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [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(6440, 262144, F64, 14) (dual of [262144, 262104, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(643, 15, F64, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,64) or 15-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(6455, 262159, F64, 18) (dual of [262159, 262104, 19]-code), using
(37, 55, large)-Net in Base 64 — Upper bound on s
There is no (37, 55, large)-net in base 64, because
- 16 times m-reduction [i] would yield (37, 39, large)-net in base 64, but