Best Known (58−19, 58, s)-Nets in Base 64
(58−19, 58, 29128)-Net over F64 — Constructive and digital
Digital (39, 58, 29128)-net over F64, using
- 641 times duplication [i] based on digital (38, 57, 29128)-net over F64, using
- net defined by OOA [i] based on linear OOA(6457, 29128, F64, 19, 19) (dual of [(29128, 19), 553375, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6457, 262153, F64, 19) (dual of [262153, 262096, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(6457, 262155, F64, 19) (dual of [262155, 262098, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(6455, 262144, F64, 19) (dual of [262144, 262089, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(6446, 262144, F64, 16) (dual of [262144, 262098, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(6457, 262155, F64, 19) (dual of [262155, 262098, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6457, 262153, F64, 19) (dual of [262153, 262096, 20]-code), using
- net defined by OOA [i] based on linear OOA(6457, 29128, F64, 19, 19) (dual of [(29128, 19), 553375, 20]-NRT-code), using
(58−19, 58, 131080)-Net over F64 — Digital
Digital (39, 58, 131080)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6458, 131080, F64, 2, 19) (dual of [(131080, 2), 262102, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6458, 262160, F64, 19) (dual of [262160, 262102, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(6455, 262145, F64, 19) (dual of [262145, 262090, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(6443, 262145, F64, 15) (dual of [262145, 262102, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [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 C([0,9]) ⊂ C([0,7]) [i] based on
- OOA 2-folding [i] based on linear OA(6458, 262160, F64, 19) (dual of [262160, 262102, 20]-code), using
(58−19, 58, large)-Net in Base 64 — Upper bound on s
There is no (39, 58, large)-net in base 64, because
- 17 times m-reduction [i] would yield (39, 41, large)-net in base 64, but