Best Known (86−27, 86, s)-Nets in Base 64
(86−27, 86, 20167)-Net over F64 — Constructive and digital
Digital (59, 86, 20167)-net over F64, using
- net defined by OOA [i] based on linear OOA(6486, 20167, F64, 27, 27) (dual of [(20167, 27), 544423, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(6486, 262172, F64, 27) (dual of [262172, 262086, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(6486, 262176, F64, 27) (dual of [262176, 262090, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- linear OA(6479, 262145, F64, 27) (dual of [262145, 262066, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- 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(647, 31, F64, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,64)), using
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- Reed–Solomon code RS(57,64) [i]
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6486, 262176, F64, 27) (dual of [262176, 262090, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(6486, 262172, F64, 27) (dual of [262172, 262086, 28]-code), using
(86−27, 86, 223498)-Net over F64 — Digital
Digital (59, 86, 223498)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6486, 223498, F64, 27) (dual of [223498, 223412, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(6486, 262176, F64, 27) (dual of [262176, 262090, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- linear OA(6479, 262145, F64, 27) (dual of [262145, 262066, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- 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(647, 31, F64, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,64)), using
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- Reed–Solomon code RS(57,64) [i]
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6486, 262176, F64, 27) (dual of [262176, 262090, 28]-code), using
(86−27, 86, large)-Net in Base 64 — Upper bound on s
There is no (59, 86, large)-net in base 64, because
- 25 times m-reduction [i] would yield (59, 61, large)-net in base 64, but