Best Known (46−15, 46, s)-Nets in Base 64
(46−15, 46, 37451)-Net over F64 — Constructive and digital
Digital (31, 46, 37451)-net over F64, using
- net defined by OOA [i] based on linear OOA(6446, 37451, F64, 15, 15) (dual of [(37451, 15), 561719, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6446, 262158, F64, 15) (dual of [262158, 262112, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6446, 262160, F64, 15) (dual of [262160, 262114, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- 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(6431, 262145, F64, 11) (dual of [262145, 262114, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6446, 262160, F64, 15) (dual of [262160, 262114, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6446, 262158, F64, 15) (dual of [262158, 262112, 16]-code), using
(46−15, 46, 160772)-Net over F64 — Digital
Digital (31, 46, 160772)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6446, 160772, F64, 15) (dual of [160772, 160726, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6446, 262160, F64, 15) (dual of [262160, 262114, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- 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(6431, 262145, F64, 11) (dual of [262145, 262114, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6446, 262160, F64, 15) (dual of [262160, 262114, 16]-code), using
(46−15, 46, large)-Net in Base 64 — Upper bound on s
There is no (31, 46, large)-net in base 64, because
- 13 times m-reduction [i] would yield (31, 33, large)-net in base 64, but