Best Known (49−15, 49, s)-Nets in Base 64
(49−15, 49, 37452)-Net over F64 — Constructive and digital
Digital (34, 49, 37452)-net over F64, using
- 641 times duplication [i] based on digital (33, 48, 37452)-net over F64, using
- net defined by OOA [i] based on linear OOA(6448, 37452, F64, 15, 15) (dual of [(37452, 15), 561732, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6448, 262165, F64, 15) (dual of [262165, 262117, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6448, 262168, F64, 15) (dual of [262168, 262120, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [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(6425, 262145, F64, 9) (dual of [262145, 262120, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [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 C([0,7]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6448, 262168, F64, 15) (dual of [262168, 262120, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6448, 262165, F64, 15) (dual of [262165, 262117, 16]-code), using
- net defined by OOA [i] based on linear OOA(6448, 37452, F64, 15, 15) (dual of [(37452, 15), 561732, 16]-NRT-code), using
(49−15, 49, 262171)-Net over F64 — Digital
Digital (34, 49, 262171)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6449, 262171, F64, 15) (dual of [262171, 262122, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(7) [i] based on
- 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(6422, 262144, F64, 8) (dual of [262144, 262122, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(646, 27, F64, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,64)), using
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- Reed–Solomon code RS(58,64) [i]
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- construction X applied to Ce(14) ⊂ Ce(7) [i] based on
(49−15, 49, large)-Net in Base 64 — Upper bound on s
There is no (34, 49, large)-net in base 64, because
- 13 times m-reduction [i] would yield (34, 36, large)-net in base 64, but