Best Known (40−13, 40, s)-Nets in Base 64
(40−13, 40, 43693)-Net over F64 — Constructive and digital
Digital (27, 40, 43693)-net over F64, using
- net defined by OOA [i] based on linear OOA(6440, 43693, F64, 13, 13) (dual of [(43693, 13), 567969, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6440, 262159, F64, 13) (dual of [262159, 262119, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6440, 262160, F64, 13) (dual of [262160, 262120, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(6437, 262145, F64, 13) (dual of [262145, 262108, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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(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([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6440, 262160, F64, 13) (dual of [262160, 262120, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6440, 262159, F64, 13) (dual of [262159, 262119, 14]-code), using
(40−13, 40, 197421)-Net over F64 — Digital
Digital (27, 40, 197421)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6440, 197421, F64, 13) (dual of [197421, 197381, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6440, 262160, F64, 13) (dual of [262160, 262120, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(6437, 262145, F64, 13) (dual of [262145, 262108, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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(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([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6440, 262160, F64, 13) (dual of [262160, 262120, 14]-code), using
(40−13, 40, large)-Net in Base 64 — Upper bound on s
There is no (27, 40, large)-net in base 64, because
- 11 times m-reduction [i] would yield (27, 29, large)-net in base 64, but