Best Known (28−13, 28, s)-Nets in Base 64
(28−13, 28, 684)-Net over F64 — Constructive and digital
Digital (15, 28, 684)-net over F64, using
- net defined by OOA [i] based on linear OOA(6428, 684, F64, 13, 13) (dual of [(684, 13), 8864, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6428, 4105, F64, 13) (dual of [4105, 4077, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6428, 4108, F64, 13) (dual of [4108, 4080, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-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(6428, 4108, F64, 13) (dual of [4108, 4080, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6428, 4105, F64, 13) (dual of [4105, 4077, 14]-code), using
(28−13, 28, 2109)-Net over F64 — Digital
Digital (15, 28, 2109)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6428, 2109, F64, 13) (dual of [2109, 2081, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6428, 4108, F64, 13) (dual of [4108, 4080, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-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(6428, 4108, F64, 13) (dual of [4108, 4080, 14]-code), using
(28−13, 28, 6378098)-Net in Base 64 — Upper bound on s
There is no (15, 28, 6378099)-net in base 64, because
- 1 times m-reduction [i] would yield (15, 27, 6378099)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 5 846007 883680 238348 460591 952185 953980 450835 113628 > 6427 [i]