Best Known (74−26, 74, s)-Nets in Base 64
(74−26, 74, 715)-Net over F64 — Constructive and digital
Digital (48, 74, 715)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 65)-net over F64, using
- digital (0, 2, 65)-net over F64 (see above)
- digital (0, 2, 65)-net over F64 (see above)
- digital (0, 3, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (0, 3, 65)-net over F64 (see above)
- digital (0, 4, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 5, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 6, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 8, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 13, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 26, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
(74−26, 74, 5042)-Net in Base 64 — Constructive
(48, 74, 5042)-net in base 64, using
- 1 times m-reduction [i] based on (48, 75, 5042)-net in base 64, using
- net defined by OOA [i] based on OOA(6475, 5042, S64, 27, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(6475, 65547, S64, 27), using
- discarding factors based on OA(6475, 65548, S64, 27), using
- discarding parts of the base [i] based on linear OA(25656, 65548, F256, 27) (dual of [65548, 65492, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- discarding parts of the base [i] based on linear OA(25656, 65548, F256, 27) (dual of [65548, 65492, 28]-code), using
- discarding factors based on OA(6475, 65548, S64, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(6475, 65547, S64, 27), using
- net defined by OOA [i] based on OOA(6475, 5042, S64, 27, 27), using
(74−26, 74, 35870)-Net over F64 — Digital
Digital (48, 74, 35870)-net over F64, using
(74−26, 74, large)-Net in Base 64 — Upper bound on s
There is no (48, 74, large)-net in base 64, because
- 24 times m-reduction [i] would yield (48, 50, large)-net in base 64, but