Best Known (44, 44+24, s)-Nets in Base 64
(44, 44+24, 650)-Net over F64 — Constructive and digital
Digital (44, 68, 650)-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, 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, 4, 65)-net 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, 12, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 24, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
(44, 44+24, 5462)-Net in Base 64 — Constructive
(44, 68, 5462)-net in base 64, using
- base change [i] based on digital (27, 51, 5462)-net over F256, using
- 1 times m-reduction [i] based on digital (27, 52, 5462)-net over F256, using
- net defined by OOA [i] based on linear OOA(25652, 5462, F256, 25, 25) (dual of [(5462, 25), 136498, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25652, 65545, F256, 25) (dual of [65545, 65493, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(25652, 65548, F256, 25) (dual of [65548, 65496, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25652, 65548, F256, 25) (dual of [65548, 65496, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25652, 65545, F256, 25) (dual of [65545, 65493, 26]-code), using
- net defined by OOA [i] based on linear OOA(25652, 5462, F256, 25, 25) (dual of [(5462, 25), 136498, 26]-NRT-code), using
- 1 times m-reduction [i] based on digital (27, 52, 5462)-net over F256, using
(44, 44+24, 32755)-Net over F64 — Digital
Digital (44, 68, 32755)-net over F64, using
(44, 44+24, large)-Net in Base 64 — Upper bound on s
There is no (44, 68, large)-net in base 64, because
- 22 times m-reduction [i] would yield (44, 46, large)-net in base 64, but