Best Known (40, 55, s)-Nets in Base 64
(40, 55, 37644)-Net over F64 — Constructive and digital
Digital (40, 55, 37644)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (5, 12, 195)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 65)-net over F64, using
- 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, 7, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- generalized (u, u+v)-construction [i] based on
- digital (28, 43, 37449)-net over F64, using
- net defined by OOA [i] based on linear OOA(6443, 37449, F64, 15, 15) (dual of [(37449, 15), 561692, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [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]
- OOA 7-folding and stacking with additional row [i] based on linear OA(6443, 262144, F64, 15) (dual of [262144, 262101, 16]-code), using
- net defined by OOA [i] based on linear OOA(6443, 37449, F64, 15, 15) (dual of [(37449, 15), 561692, 16]-NRT-code), using
- digital (5, 12, 195)-net over F64, using
(40, 55, 299595)-Net in Base 64 — Constructive
(40, 55, 299595)-net in base 64, using
- 641 times duplication [i] based on (39, 54, 299595)-net in base 64, using
- net defined by OOA [i] based on OOA(6454, 299595, S64, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(6454, 2097166, S64, 15), using
- discarding factors based on OA(6454, 2097168, S64, 15), using
- discarding parts of the base [i] based on linear OA(12846, 2097168, F128, 15) (dual of [2097168, 2097122, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding parts of the base [i] based on linear OA(12846, 2097168, F128, 15) (dual of [2097168, 2097122, 16]-code), using
- discarding factors based on OA(6454, 2097168, S64, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(6454, 2097166, S64, 15), using
- net defined by OOA [i] based on OOA(6454, 299595, S64, 15, 15), using
(40, 55, 1196263)-Net over F64 — Digital
Digital (40, 55, 1196263)-net over F64, using
(40, 55, large)-Net in Base 64 — Upper bound on s
There is no (40, 55, large)-net in base 64, because
- 13 times m-reduction [i] would yield (40, 42, large)-net in base 64, but