Best Known (45, 62, s)-Nets in Base 64
(45, 62, 32913)-Net over F64 — Constructive and digital
Digital (45, 62, 32913)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (5, 13, 145)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (1, 9, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- digital (0, 4, 65)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (32, 49, 32768)-net over F64, using
- net defined by OOA [i] based on linear OOA(6449, 32768, F64, 17, 17) (dual of [(32768, 17), 557007, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6449, 262145, F64, 17) (dual of [262145, 262096, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(6449, 262145, F64, 17) (dual of [262145, 262096, 18]-code), using
- net defined by OOA [i] based on linear OOA(6449, 32768, F64, 17, 17) (dual of [(32768, 17), 557007, 18]-NRT-code), using
- digital (5, 13, 145)-net over F64, using
(45, 62, 262146)-Net in Base 64 — Constructive
(45, 62, 262146)-net in base 64, using
- net defined by OOA [i] based on OOA(6462, 262146, S64, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(6462, 2097169, S64, 17), using
- discarding factors based on OA(6462, 2097171, S64, 17), using
- discarding parts of the base [i] based on linear OA(12853, 2097171, F128, 17) (dual of [2097171, 2097118, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(12849, 2097152, F128, 17) (dual of [2097152, 2097103, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(12853, 2097171, F128, 17) (dual of [2097171, 2097118, 18]-code), using
- discarding factors based on OA(6462, 2097171, S64, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(6462, 2097169, S64, 17), using
(45, 62, 1076834)-Net over F64 — Digital
Digital (45, 62, 1076834)-net over F64, using
(45, 62, large)-Net in Base 64 — Upper bound on s
There is no (45, 62, large)-net in base 64, because
- 15 times m-reduction [i] would yield (45, 47, large)-net in base 64, but