Best Known (61−17, 61, s)-Nets in Base 64
(61−17, 61, 32898)-Net over F64 — Constructive and digital
Digital (44, 61, 32898)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 130)-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 (0, 8, 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, 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 (4, 12, 130)-net over F64, using
(61−17, 61, 262145)-Net in Base 64 — Constructive
(44, 61, 262145)-net in base 64, using
- 641 times duplication [i] based on (43, 60, 262145)-net in base 64, using
- net defined by OOA [i] based on OOA(6460, 262145, S64, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(6460, 2097161, S64, 17), using
- discarding factors based on OA(6460, 2097163, S64, 17), using
- discarding parts of the base [i] based on linear OA(12851, 2097163, F128, 17) (dual of [2097163, 2097112, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [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(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(12851, 2097163, F128, 17) (dual of [2097163, 2097112, 18]-code), using
- discarding factors based on OA(6460, 2097163, S64, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(6460, 2097161, S64, 17), using
- net defined by OOA [i] based on OOA(6460, 262145, S64, 17, 17), using
(61−17, 61, 830354)-Net over F64 — Digital
Digital (44, 61, 830354)-net over F64, using
(61−17, 61, large)-Net in Base 64 — Upper bound on s
There is no (44, 61, large)-net in base 64, because
- 15 times m-reduction [i] would yield (44, 46, large)-net in base 64, but