Best Known (56, 56+20, s)-Nets in Base 64
(56, 56+20, 26409)-Net over F64 — Constructive and digital
Digital (56, 76, 26409)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (8, 18, 195)-net over F64, using
- 1 times m-reduction [i] based on digital (8, 19, 195)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- 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, 5, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 11, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 3, 65)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- 1 times m-reduction [i] based on digital (8, 19, 195)-net over F64, using
- digital (38, 58, 26214)-net over F64, using
- net defined by OOA [i] based on linear OOA(6458, 26214, F64, 20, 20) (dual of [(26214, 20), 524222, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(6458, 262140, F64, 20) (dual of [262140, 262082, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(6458, 262140, F64, 20) (dual of [262140, 262082, 21]-code), using
- net defined by OOA [i] based on linear OOA(6458, 26214, F64, 20, 20) (dual of [(26214, 20), 524222, 21]-NRT-code), using
- digital (8, 18, 195)-net over F64, using
(56, 56+20, 209718)-Net in Base 64 — Constructive
(56, 76, 209718)-net in base 64, using
- net defined by OOA [i] based on OOA(6476, 209718, S64, 20, 20), using
- OA 10-folding and stacking [i] based on OA(6476, 2097180, S64, 20), using
- discarding factors based on OA(6476, 2097183, S64, 20), using
- discarding parts of the base [i] based on linear OA(12865, 2097183, F128, 20) (dual of [2097183, 2097118, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(11) [i] based on
- linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(1287, 31, F128, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to Ce(19) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(12865, 2097183, F128, 20) (dual of [2097183, 2097118, 21]-code), using
- discarding factors based on OA(6476, 2097183, S64, 20), using
- OA 10-folding and stacking [i] based on OA(6476, 2097180, S64, 20), using
(56, 56+20, 2111528)-Net over F64 — Digital
Digital (56, 76, 2111528)-net over F64, using
(56, 56+20, large)-Net in Base 64 — Upper bound on s
There is no (56, 76, large)-net in base 64, because
- 18 times m-reduction [i] would yield (56, 58, large)-net in base 64, but