Best Known (64, 64+24, s)-Nets in Base 64
(64, 64+24, 21975)-Net over F64 — Constructive and digital
Digital (64, 88, 21975)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (6, 18, 130)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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, 12, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 6, 65)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (46, 70, 21845)-net over F64, using
- net defined by OOA [i] based on linear OOA(6470, 21845, F64, 24, 24) (dual of [(21845, 24), 524210, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(6470, 262140, F64, 24) (dual of [262140, 262070, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(6470, 262144, F64, 24) (dual of [262144, 262074, 25]-code), using
- an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(6470, 262144, F64, 24) (dual of [262144, 262074, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(6470, 262140, F64, 24) (dual of [262140, 262070, 25]-code), using
- net defined by OOA [i] based on linear OOA(6470, 21845, F64, 24, 24) (dual of [(21845, 24), 524210, 25]-NRT-code), using
- digital (6, 18, 130)-net over F64, using
(64, 64+24, 174764)-Net in Base 64 — Constructive
(64, 88, 174764)-net in base 64, using
- 641 times duplication [i] based on (63, 87, 174764)-net in base 64, using
- net defined by OOA [i] based on OOA(6487, 174764, S64, 24, 24), using
- OA 12-folding and stacking [i] based on OA(6487, 2097168, S64, 24), using
- discarding factors based on OA(6487, 2097171, S64, 24), using
- discarding parts of the base [i] based on linear OA(12874, 2097171, F128, 24) (dual of [2097171, 2097097, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(23) ⊂ Ce(18) [i] based on
- discarding parts of the base [i] based on linear OA(12874, 2097171, F128, 24) (dual of [2097171, 2097097, 25]-code), using
- discarding factors based on OA(6487, 2097171, S64, 24), using
- OA 12-folding and stacking [i] based on OA(6487, 2097168, S64, 24), using
- net defined by OOA [i] based on OOA(6487, 174764, S64, 24, 24), using
(64, 64+24, 1218208)-Net over F64 — Digital
Digital (64, 88, 1218208)-net over F64, using
(64, 64+24, large)-Net in Base 64 — Upper bound on s
There is no (64, 88, large)-net in base 64, because
- 22 times m-reduction [i] would yield (64, 66, large)-net in base 64, but