Best Known (80−22, 80, s)-Nets in Base 64
(80−22, 80, 23961)-Net over F64 — Constructive and digital
Digital (58, 80, 23961)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (5, 16, 130)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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, 11, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 5, 65)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (42, 64, 23831)-net over F64, using
- net defined by OOA [i] based on linear OOA(6464, 23831, F64, 22, 22) (dual of [(23831, 22), 524218, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(6464, 262141, F64, 22) (dual of [262141, 262077, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(6464, 262141, F64, 22) (dual of [262141, 262077, 23]-code), using
- net defined by OOA [i] based on linear OOA(6464, 23831, F64, 22, 22) (dual of [(23831, 22), 524218, 23]-NRT-code), using
- digital (5, 16, 130)-net over F64, using
(80−22, 80, 190651)-Net in Base 64 — Constructive
(58, 80, 190651)-net in base 64, using
- 1 times m-reduction [i] based on (58, 81, 190651)-net in base 64, using
- net defined by OOA [i] based on OOA(6481, 190651, S64, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(6481, 2097162, S64, 23), using
- discarding factors based on OA(6481, 2097163, S64, 23), using
- discarding parts of the base [i] based on linear OA(12869, 2097163, F128, 23) (dual of [2097163, 2097094, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(12867, 2097152, F128, 23) (dual of [2097152, 2097085, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(22) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(12869, 2097163, F128, 23) (dual of [2097163, 2097094, 24]-code), using
- discarding factors based on OA(6481, 2097163, S64, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(6481, 2097162, S64, 23), using
- net defined by OOA [i] based on OOA(6481, 190651, S64, 23, 23), using
(80−22, 80, 1046748)-Net over F64 — Digital
Digital (58, 80, 1046748)-net over F64, using
(80−22, 80, large)-Net in Base 64 — Upper bound on s
There is no (58, 80, large)-net in base 64, because
- 20 times m-reduction [i] would yield (58, 60, large)-net in base 64, but