Best Known (82−23, 82, s)-Nets in Base 64
(82−23, 82, 23935)-Net over F64 — Constructive and digital
Digital (59, 82, 23935)-net over F64, using
- 641 times duplication [i] based on digital (58, 81, 23935)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 14, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- digital (44, 67, 23831)-net over F64, using
- net defined by OOA [i] based on linear OOA(6467, 23831, F64, 23, 23) (dual of [(23831, 23), 548046, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6467, 262142, F64, 23) (dual of [262142, 262075, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(6467, 262144, F64, 23) (dual of [262144, 262077, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(6467, 262144, F64, 23) (dual of [262144, 262077, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6467, 262142, F64, 23) (dual of [262142, 262075, 24]-code), using
- net defined by OOA [i] based on linear OOA(6467, 23831, F64, 23, 23) (dual of [(23831, 23), 548046, 24]-NRT-code), using
- digital (3, 14, 104)-net over F64, using
- (u, u+v)-construction [i] based on
(82−23, 82, 190651)-Net in Base 64 — Constructive
(59, 82, 190651)-net in base 64, using
- 641 times duplication [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
(82−23, 82, 775611)-Net over F64 — Digital
Digital (59, 82, 775611)-net over F64, using
(82−23, 82, large)-Net in Base 64 — Upper bound on s
There is no (59, 82, large)-net in base 64, because
- 21 times m-reduction [i] would yield (59, 61, large)-net in base 64, but