Best Known (22, 22+9, s)-Nets in Base 64
(22, 22+9, 67552)-Net over F64 — Constructive and digital
Digital (22, 31, 67552)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (2, 6, 2016)-net over F64, using
- net defined by OOA [i] based on linear OOA(646, 2016, F64, 4, 4) (dual of [(2016, 4), 8058, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(646, 2016, F64, 3, 4) (dual of [(2016, 3), 6042, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(646, 4032, F64, 4) (dual of [4032, 4026, 5]-code), using
- 1 times truncation [i] based on linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(646, 4032, F64, 4) (dual of [4032, 4026, 5]-code), using
- appending kth column [i] based on linear OOA(646, 2016, F64, 3, 4) (dual of [(2016, 3), 6042, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(646, 2016, F64, 4, 4) (dual of [(2016, 4), 8058, 5]-NRT-code), using
- digital (16, 25, 65536)-net over F64, using
- net defined by OOA [i] based on linear OOA(6425, 65536, F64, 9, 9) (dual of [(65536, 9), 589799, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6425, 262145, F64, 9) (dual of [262145, 262120, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(6425, 262145, F64, 9) (dual of [262145, 262120, 10]-code), using
- net defined by OOA [i] based on linear OOA(6425, 65536, F64, 9, 9) (dual of [(65536, 9), 589799, 10]-NRT-code), using
- digital (2, 6, 2016)-net over F64, using
(22, 22+9, 524289)-Net in Base 64 — Constructive
(22, 31, 524289)-net in base 64, using
- net defined by OOA [i] based on OOA(6431, 524289, S64, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(6431, 2097157, S64, 9), using
- discarding factors based on OA(6431, 2097160, S64, 9), using
- discarding parts of the base [i] based on linear OA(12826, 2097160, F128, 9) (dual of [2097160, 2097134, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(12819, 2097153, F128, 7) (dual of [2097153, 2097134, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- discarding parts of the base [i] based on linear OA(12826, 2097160, F128, 9) (dual of [2097160, 2097134, 10]-code), using
- discarding factors based on OA(6431, 2097160, S64, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(6431, 2097157, S64, 9), using
(22, 22+9, 596074)-Net over F64 — Digital
Digital (22, 31, 596074)-net over F64, using
(22, 22+9, large)-Net in Base 64 — Upper bound on s
There is no (22, 31, large)-net in base 64, because
- 7 times m-reduction [i] would yield (22, 24, large)-net in base 64, but