Best Known (46−9, 46, s)-Nets in Base 16
(46−9, 46, 262169)-Net over F16 — Constructive and digital
Digital (37, 46, 262169)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- digital (32, 41, 262145)-net over F16, using
- net defined by OOA [i] based on linear OOA(1641, 262145, F16, 9, 9) (dual of [(262145, 9), 2359264, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(1641, 1048581, F16, 9) (dual of [1048581, 1048540, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(1641, 1048576, F16, 9) (dual of [1048576, 1048535, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(1636, 1048576, F16, 8) (dual of [1048576, 1048540, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(1641, 1048581, F16, 9) (dual of [1048581, 1048540, 10]-code), using
- net defined by OOA [i] based on linear OOA(1641, 262145, F16, 9, 9) (dual of [(262145, 9), 2359264, 10]-NRT-code), using
- digital (1, 5, 24)-net over F16, using
(46−9, 46, 524289)-Net in Base 16 — Constructive
(37, 46, 524289)-net in base 16, using
- net defined by OOA [i] based on OOA(1646, 524289, S16, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(1646, 2097157, S16, 9), using
- discarding factors based on OA(1646, 2097160, S16, 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(1646, 2097160, S16, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(1646, 2097157, S16, 9), using
(46−9, 46, 2105181)-Net over F16 — Digital
Digital (37, 46, 2105181)-net over F16, using
(46−9, 46, large)-Net in Base 16 — Upper bound on s
There is no (37, 46, large)-net in base 16, because
- 7 times m-reduction [i] would yield (37, 39, large)-net in base 16, but