Best Known (31−9, 31, s)-Nets in Base 16
(31−9, 31, 1542)-Net over F16 — Constructive and digital
Digital (22, 31, 1542)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 5, 514)-net over F16, using
- s-reduction based on digital (2, 5, 629)-net over F16, using
- net defined by OOA [i] based on linear OOA(165, 629, F16, 3, 3) (dual of [(629, 3), 1882, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(165, 629, F16, 2, 3) (dual of [(629, 2), 1253, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(165, 629, F16, 3, 3) (dual of [(629, 3), 1882, 4]-NRT-code), using
- s-reduction based on digital (2, 5, 629)-net over F16, using
- digital (4, 8, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 4, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 4, 257)-net over F256, using
- digital (9, 18, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 9, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 9, 257)-net over F256, using
- digital (2, 5, 514)-net over F16, using
(31−9, 31, 4096)-Net in Base 16 — Constructive
(22, 31, 4096)-net in base 16, using
- 161 times duplication [i] based on (21, 30, 4096)-net in base 16, using
- net defined by OOA [i] based on OOA(1630, 4096, S16, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(1630, 16385, S16, 9), using
- discarding factors based on OA(1630, 16386, S16, 9), using
- discarding parts of the base [i] based on linear OA(12817, 16386, F128, 9) (dual of [16386, 16369, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(12817, 16384, F128, 9) (dual of [16384, 16367, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- discarding parts of the base [i] based on linear OA(12817, 16386, F128, 9) (dual of [16386, 16369, 10]-code), using
- discarding factors based on OA(1630, 16386, S16, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(1630, 16385, S16, 9), using
- net defined by OOA [i] based on OOA(1630, 4096, S16, 9, 9), using
(31−9, 31, 11634)-Net over F16 — Digital
Digital (22, 31, 11634)-net over F16, using
(31−9, 31, large)-Net in Base 16 — Upper bound on s
There is no (22, 31, large)-net in base 16, because
- 7 times m-reduction [i] would yield (22, 24, large)-net in base 16, but