Best Known (31, 31+13, s)-Nets in Base 16
(31, 31+13, 1148)-Net over F16 — Constructive and digital
Digital (31, 44, 1148)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 6, 120)-net over F16, using
- net defined by OOA [i] based on linear OOA(166, 120, F16, 4, 4) (dual of [(120, 4), 474, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- 1 times truncation [i] based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- net defined by OOA [i] based on linear OOA(166, 120, F16, 4, 4) (dual of [(120, 4), 474, 5]-NRT-code), using
- digital (6, 12, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 6, 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, 6, 257)-net over F256, using
- digital (13, 26, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 13, 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, 13, 257)-net over F256, using
- digital (2, 6, 120)-net over F16, using
(31, 31+13, 2730)-Net in Base 16 — Constructive
(31, 44, 2730)-net in base 16, using
- net defined by OOA [i] based on OOA(1644, 2730, S16, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1644, 16381, S16, 13), using
- discarding factors based on OA(1644, 16386, S16, 13), using
- discarding parts of the base [i] based on linear OA(12825, 16386, F128, 13) (dual of [16386, 16361, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(12825, 16386, F128, 13) (dual of [16386, 16361, 14]-code), using
- discarding factors based on OA(1644, 16386, S16, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1644, 16381, S16, 13), using
(31, 31+13, 9176)-Net over F16 — Digital
Digital (31, 44, 9176)-net over F16, using
(31, 31+13, large)-Net in Base 16 — Upper bound on s
There is no (31, 44, large)-net in base 16, because
- 11 times m-reduction [i] would yield (31, 33, large)-net in base 16, but