Best Known (52−15, 52, s)-Nets in Base 16
(52−15, 52, 1148)-Net over F16 — Constructive and digital
Digital (37, 52, 1148)-net over F16, using
- 161 times duplication [i] based on digital (36, 51, 1148)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 7, 120)-net over F16, using
- net defined by OOA [i] based on linear OOA(167, 120, F16, 5, 5) (dual of [(120, 5), 593, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- net defined by OOA [i] based on linear OOA(167, 120, F16, 5, 5) (dual of [(120, 5), 593, 6]-NRT-code), using
- digital (7, 14, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 7, 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, 7, 257)-net over F256, using
- digital (15, 30, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 15, 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, 15, 257)-net over F256, using
- digital (2, 7, 120)-net over F16, using
- generalized (u, u+v)-construction [i] based on
(52−15, 52, 2340)-Net in Base 16 — Constructive
(37, 52, 2340)-net in base 16, using
- 161 times duplication [i] based on (36, 51, 2340)-net in base 16, using
- net defined by OOA [i] based on OOA(1651, 2340, S16, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(1651, 16381, S16, 15), using
- discarding factors based on OA(1651, 16386, S16, 15), using
- discarding parts of the base [i] based on linear OA(12829, 16386, F128, 15) (dual of [16386, 16357, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(12829, 16384, F128, 15) (dual of [16384, 16355, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(12827, 16384, F128, 14) (dual of [16384, 16357, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(14) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(12829, 16386, F128, 15) (dual of [16386, 16357, 16]-code), using
- discarding factors based on OA(1651, 16386, S16, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(1651, 16381, S16, 15), using
- net defined by OOA [i] based on OOA(1651, 2340, S16, 15, 15), using
(52−15, 52, 11969)-Net over F16 — Digital
Digital (37, 52, 11969)-net over F16, using
(52−15, 52, large)-Net in Base 16 — Upper bound on s
There is no (37, 52, large)-net in base 16, because
- 13 times m-reduction [i] would yield (37, 39, large)-net in base 16, but