Best Known (38, 38+15, s)-Nets in Base 16
(38, 38+15, 1285)-Net over F16 — Constructive and digital
Digital (38, 53, 1285)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 9, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(4,256) in PG(8,16)) for nets [i] based on digital (0, 5, 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
- base reduction for projective spaces (embedding PG(4,256) in PG(8,16)) for nets [i] based on digital (0, 5, 257)-net over F256, 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 (see above)
- 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 (4, 9, 257)-net over F16, using
(38, 38+15, 2341)-Net in Base 16 — Constructive
(38, 53, 2341)-net in base 16, using
- net defined by OOA [i] based on OOA(1653, 2341, S16, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(1653, 16388, S16, 15), using
- discarding factors based on OA(1653, 16390, S16, 15), using
- discarding parts of the base [i] based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 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([0,7]) ⊂ C([0,6]) [i] based on
- discarding parts of the base [i] based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- discarding factors based on OA(1653, 16390, S16, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(1653, 16388, S16, 15), using
(38, 38+15, 14589)-Net over F16 — Digital
Digital (38, 53, 14589)-net over F16, using
(38, 38+15, large)-Net in Base 16 — Upper bound on s
There is no (38, 53, large)-net in base 16, because
- 13 times m-reduction [i] would yield (38, 40, large)-net in base 16, but