Best Known (32, 45, s)-Nets in Base 16
(32, 45, 1285)-Net over F16 — Constructive and digital
Digital (32, 45, 1285)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (3, 7, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(3,256) in PG(6,16)) 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
- base reduction for projective spaces (embedding PG(3,256) in PG(6,16)) for nets [i] based on digital (0, 4, 257)-net over F256, 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 (see above)
- 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 (3, 7, 257)-net over F16, using
(32, 45, 2731)-Net in Base 16 — Constructive
(32, 45, 2731)-net in base 16, using
- net defined by OOA [i] based on OOA(1645, 2731, S16, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1645, 16387, S16, 13), using
- 1 times code embedding in larger space [i] 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
- 1 times code embedding in larger space [i] based on OA(1644, 16386, S16, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(1645, 16387, S16, 13), using
(32, 45, 11560)-Net over F16 — Digital
Digital (32, 45, 11560)-net over F16, using
(32, 45, large)-Net in Base 16 — Upper bound on s
There is no (32, 45, large)-net in base 16, because
- 11 times m-reduction [i] would yield (32, 34, large)-net in base 16, but