Best Known (31, 31+12, s)-Nets in Base 16
(31, 31+12, 1285)-Net over F16 — Constructive and digital
Digital (31, 43, 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 (12, 24, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 12, 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, 12, 257)-net over F256, using
- digital (3, 7, 257)-net over F16, using
(31, 31+12, 5461)-Net in Base 16 — Constructive
(31, 43, 5461)-net in base 16, using
- net defined by OOA [i] based on OOA(1643, 5461, S16, 12, 12), using
- OA 6-folding and stacking [i] based on OA(1643, 32766, S16, 12), using
- discarding factors based on OA(1643, 32771, S16, 12), using
- discarding parts of the base [i] based on linear OA(3234, 32771, F32, 12) (dual of [32771, 32737, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(3234, 32768, F32, 12) (dual of [32768, 32734, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(3231, 32768, F32, 11) (dual of [32768, 32737, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(320, 3, F32, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding parts of the base [i] based on linear OA(3234, 32771, F32, 12) (dual of [32771, 32737, 13]-code), using
- discarding factors based on OA(1643, 32771, S16, 12), using
- OA 6-folding and stacking [i] based on OA(1643, 32766, S16, 12), using
(31, 31+12, 16676)-Net over F16 — Digital
Digital (31, 43, 16676)-net over F16, using
(31, 31+12, large)-Net in Base 16 — Upper bound on s
There is no (31, 43, large)-net in base 16, because
- 10 times m-reduction [i] would yield (31, 33, large)-net in base 16, but