Best Known (57, 79, s)-Nets in Base 16
(57, 79, 1285)-Net over F16 — Constructive and digital
Digital (57, 79, 1285)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (6, 13, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(6,256) in PG(12,16)) 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
- base reduction for projective spaces (embedding PG(6,256) in PG(12,16)) for nets [i] based on digital (0, 7, 257)-net over F256, using
- digital (11, 22, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 11, 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, 11, 257)-net over F256, using
- digital (22, 44, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 22, 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, 22, 257)-net over F256, using
- digital (6, 13, 257)-net over F16, using
(57, 79, 1490)-Net in Base 16 — Constructive
(57, 79, 1490)-net in base 16, using
- 161 times duplication [i] based on (56, 78, 1490)-net in base 16, using
- net defined by OOA [i] based on OOA(1678, 1490, S16, 22, 22), using
- OA 11-folding and stacking [i] based on OA(1678, 16390, S16, 22), using
- 1 times code embedding in larger space [i] based on OA(1677, 16389, S16, 22), using
- discarding parts of the base [i] based on linear OA(12844, 16389, F128, 22) (dual of [16389, 16345, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(12843, 16384, F128, 22) (dual of [16384, 16341, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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 Ce(21) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(12844, 16389, F128, 22) (dual of [16389, 16345, 23]-code), using
- 1 times code embedding in larger space [i] based on OA(1677, 16389, S16, 22), using
- OA 11-folding and stacking [i] based on OA(1678, 16390, S16, 22), using
- net defined by OOA [i] based on OOA(1678, 1490, S16, 22, 22), using
(57, 79, 19607)-Net over F16 — Digital
Digital (57, 79, 19607)-net over F16, using
(57, 79, large)-Net in Base 16 — Upper bound on s
There is no (57, 79, large)-net in base 16, because
- 20 times m-reduction [i] would yield (57, 59, large)-net in base 16, but