Best Known (42, 58, s)-Nets in Base 16
(42, 58, 1542)-Net over F16 — Constructive and digital
Digital (42, 58, 1542)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (5, 10, 514)-net over F16, using
- trace code 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
- trace code for nets [i] based on digital (0, 5, 257)-net over F256, using
- digital (8, 16, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 8, 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, 8, 257)-net over F256, using
- digital (16, 32, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 16, 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, 16, 257)-net over F256, using
- digital (5, 10, 514)-net over F16, using
(42, 58, 4096)-Net in Base 16 — Constructive
(42, 58, 4096)-net in base 16, using
- net defined by OOA [i] based on OOA(1658, 4096, S16, 16, 16), using
- OA 8-folding and stacking [i] based on OA(1658, 32768, S16, 16), using
- discarding factors based on OA(1658, 32771, S16, 16), using
- discarding parts of the base [i] based on linear OA(3246, 32771, F32, 16) (dual of [32771, 32725, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(3246, 32768, F32, 16) (dual of [32768, 32722, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(3243, 32768, F32, 15) (dual of [32768, 32725, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(15) ⊂ Ce(14) [i] based on
- discarding parts of the base [i] based on linear OA(3246, 32771, F32, 16) (dual of [32771, 32725, 17]-code), using
- discarding factors based on OA(1658, 32771, S16, 16), using
- OA 8-folding and stacking [i] based on OA(1658, 32768, S16, 16), using
(42, 58, 19399)-Net over F16 — Digital
Digital (42, 58, 19399)-net over F16, using
(42, 58, large)-Net in Base 16 — Upper bound on s
There is no (42, 58, large)-net in base 16, because
- 14 times m-reduction [i] would yield (42, 44, large)-net in base 16, but