Best Known (51, 51+21, s)-Nets in Base 16
(51, 51+21, 1066)-Net over F16 — Constructive and digital
Digital (51, 72, 1066)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (3, 10, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- digital (10, 20, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 10, 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, 10, 257)-net over F256, using
- digital (21, 42, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 21, 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, 21, 257)-net over F256, using
- digital (3, 10, 38)-net over F16, using
(51, 51+21, 1638)-Net in Base 16 — Constructive
(51, 72, 1638)-net in base 16, using
- net defined by OOA [i] based on OOA(1672, 1638, S16, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(1672, 16381, S16, 21), using
- discarding factors based on OA(1672, 16386, S16, 21), using
- discarding parts of the base [i] based on linear OA(12841, 16386, F128, 21) (dual of [16386, 16345, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(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(20) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(12841, 16386, F128, 21) (dual of [16386, 16345, 22]-code), using
- discarding factors based on OA(1672, 16386, S16, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(1672, 16381, S16, 21), using
(51, 51+21, 11979)-Net over F16 — Digital
Digital (51, 72, 11979)-net over F16, using
(51, 51+21, large)-Net in Base 16 — Upper bound on s
There is no (51, 72, large)-net in base 16, because
- 19 times m-reduction [i] would yield (51, 53, large)-net in base 16, but