Best Known (28, 28+11, s)-Nets in Base 16
(28, 28+11, 1559)-Net over F16 — Constructive and digital
Digital (28, 39, 1559)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 17)-net over F16, using
- digital (2, 5, 514)-net over F16, using
- s-reduction based on digital (2, 5, 629)-net over F16, using
- net defined by OOA [i] based on linear OOA(165, 629, F16, 3, 3) (dual of [(629, 3), 1882, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(165, 629, F16, 2, 3) (dual of [(629, 2), 1253, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(165, 629, F16, 3, 3) (dual of [(629, 3), 1882, 4]-NRT-code), using
- s-reduction based on digital (2, 5, 629)-net over F16, using
- 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 (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
(28, 28+11, 6554)-Net in Base 16 — Constructive
(28, 39, 6554)-net in base 16, using
- net defined by OOA [i] based on OOA(1639, 6554, S16, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(1639, 32771, S16, 11), using
- discarding parts of the base [i] based on linear OA(3231, 32771, F32, 11) (dual of [32771, 32740, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 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(3228, 32768, F32, 10) (dual of [32768, 32740, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- discarding parts of the base [i] based on linear OA(3231, 32771, F32, 11) (dual of [32771, 32740, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on OA(1639, 32771, S16, 11), using
(28, 28+11, 15000)-Net over F16 — Digital
Digital (28, 39, 15000)-net over F16, using
(28, 28+11, large)-Net in Base 16 — Upper bound on s
There is no (28, 39, large)-net in base 16, because
- 9 times m-reduction [i] would yield (28, 30, large)-net in base 16, but