Best Known (34−10, 34, s)-Nets in Base 16
(34−10, 34, 1285)-Net over F16 — Constructive and digital
Digital (24, 34, 1285)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (1, 4, 257)-net over F16, using
- net defined by OOA [i] based on linear OOA(164, 257, F16, 3, 3) (dual of [(257, 3), 767, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(164, 257, F16, 2, 3) (dual of [(257, 2), 510, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(164, 257, F16, 3, 3) (dual of [(257, 3), 767, 4]-NRT-code), 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 (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 (see above)
- trace code for nets [i] based on digital (0, 10, 257)-net over F256, using
- digital (1, 4, 257)-net over F16, using
(34−10, 34, 3277)-Net in Base 16 — Constructive
(24, 34, 3277)-net in base 16, using
- net defined by OOA [i] based on OOA(1634, 3277, S16, 10, 10), using
- OA 5-folding and stacking [i] based on OA(1634, 16385, S16, 10), using
- discarding factors based on OA(1634, 16386, S16, 10), using
- discarding parts of the base [i] based on linear OA(12819, 16386, F128, 10) (dual of [16386, 16367, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(12819, 16384, F128, 10) (dual of [16384, 16365, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(12817, 16384, F128, 9) (dual of [16384, 16367, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- discarding parts of the base [i] based on linear OA(12819, 16386, F128, 10) (dual of [16386, 16367, 11]-code), using
- discarding factors based on OA(1634, 16386, S16, 10), using
- OA 5-folding and stacking [i] based on OA(1634, 16385, S16, 10), using
(34−10, 34, 9789)-Net over F16 — Digital
Digital (24, 34, 9789)-net over F16, using
(34−10, 34, large)-Net in Base 16 — Upper bound on s
There is no (24, 34, large)-net in base 16, because
- 8 times m-reduction [i] would yield (24, 26, large)-net in base 16, but