Best Known (69, 69+28, s)-Nets in Base 16
(69, 69+28, 1073)-Net over F16 — Constructive and digital
Digital (69, 97, 1073)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 13, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- digital (14, 28, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 14, 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, 14, 257)-net over F256, using
- digital (28, 56, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 28, 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, 28, 257)-net over F256, using
- digital (4, 13, 45)-net over F16, using
(69, 69+28, 1170)-Net in Base 16 — Constructive
(69, 97, 1170)-net in base 16, using
- net defined by OOA [i] based on OOA(1697, 1170, S16, 28, 28), using
- OA 14-folding and stacking [i] based on OA(1697, 16380, S16, 28), using
- discarding factors based on OA(1697, 16386, S16, 28), using
- discarding parts of the base [i] based on linear OA(12855, 16386, F128, 28) (dual of [16386, 16331, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(26) [i] based on
- linear OA(12855, 16384, F128, 28) (dual of [16384, 16329, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(12853, 16384, F128, 27) (dual of [16384, 16331, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(27) ⊂ Ce(26) [i] based on
- discarding parts of the base [i] based on linear OA(12855, 16386, F128, 28) (dual of [16386, 16331, 29]-code), using
- discarding factors based on OA(1697, 16386, S16, 28), using
- OA 14-folding and stacking [i] based on OA(1697, 16380, S16, 28), using
(69, 69+28, 15439)-Net over F16 — Digital
Digital (69, 97, 15439)-net over F16, using
(69, 69+28, large)-Net in Base 16 — Upper bound on s
There is no (69, 97, large)-net in base 16, because
- 26 times m-reduction [i] would yield (69, 71, large)-net in base 16, but