Best Known (86, 86+32, s)-Nets in Base 16
(86, 86+32, 1544)-Net over F16 — Constructive and digital
Digital (86, 118, 1544)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- 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 (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 (34, 66, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 33, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 33, 258)-net over F256, using
- digital (10, 20, 514)-net over F16, using
(86, 86+32, 2048)-Net in Base 16 — Constructive
(86, 118, 2048)-net in base 16, using
- net defined by OOA [i] based on OOA(16118, 2048, S16, 32, 32), using
- OA 16-folding and stacking [i] based on OA(16118, 32768, S16, 32), using
- discarding factors based on OA(16118, 32771, S16, 32), using
- discarding parts of the base [i] based on linear OA(3294, 32771, F32, 32) (dual of [32771, 32677, 33]-code), using
- 1 times truncation [i] based on linear OA(3295, 32772, F32, 33) (dual of [32772, 32677, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(3294, 32768, F32, 33) (dual of [32768, 32674, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3291, 32768, F32, 31) (dual of [32768, 32677, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(321, 4, F32, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- 1 times truncation [i] based on linear OA(3295, 32772, F32, 33) (dual of [32772, 32677, 34]-code), using
- discarding parts of the base [i] based on linear OA(3294, 32771, F32, 32) (dual of [32771, 32677, 33]-code), using
- discarding factors based on OA(16118, 32771, S16, 32), using
- OA 16-folding and stacking [i] based on OA(16118, 32768, S16, 32), using
(86, 86+32, 31738)-Net over F16 — Digital
Digital (86, 118, 31738)-net over F16, using
(86, 86+32, large)-Net in Base 16 — Upper bound on s
There is no (86, 118, large)-net in base 16, because
- 30 times m-reduction [i] would yield (86, 88, large)-net in base 16, but