Best Known (82, 82+30, s)-Nets in Base 16
(82, 82+30, 1544)-Net over F16 — Constructive and digital
Digital (82, 112, 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 (15, 30, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 15, 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, 15, 257)-net over F256, using
- digital (32, 62, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 31, 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, 31, 258)-net over F256, using
- digital (10, 20, 514)-net over F16, using
(82, 82+30, 2185)-Net in Base 16 — Constructive
(82, 112, 2185)-net in base 16, using
- net defined by OOA [i] based on OOA(16112, 2185, S16, 30, 30), using
- OA 15-folding and stacking [i] based on OA(16112, 32775, S16, 30), using
- discarding parts of the base [i] based on linear OA(3289, 32775, F32, 30) (dual of [32775, 32686, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(27) [i] based on
- linear OA(3288, 32768, F32, 30) (dual of [32768, 32680, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(3282, 32768, F32, 28) (dual of [32768, 32686, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(321, 7, F32, 1) (dual of [7, 6, 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(29) ⊂ Ce(27) [i] based on
- discarding parts of the base [i] based on linear OA(3289, 32775, F32, 30) (dual of [32775, 32686, 31]-code), using
- OA 15-folding and stacking [i] based on OA(16112, 32775, S16, 30), using
(82, 82+30, 34802)-Net over F16 — Digital
Digital (82, 112, 34802)-net over F16, using
(82, 82+30, large)-Net in Base 16 — Upper bound on s
There is no (82, 112, large)-net in base 16, because
- 28 times m-reduction [i] would yield (82, 84, large)-net in base 16, but