Best Known (68, 86, s)-Nets in Base 4
(68, 86, 1045)-Net over F4 — Constructive and digital
Digital (68, 86, 1045)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 14, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (54, 72, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 18, 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, 18, 257)-net over F256, using
- digital (5, 14, 17)-net over F4, using
(68, 86, 3568)-Net over F4 — Digital
Digital (68, 86, 3568)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(486, 3568, F4, 18) (dual of [3568, 3482, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(486, 4127, F4, 18) (dual of [4127, 4041, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(479, 4096, F4, 18) (dual of [4096, 4017, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(455, 4096, F4, 13) (dual of [4096, 4041, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(47, 31, F4, 4) (dual of [31, 24, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(486, 4127, F4, 18) (dual of [4127, 4041, 19]-code), using
(68, 86, 782787)-Net in Base 4 — Upper bound on s
There is no (68, 86, 782788)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 5986 329897 105921 952383 928993 784687 151933 799509 225160 > 486 [i]