Best Known (69, 69+18, s)-Nets in Base 4
(69, 69+18, 1046)-Net over F4 — Constructive and digital
Digital (69, 87, 1046)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (6, 15, 18)-net over F4, using
- 3 times m-reduction [i] based on digital (6, 18, 18)-net 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 (6, 15, 18)-net over F4, using
(69, 69+18, 3892)-Net over F4 — Digital
Digital (69, 87, 3892)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(487, 3892, F4, 18) (dual of [3892, 3805, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(487, 4128, F4, 18) (dual of [4128, 4041, 19]-code), using
- 1 times code embedding in larger space [i] 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
- 1 times code embedding in larger space [i] based on linear OA(486, 4127, F4, 18) (dual of [4127, 4041, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(487, 4128, F4, 18) (dual of [4128, 4041, 19]-code), using
(69, 69+18, 913145)-Net in Base 4 — Upper bound on s
There is no (69, 87, 913146)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 23945 296099 571746 276040 374121 186765 823961 553375 092261 > 487 [i]