Best Known (85−18, 85, s)-Nets in Base 4
(85−18, 85, 1043)-Net over F4 — Constructive and digital
Digital (67, 85, 1043)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-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 (4, 13, 15)-net over F4, using
(85−18, 85, 3271)-Net over F4 — Digital
Digital (67, 85, 3271)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(485, 3271, F4, 18) (dual of [3271, 3186, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(485, 4121, F4, 18) (dual of [4121, 4036, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ 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(461, 4096, F4, 14) (dual of [4096, 4035, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(45, 24, F4, 3) (dual of [24, 19, 4]-code or 24-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(485, 4121, F4, 18) (dual of [4121, 4036, 19]-code), using
(85−18, 85, 671039)-Net in Base 4 — Upper bound on s
There is no (67, 85, 671040)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1496 596453 176525 537522 449275 408215 774149 305310 842825 > 485 [i]