Best Known (84−18, 84, s)-Nets in Base 4
(84−18, 84, 1042)-Net over F4 — Constructive and digital
Digital (66, 84, 1042)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-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 (3, 12, 14)-net over F4, using
(84−18, 84, 2998)-Net over F4 — Digital
Digital (66, 84, 2998)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(484, 2998, F4, 18) (dual of [2998, 2914, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(484, 4119, F4, 18) (dual of [4119, 4035, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [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(45, 23, F4, 3) (dual of [23, 18, 4]-code or 23-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
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(484, 4119, F4, 18) (dual of [4119, 4035, 19]-code), using
(84−18, 84, 575243)-Net in Base 4 — Upper bound on s
There is no (66, 84, 575244)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 374 149095 905643 308089 888771 448746 468417 289053 564057 > 484 [i]