Best Known (199, 199+46, s)-Nets in Base 4
(199, 199+46, 1544)-Net over F4 — Constructive and digital
Digital (199, 245, 1544)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 23, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (176, 222, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 74, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 74, 513)-net over F64, using
- digital (0, 23, 5)-net over F4, using
(199, 199+46, 12510)-Net over F4 — Digital
Digital (199, 245, 12510)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4245, 12510, F4, 46) (dual of [12510, 12265, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(4245, 16412, F4, 46) (dual of [16412, 16167, 47]-code), using
- construction XX applied to Ce(45) ⊂ Ce(41) ⊂ Ce(40) [i] based on
- linear OA(4239, 16384, F4, 46) (dual of [16384, 16145, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(4218, 16384, F4, 42) (dual of [16384, 16166, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(4211, 16384, F4, 41) (dual of [16384, 16173, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(45, 27, F4, 3) (dual of [27, 22, 4]-code or 27-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(45) ⊂ Ce(41) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(4245, 16412, F4, 46) (dual of [16412, 16167, 47]-code), using
(199, 199+46, 8139200)-Net in Base 4 — Upper bound on s
There is no (199, 245, 8139201)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 3196 672407 784446 524771 019476 780387 602099 206439 241068 374781 550130 397892 550033 411837 286742 703225 966649 655914 024068 400368 997494 364977 696719 252788 853824 > 4245 [i]