Best Known (51, 65, s)-Nets in Base 4
(51, 65, 1038)-Net over F4 — Constructive and digital
Digital (51, 65, 1038)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (42, 56, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 14, 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, 14, 257)-net over F256, using
- digital (2, 9, 10)-net over F4, using
(51, 65, 2857)-Net over F4 — Digital
Digital (51, 65, 2857)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(465, 2857, F4, 14) (dual of [2857, 2792, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(465, 4113, F4, 14) (dual of [4113, 4048, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- 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(443, 4096, F4, 10) (dual of [4096, 4053, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(44, 17, F4, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(465, 4113, F4, 14) (dual of [4113, 4048, 15]-code), using
(51, 65, 438883)-Net in Base 4 — Upper bound on s
There is no (51, 65, 438884)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1361 141978 968661 108404 188426 764271 057160 > 465 [i]