Best Known (50, 50+14, s)-Nets in Base 4
(50, 50+14, 1037)-Net over F4 — Constructive and digital
Digital (50, 64, 1037)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-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 (1, 8, 9)-net over F4, using
(50, 50+14, 2544)-Net over F4 — Digital
Digital (50, 64, 2544)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(464, 2544, F4, 14) (dual of [2544, 2480, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(464, 4111, F4, 14) (dual of [4111, 4047, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [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(449, 4096, F4, 11) (dual of [4096, 4047, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(43, 15, F4, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(464, 4111, F4, 14) (dual of [4111, 4047, 15]-code), using
(50, 50+14, 360030)-Net in Base 4 — Upper bound on s
There is no (50, 64, 360031)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 340 284805 456110 999431 987691 921005 999212 > 464 [i]