Best Known (74−16, 74, s)-Nets in Base 4
(74−16, 74, 1038)-Net over F4 — Constructive and digital
Digital (58, 74, 1038)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 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
- a shift-net [i]
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (48, 64, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 16, 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, 16, 257)-net over F256, using
- digital (2, 10, 10)-net over F4, using
(74−16, 74, 2767)-Net over F4 — Digital
Digital (58, 74, 2767)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(474, 2767, F4, 16) (dual of [2767, 2693, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(474, 4109, F4, 16) (dual of [4109, 4035, 17]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(473, 4096, F4, 17) (dual of [4096, 4023, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(41, 13, F4, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(474, 4109, F4, 16) (dual of [4109, 4035, 17]-code), using
(74−16, 74, 465176)-Net in Base 4 — Upper bound on s
There is no (58, 74, 465177)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 356 813633 636859 427305 588019 582126 675326 581167 > 474 [i]