Best Known (57, 57+16, s)-Nets in Base 4
(57, 57+16, 1037)-Net over F4 — Constructive and digital
Digital (57, 73, 1037)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 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 (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 (1, 9, 9)-net over F4, using
(57, 57+16, 2505)-Net over F4 — Digital
Digital (57, 73, 2505)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(473, 2505, F4, 16) (dual of [2505, 2432, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(473, 4102, F4, 16) (dual of [4102, 4029, 17]-code), using
- 1 times truncation [i] based on linear OA(474, 4103, F4, 17) (dual of [4103, 4029, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [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(467, 4096, F4, 15) (dual of [4096, 4029, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 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(14) [i] based on
- 1 times truncation [i] based on linear OA(474, 4103, F4, 17) (dual of [4103, 4029, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(473, 4102, F4, 16) (dual of [4102, 4029, 17]-code), using
(57, 57+16, 391164)-Net in Base 4 — Upper bound on s
There is no (57, 73, 391165)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 89 203972 803953 775625 818738 695476 910880 943045 > 473 [i]