Best Known (59−13, 59, s)-Nets in Base 4
(59−13, 59, 1037)-Net over F4 — Constructive and digital
Digital (46, 59, 1037)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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 (39, 52, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 13, 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, 13, 257)-net over F256, using
- digital (1, 7, 9)-net over F4, using
(59−13, 59, 2438)-Net over F4 — Digital
Digital (46, 59, 2438)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(459, 2438, F4, 13) (dual of [2438, 2379, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(459, 4113, F4, 13) (dual of [4113, 4054, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(455, 4096, F4, 13) (dual of [4096, 4041, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(437, 4096, F4, 9) (dual of [4096, 4059, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(12) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(459, 4113, F4, 13) (dual of [4113, 4054, 14]-code), using
(59−13, 59, 659190)-Net in Base 4 — Upper bound on s
There is no (46, 59, 659191)-net in base 4, because
- 1 times m-reduction [i] would yield (46, 58, 659191)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 83076 929308 760998 407542 852040 079237 > 458 [i]