Best Known (42, 42+12, s)-Nets in Base 4
(42, 42+12, 1033)-Net over F4 — Constructive and digital
Digital (42, 54, 1033)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (36, 48, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 12, 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, 12, 257)-net over F256, using
- digital (0, 6, 5)-net over F4, using
(42, 42+12, 2336)-Net over F4 — Digital
Digital (42, 54, 2336)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(454, 2336, F4, 12) (dual of [2336, 2282, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(454, 4095, F4, 12) (dual of [4095, 4041, 13]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(455, 4096, F4, 13) (dual of [4096, 4041, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(454, 4095, F4, 12) (dual of [4095, 4041, 13]-code), using
(42, 42+12, 261597)-Net in Base 4 — Upper bound on s
There is no (42, 54, 261598)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 324 523664 664292 279025 639983 170100 > 454 [i]