Best Known (40, 40+19, s)-Nets in Base 4
(40, 40+19, 195)-Net over F4 — Constructive and digital
Digital (40, 59, 195)-net over F4, using
- 1 times m-reduction [i] based on digital (40, 60, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 20, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 20, 65)-net over F64, using
(40, 40+19, 259)-Net over F4 — Digital
Digital (40, 59, 259)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(459, 259, F4, 19) (dual of [259, 200, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(459, 270, F4, 19) (dual of [270, 211, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(455, 256, F4, 19) (dual of [256, 201, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(445, 256, F4, 15) (dual of [256, 211, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(44, 14, F4, 3) (dual of [14, 10, 4]-code or 14-cap in PG(3,4)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(459, 270, F4, 19) (dual of [270, 211, 20]-code), using
(40, 40+19, 10478)-Net in Base 4 — Upper bound on s
There is no (40, 59, 10479)-net in base 4, because
- 1 times m-reduction [i] would yield (40, 58, 10479)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 83141 193194 973785 883916 213274 579626 > 458 [i]