Best Known (42, 42+20, s)-Nets in Base 4
(42, 42+20, 195)-Net over F4 — Constructive and digital
Digital (42, 62, 195)-net over F4, using
- 1 times m-reduction [i] based on digital (42, 63, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 21, 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, 21, 65)-net over F64, using
(42, 42+20, 264)-Net over F4 — Digital
Digital (42, 62, 264)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(462, 264, F4, 20) (dual of [264, 202, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(462, 269, F4, 20) (dual of [269, 207, 21]-code), using
- 1 times truncation [i] based on linear OA(463, 270, F4, 21) (dual of [270, 207, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(459, 256, F4, 21) (dual of [256, 197, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(449, 256, F4, 17) (dual of [256, 207, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(20) ⊂ Ce(16) [i] based on
- 1 times truncation [i] based on linear OA(463, 270, F4, 21) (dual of [270, 207, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(462, 269, F4, 20) (dual of [269, 207, 21]-code), using
(42, 42+20, 8150)-Net in Base 4 — Upper bound on s
There is no (42, 62, 8151)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 21 268093 290613 653703 582155 561671 787886 > 462 [i]