Best Known (47, 47+13, s)-Nets in Base 4
(47, 47+13, 1040)-Net over F4 — Constructive and digital
Digital (47, 60, 1040)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 12)-net 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
(47, 47+13, 2766)-Net over F4 — Digital
Digital (47, 60, 2766)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(460, 2766, F4, 13) (dual of [2766, 2706, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(460, 4119, F4, 13) (dual of [4119, 4059, 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(45, 23, F4, 3) (dual of [23, 18, 4]-code or 23-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(460, 4119, F4, 13) (dual of [4119, 4059, 14]-code), using
(47, 47+13, 830529)-Net in Base 4 — Upper bound on s
There is no (47, 60, 830530)-net in base 4, because
- 1 times m-reduction [i] would yield (47, 59, 830530)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 332308 229923 851437 602091 045170 916704 > 459 [i]