Best Known (171−31, 171, s)-Nets in Base 4
(171−31, 171, 1223)-Net over F4 — Constructive and digital
Digital (140, 171, 1223)-net over F4, using
- 42 times duplication [i] based on digital (138, 169, 1223)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (30, 45, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 15, 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, 15, 65)-net over F64, using
- digital (93, 124, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 31, 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, 31, 257)-net over F256, using
- digital (30, 45, 195)-net over F4, using
- (u, u+v)-construction [i] based on
(171−31, 171, 13139)-Net over F4 — Digital
Digital (140, 171, 13139)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4171, 13139, F4, 31) (dual of [13139, 12968, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4171, 16421, F4, 31) (dual of [16421, 16250, 32]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4169, 16419, F4, 31) (dual of [16419, 16250, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(4162, 16384, F4, 31) (dual of [16384, 16222, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4134, 16384, F4, 26) (dual of [16384, 16250, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4169, 16419, F4, 31) (dual of [16419, 16250, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4171, 16421, F4, 31) (dual of [16421, 16250, 32]-code), using
(171−31, 171, large)-Net in Base 4 — Upper bound on s
There is no (140, 171, large)-net in base 4, because
- 29 times m-reduction [i] would yield (140, 142, large)-net in base 4, but