Best Known (137, 168, s)-Nets in Base 4
(137, 168, 1158)-Net over F4 — Constructive and digital
Digital (137, 168, 1158)-net over F4, using
- 42 times duplication [i] based on digital (135, 166, 1158)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (27, 42, 130)-net over F4, using
- trace code for nets [i] based on digital (6, 21, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 21, 65)-net over F16, 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 (27, 42, 130)-net over F4, using
- (u, u+v)-construction [i] based on
(137, 168, 11380)-Net over F4 — Digital
Digital (137, 168, 11380)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4168, 11380, F4, 31) (dual of [11380, 11212, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4168, 16412, F4, 31) (dual of [16412, 16244, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(26) ⊂ 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(4141, 16384, F4, 27) (dual of [16384, 16243, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(45, 27, F4, 3) (dual of [27, 22, 4]-code or 27-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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(30) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4168, 16412, F4, 31) (dual of [16412, 16244, 32]-code), using
(137, 168, large)-Net in Base 4 — Upper bound on s
There is no (137, 168, large)-net in base 4, because
- 29 times m-reduction [i] would yield (137, 139, large)-net in base 4, but