Best Known (161−30, 161, s)-Nets in Base 4
(161−30, 161, 1126)-Net over F4 — Constructive and digital
Digital (131, 161, 1126)-net over F4, using
- 41 times duplication [i] based on digital (130, 160, 1126)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (25, 40, 98)-net over F4, using
- trace code for nets [i] based on digital (5, 20, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- trace code for nets [i] based on digital (5, 20, 49)-net over F16, using
- digital (90, 120, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 30, 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, 30, 257)-net over F256, using
- digital (25, 40, 98)-net over F4, using
- (u, u+v)-construction [i] based on
(161−30, 161, 10359)-Net over F4 — Digital
Digital (131, 161, 10359)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4161, 10359, F4, 30) (dual of [10359, 10198, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4161, 16412, F4, 30) (dual of [16412, 16251, 31]-code), using
- construction XX applied to Ce(29) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4155, 16384, F4, 30) (dual of [16384, 16229, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(29) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4161, 16412, F4, 30) (dual of [16412, 16251, 31]-code), using
(161−30, 161, 6205190)-Net in Base 4 — Upper bound on s
There is no (131, 161, 6205191)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 8 543958 229151 396968 789271 771064 233665 883002 450358 302785 835055 494971 967921 549045 345917 946762 352512 > 4161 [i]