Best Known (20−8, 20, s)-Nets in Base 5
(20−8, 20, 58)-Net over F5 — Constructive and digital
Digital (12, 20, 58)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (8, 16, 52)-net over F5, using
- trace code for nets [i] based on digital (0, 8, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using
- the rational function field F25(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- trace code for nets [i] based on digital (0, 8, 26)-net over F25, using
- digital (0, 4, 6)-net over F5, using
(20−8, 20, 119)-Net over F5 — Digital
Digital (12, 20, 119)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(520, 119, F5, 8) (dual of [119, 99, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(520, 132, F5, 8) (dual of [132, 112, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(519, 125, F5, 8) (dual of [125, 106, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(513, 125, F5, 6) (dual of [125, 112, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(51, 7, F5, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(520, 132, F5, 8) (dual of [132, 112, 9]-code), using
(20−8, 20, 1726)-Net in Base 5 — Upper bound on s
There is no (12, 20, 1727)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 95 435408 701425 > 520 [i]