Best Known (20−8, 20, s)-Nets in Base 7
(20−8, 20, 108)-Net over F7 — Constructive and digital
Digital (12, 20, 108)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (8, 16, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 8, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- trace code for nets [i] based on digital (0, 8, 50)-net over F49, using
- digital (0, 4, 8)-net over F7, using
(20−8, 20, 234)-Net over F7 — Digital
Digital (12, 20, 234)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(720, 234, F7, 8) (dual of [234, 214, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(720, 347, F7, 8) (dual of [347, 327, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(719, 343, F7, 8) (dual of [343, 324, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(716, 343, F7, 6) (dual of [343, 327, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(71, 4, F7, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 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(720, 347, F7, 8) (dual of [347, 327, 9]-code), using
(20−8, 20, 6197)-Net in Base 7 — Upper bound on s
There is no (12, 20, 6198)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 79800 779678 275369 > 720 [i]