Best Known (20, 20+12, s)-Nets in Base 7
(20, 20+12, 113)-Net over F7 — Constructive and digital
Digital (20, 32, 113)-net over F7, using
- 1 times m-reduction [i] based on digital (20, 33, 113)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 13)-net over F7, using
- 6 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (13, 26, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 13, 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, 13, 50)-net over F49, using
- digital (1, 7, 13)-net over F7, using
- (u, u+v)-construction [i] based on
(20, 20+12, 309)-Net over F7 — Digital
Digital (20, 32, 309)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(732, 309, F7, 12) (dual of [309, 277, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(732, 350, F7, 12) (dual of [350, 318, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(731, 343, F7, 12) (dual of [343, 312, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(725, 343, F7, 10) (dual of [343, 318, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(71, 7, F7, 1) (dual of [7, 6, 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(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(732, 350, F7, 12) (dual of [350, 318, 13]-code), using
(20, 20+12, 16038)-Net in Base 7 — Upper bound on s
There is no (20, 32, 16039)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 1104 605622 199780 321633 039405 > 732 [i]