Best Known (56−22, 56, s)-Nets in Base 7
(56−22, 56, 113)-Net over F7 — Constructive and digital
Digital (34, 56, 113)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 13)-net over F7, using
- 1 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (22, 44, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 22, 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, 22, 50)-net over F49, using
- digital (1, 12, 13)-net over F7, using
(56−22, 56, 281)-Net over F7 — Digital
Digital (34, 56, 281)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(756, 281, F7, 22) (dual of [281, 225, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(756, 347, F7, 22) (dual of [347, 291, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(755, 343, F7, 22) (dual of [343, 288, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(752, 343, F7, 20) (dual of [343, 291, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(756, 347, F7, 22) (dual of [347, 291, 23]-code), using
(56−22, 56, 16405)-Net in Base 7 — Upper bound on s
There is no (34, 56, 16406)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 211605 480418 193904 919772 996193 400991 630150 937721 > 756 [i]