Best Known (83−24, 83, s)-Nets in Base 7
(83−24, 83, 215)-Net over F7 — Constructive and digital
Digital (59, 83, 215)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (1, 9, 13)-net over F7, using
- 4 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (12, 24, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 12, 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, 12, 50)-net over F49, using
- digital (26, 50, 102)-net over F7, using
- trace code for nets [i] based on digital (1, 25, 51)-net over F49, using
- net from sequence [i] based on digital (1, 50)-sequence over F49, using
- trace code for nets [i] based on digital (1, 25, 51)-net over F49, using
- digital (1, 9, 13)-net over F7, using
(83−24, 83, 2118)-Net over F7 — Digital
Digital (59, 83, 2118)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(783, 2118, F7, 24) (dual of [2118, 2035, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(783, 2411, F7, 24) (dual of [2411, 2328, 25]-code), using
- 1 times code embedding in larger space [i] based on linear OA(782, 2410, F7, 24) (dual of [2410, 2328, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(781, 2401, F7, 24) (dual of [2401, 2320, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(773, 2401, F7, 22) (dual of [2401, 2328, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(71, 9, F7, 1) (dual of [9, 8, 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(23) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(782, 2410, F7, 24) (dual of [2410, 2328, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(783, 2411, F7, 24) (dual of [2411, 2328, 25]-code), using
(83−24, 83, 617257)-Net in Base 7 — Upper bound on s
There is no (59, 83, 617258)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 13904 074645 909252 314360 989511 116384 304560 079798 908972 719690 037924 623129 > 783 [i]