Best Known (63, 89, s)-Nets in Base 7
(63, 89, 215)-Net over F7 — Constructive and digital
Digital (63, 89, 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 (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 (28, 54, 102)-net over F7, using
- trace code for nets [i] based on digital (1, 27, 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, 27, 51)-net over F49, using
- digital (1, 9, 13)-net over F7, using
(63, 89, 2036)-Net over F7 — Digital
Digital (63, 89, 2036)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(789, 2036, F7, 26) (dual of [2036, 1947, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(789, 2401, F7, 26) (dual of [2401, 2312, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(789, 2401, F7, 26) (dual of [2401, 2312, 27]-code), using
(63, 89, 576660)-Net in Base 7 — Upper bound on s
There is no (63, 89, 576661)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 1635 812202 848525 575302 860271 772859 338912 514784 020185 904907 381026 868115 772479 > 789 [i]