Best Known (64, 64+26, s)-Nets in Base 7
(64, 64+26, 216)-Net over F7 — Constructive and digital
Digital (64, 90, 216)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 12, 16)-net over F7, using
- 4 times m-reduction [i] based on digital (4, 16, 16)-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 (26, 52, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 26, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49 (see above)
- trace code for nets [i] based on digital (0, 26, 50)-net over F49, using
- digital (4, 12, 16)-net over F7, using
(64, 64+26, 2209)-Net over F7 — Digital
Digital (64, 90, 2209)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(790, 2209, F7, 26) (dual of [2209, 2119, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(790, 2410, F7, 26) (dual of [2410, 2320, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [i] 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]
- 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(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(25) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(790, 2410, F7, 26) (dual of [2410, 2320, 27]-code), using
(64, 64+26, 669774)-Net in Base 7 — Upper bound on s
There is no (64, 90, 669775)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 11450 670240 176045 569016 628024 726531 328331 241454 340566 352085 025934 135054 879251 > 790 [i]