Best Known (18−7, 18, s)-Nets in Base 7
(18−7, 18, 150)-Net over F7 — Constructive and digital
Digital (11, 18, 150)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 50)-net over F7, using
- net defined by OOA [i] based on linear OOA(74, 50, F7, 3, 3) (dual of [(50, 3), 146, 4]-NRT-code), using
- digital (7, 14, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 7, 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, 7, 50)-net over F49, using
- digital (1, 4, 50)-net over F7, using
(18−7, 18, 322)-Net over F7 — Digital
Digital (11, 18, 322)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(718, 322, F7, 7) (dual of [322, 304, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(718, 342, F7, 7) (dual of [342, 324, 8]-code), using
- the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(718, 342, F7, 7) (dual of [342, 324, 8]-code), using
(18−7, 18, 18624)-Net in Base 7 — Upper bound on s
There is no (11, 18, 18625)-net in base 7, because
- 1 times m-reduction [i] would yield (11, 17, 18625)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 232 646363 025751 > 717 [i]