Best Known (37−15, 37, s)-Nets in Base 7
(37−15, 37, 108)-Net over F7 — Constructive and digital
Digital (22, 37, 108)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (15, 30, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 15, 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, 15, 50)-net over F49, using
- digital (0, 7, 8)-net over F7, using
(37−15, 37, 200)-Net over F7 — Digital
Digital (22, 37, 200)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(737, 200, F7, 15) (dual of [200, 163, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(737, 342, F7, 15) (dual of [342, 305, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(737, 342, F7, 15) (dual of [342, 305, 16]-code), using
(37−15, 37, 12498)-Net in Base 7 — Upper bound on s
There is no (22, 37, 12499)-net in base 7, because
- 1 times m-reduction [i] would yield (22, 36, 12499)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 2 653189 319456 898596 860272 059959 > 736 [i]