Best Known (46−18, 46, s)-Nets in Base 7
(46−18, 46, 113)-Net over F7 — Constructive and digital
Digital (28, 46, 113)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 13)-net over F7, using
- 3 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (18, 36, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 18, 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, 18, 50)-net over F49, using
- digital (1, 10, 13)-net over F7, using
(46−18, 46, 261)-Net over F7 — Digital
Digital (28, 46, 261)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(746, 261, F7, 18) (dual of [261, 215, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(746, 342, F7, 18) (dual of [342, 296, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(746, 342, F7, 18) (dual of [342, 296, 19]-code), using
(46−18, 46, 14415)-Net in Base 7 — Upper bound on s
There is no (28, 46, 14416)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 749 362463 730647 894539 647376 881049 769185 > 746 [i]