Best Known (21, 21+13, s)-Nets in Base 7
(21, 21+13, 113)-Net over F7 — Constructive and digital
Digital (21, 34, 113)-net over F7, using
- 71 times duplication [i] based on digital (20, 33, 113)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 13)-net over F7, using
- 6 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 (1, 7, 13)-net over F7, using
- (u, u+v)-construction [i] based on
(21, 21+13, 275)-Net over F7 — Digital
Digital (21, 34, 275)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(734, 275, F7, 13) (dual of [275, 241, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(734, 342, F7, 13) (dual of [342, 308, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(734, 342, F7, 13) (dual of [342, 308, 14]-code), using
(21, 21+13, 22184)-Net in Base 7 — Upper bound on s
There is no (21, 34, 22185)-net in base 7, because
- 1 times m-reduction [i] would yield (21, 33, 22185)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 7732 904477 354144 172147 369313 > 733 [i]