Best Known (37, 61, s)-Nets in Base 7
(37, 61, 113)-Net over F7 — Constructive and digital
Digital (37, 61, 113)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 13)-net over F7, using
- digital (24, 48, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 24, 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, 24, 50)-net over F49, using
(37, 61, 293)-Net over F7 — Digital
Digital (37, 61, 293)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(761, 293, F7, 24) (dual of [293, 232, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(761, 342, F7, 24) (dual of [342, 281, 25]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,23], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(761, 342, F7, 24) (dual of [342, 281, 25]-code), using
(37, 61, 17415)-Net in Base 7 — Upper bound on s
There is no (37, 61, 17416)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 3557 152745 304941 361950 503412 500506 188592 556609 475521 > 761 [i]