Best Known (30, 30+19, s)-Nets in Base 7
(30, 30+19, 113)-Net over F7 — Constructive and digital
Digital (30, 49, 113)-net over F7, using
- 71 times duplication [i] based on digital (29, 48, 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 (19, 38, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 19, 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, 19, 50)-net over F49, using
- digital (1, 10, 13)-net over F7, using
- (u, u+v)-construction [i] based on
(30, 30+19, 282)-Net over F7 — Digital
Digital (30, 49, 282)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(749, 282, F7, 19) (dual of [282, 233, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(749, 342, F7, 19) (dual of [342, 293, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(749, 342, F7, 19) (dual of [342, 293, 20]-code), using
(30, 30+19, 22216)-Net in Base 7 — Upper bound on s
There is no (30, 49, 22217)-net in base 7, because
- 1 times m-reduction [i] would yield (30, 48, 22217)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 36705 772845 878518 575338 671069 084640 462455 > 748 [i]