Best Known (42−16, 42, s)-Nets in Base 7
(42−16, 42, 113)-Net over F7 — Constructive and digital
Digital (26, 42, 113)-net over F7, using
- 1 times m-reduction [i] based on digital (26, 43, 113)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 13)-net over F7, using
- 4 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (17, 34, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 17, 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, 17, 50)-net over F49, using
- digital (1, 9, 13)-net over F7, using
- (u, u+v)-construction [i] based on
(42−16, 42, 293)-Net over F7 — Digital
Digital (26, 42, 293)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(742, 293, F7, 16) (dual of [293, 251, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(742, 342, F7, 16) (dual of [342, 300, 17]-code), using
- the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(742, 342, F7, 16) (dual of [342, 300, 17]-code), using
(42−16, 42, 17146)-Net in Base 7 — Upper bound on s
There is no (26, 42, 17147)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 312013 774891 598723 781150 344776 549585 > 742 [i]