Best Known (11−5, 11, s)-Nets in Base 7
(11−5, 11, 100)-Net over F7 — Constructive and digital
Digital (6, 11, 100)-net over F7, using
- 1 times m-reduction [i] based on digital (6, 12, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 6, 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, 6, 50)-net over F49, using
(11−5, 11, 129)-Net over F7 — Digital
Digital (6, 11, 129)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(711, 129, F7, 5) (dual of [129, 118, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(71, 43, F7, 1) (dual of [43, 42, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(73, 43, F7, 2) (dual of [43, 40, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- linear OA(71, 43, F7, 1) (dual of [43, 42, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
(11−5, 11, 3960)-Net in Base 7 — Upper bound on s
There is no (6, 11, 3961)-net in base 7, because
- 1 times m-reduction [i] would yield (6, 10, 3961)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 282 530209 > 710 [i]