Best Known (4, 9, s)-Nets in Base 7
(4, 9, 50)-Net over F7 — Constructive and digital
Digital (4, 9, 50)-net over F7, using
- base reduction for projective spaces (embedding PG(4,49) in PG(8,7)) for nets [i] based on digital (0, 5, 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
(4, 9, 53)-Net over F7 — Digital
Digital (4, 9, 53)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(79, 53, F7, 5) (dual of [53, 44, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(79, 67, F7, 5) (dual of [67, 58, 6]-code), using
- a “Gra†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(79, 67, F7, 5) (dual of [67, 58, 6]-code), using
(4, 9, 565)-Net in Base 7 — Upper bound on s
There is no (4, 9, 566)-net in base 7, because
- 1 times m-reduction [i] would yield (4, 8, 566)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 5 783389 > 78 [i]