Best Known (3, 7, s)-Nets in Base 7
(3, 7, 50)-Net over F7 — Constructive and digital
Digital (3, 7, 50)-net over F7, using
- base reduction for projective spaces (embedding PG(3,49) in PG(6,7)) for nets [i] based on digital (0, 4, 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
(3, 7, 70)-Net over F7 — Digital
Digital (3, 7, 70)-net over F7, using
- net defined by OOA [i] based on linear OOA(77, 70, F7, 4, 4) (dual of [(70, 4), 273, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(77, 70, F7, 3, 4) (dual of [(70, 3), 203, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(77, 70, F7, 4) (dual of [70, 63, 5]-code), using
- appending kth column [i] based on linear OOA(77, 70, F7, 3, 4) (dual of [(70, 3), 203, 5]-NRT-code), using
(3, 7, 212)-Net in Base 7 — Upper bound on s
There is no (3, 7, 213)-net in base 7, because
- extracting embedded orthogonal array [i] would yield OA(77, 213, S7, 4), but
- the linear programming bound shows that M ≥ 15829 806377 / 19127 > 77 [i]