Best Known (2, 18, s)-Nets in Base 7
(2, 18, 10)-Net over F7 — Constructive and digital
Digital (2, 18, 10)-net over F7, using
- net from sequence [i] based on digital (2, 9)-sequence over F7, using
- Niederreiter sequence (Bratley–Fox–Niederreiter implementation) with equidistant coordinate [i]
(2, 18, 16)-Net over F7 — Digital
Digital (2, 18, 16)-net over F7, using
- net from sequence [i] based on digital (2, 15)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 2 and N(F) ≥ 16, using
(2, 18, 23)-Net over F7 — Upper bound on s (digital)
There is no digital (2, 18, 24)-net over F7, because
- 2 times m-reduction [i] would yield digital (2, 16, 24)-net over F7, but
- extracting embedded orthogonal array [i] would yield linear OA(716, 24, F7, 14) (dual of [24, 8, 15]-code), but
- residual code [i] would yield OA(72, 9, S7, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 55 > 72 [i]
- residual code [i] would yield OA(72, 9, S7, 2), but
- extracting embedded orthogonal array [i] would yield linear OA(716, 24, F7, 14) (dual of [24, 8, 15]-code), but
(2, 18, 41)-Net in Base 7 — Upper bound on s
There is no (2, 18, 42)-net in base 7, because
- extracting embedded orthogonal array [i] would yield OA(718, 42, S7, 16), but
- the linear programming bound shows that M ≥ 3 975831 765629 104285 784425 / 2305 072363 > 718 [i]