Best Known (1, 1+∞, s)-Nets in Base 9
(1, 1+∞, 16)-Net over F9 — Constructive and digital
Digital (1, m, 16)-net over F9 for arbitrarily large m, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 1 and N(F) ≥ 16, using
(1, 1+∞, 16)-Net over F9 — Upper bound on s (digital)
There is no digital (1, m, 17)-net over F9 for arbitrarily large m, because
- m-reduction [i] would yield digital (1, 11, 17)-net over F9, but
- extracting embedded orthogonal array [i] would yield linear OA(911, 17, F9, 10) (dual of [17, 6, 11]-code), but
- “MPa†bound on codes from Brouwer’s database [i]
- extracting embedded orthogonal array [i] would yield linear OA(911, 17, F9, 10) (dual of [17, 6, 11]-code), but
(1, 1+∞, 19)-Net in Base 9 — Upper bound on s
There is no (1, m, 20)-net in base 9 for arbitrarily large m, because
- m-reduction [i] would yield (1, 19, 20)-net in base 9, but
- extracting embedded OOA [i] would yield OOA(919, 20, S9, 2, 18), but
- the linear programming bound for OOAs shows that M ≥ 498 464283 821334 080841 / 361 > 919 [i]
- extracting embedded OOA [i] would yield OOA(919, 20, S9, 2, 18), but