Best Known (2, 2+11, s)-Nets in Base 9
(2, 2+11, 20)-Net over F9 — Constructive and digital
Digital (2, 13, 20)-net over F9, using
- net from sequence [i] based on digital (2, 19)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 2 and N(F) ≥ 20, using
(2, 2+11, 52)-Net over F9 — Upper bound on s (digital)
There is no digital (2, 13, 53)-net over F9, because
- extracting embedded orthogonal array [i] would yield linear OA(913, 53, F9, 11) (dual of [53, 40, 12]-code), but
- construction Y1 [i] would yield
- linear OA(912, 17, F9, 11) (dual of [17, 5, 12]-code), but
- “MPa†bound on codes from Brouwer’s database [i]
- linear OA(940, 53, F9, 36) (dual of [53, 13, 37]-code), but
- discarding factors / shortening the dual code would yield linear OA(940, 48, F9, 36) (dual of [48, 8, 37]-code), but
- linear OA(912, 17, F9, 11) (dual of [17, 5, 12]-code), but
- construction Y1 [i] would yield
(2, 2+11, 60)-Net in Base 9 — Upper bound on s
There is no (2, 13, 61)-net in base 9, because
- 1 times m-reduction [i] would yield (2, 12, 61)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 283877 043465 > 912 [i]