MinT - Best Known (25−23, 25, s)-Nets in Base 9

Best Known (25−23, 25, s)-Nets in Base 9

(25−23, 25, 20)-Net over F₉ — Constructive and digital

Digital (2, 25, 20)-net over F₉, using

net from sequence [i] based on digital (2, 19)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
  - a function field by GarcÃa and Quoos [i]

(25−23, 25, 29)-Net over F₉ — Upper bound on s (digital)

There is no digital (2, 25, 30)-net over F₉, because

5 times m-reduction [i] would yield digital (2, 20, 30)-net over F₉, but
- extracting embedded orthogonal array [i] would yield linear OA(9²⁰, 30, F₉, 18) (dual of [30, 10, 19]-code), but
  - residual code [i] would yield OA(9², 11, S₉, 2), but
    - bound for OAs with strength kÂ =Â 2 [i]
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 89 > 9² [i]

(25−23, 25, 41)-Net in Base 9 — Upper bound on s

There is no (2, 25, 42)-net in base 9, because

extracting embedded orthogonal array [i] would yield OA(9²⁵, 42, S₉, 23), but
- the linear programming bound shows that MÂ â‰¥ 4250 195386 464997 943049 464163 / 5831 > 9²⁵ [i]