MinT - Best Known (2, 2+20, s)-Nets in Base 9

Best Known (2, 2+20, s)-Nets in Base 9

(2, 2+20, 20)-Net over F₉ — Constructive and digital

Digital (2, 22, 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]

(2, 2+20, 29)-Net over F₉ — Upper bound on s (digital)

There is no digital (2, 22, 30)-net over F₉, because

2 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]

(2, 2+20, 54)-Net in Base 9 — Upper bound on s

There is no (2, 22, 55)-net in base 9, because

extracting embedded orthogonal array [i] would yield OA(9²², 55, S₉, 20), but
- the linear programming bound shows that MÂ â‰¥ 760 385024 469179 203776 643903 368285 / 765466 314941 > 9²² [i]