MinT - Best Known (33−31, 33, s)-Nets in Base 9

Best Known (33−31, 33, s)-Nets in Base 9

(33−31, 33, 20)-Net over F₉ — Constructive and digital

Digital (2, 33, 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]

(33−31, 33, 29)-Net over F₉ — Upper bound on s (digital)

There is no digital (2, 33, 30)-net over F₉, because

13 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]

(33−31, 33, 30)-Net in Base 9 — Upper bound on s

There is no (2, 33, 31)-net in base 9, because

6 times m-reduction [i] would yield (2, 27, 31)-net in base 9, but
- extracting embedded orthogonal array [i] would yield OA(9²⁷, 31, S₉, 25), but
  - the linear programming bound shows that MÂ â‰¥ 25644 034018 340666 323462 064529 / 377 > 9²⁷ [i]