MinT - Best Known (2, 2+15, s)-Nets in Base 9

Best Known (2, 2+15, s)-Nets in Base 9

(2, 2+15, 20)-Net over F₉ — Constructive and digital

Digital (2, 17, 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+15, 43)-Net over F₉ — Upper bound on s (digital)

There is no digital (2, 17, 44)-net over F₉, because

3 times m-reduction [i] would yield digital (2, 14, 44)-net over F₉, but
- extracting embedded orthogonal array [i] would yield linear OA(9¹⁴, 44, F₉, 12) (dual of [44, 30, 13]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(9¹³, 17, F₉, 12) (dual of [17, 4, 13]-code), but
      - â€œMPaâ€ bound on codes from Brouwerâ€™s database [i]
    2. linear OA(9³⁰, 44, F₉, 27) (dual of [44, 14, 28]-code), but
      - discarding factors / shortening the dual code would yield linear OA(9³⁰, 39, F₉, 27) (dual of [39, 9, 28]-code), but
        residual code [i] would yield OA(9³, 11, S₉, 3), but
        bound for OAs with index unity [i]

(2, 2+15, 59)-Net in Base 9 — Upper bound on s

There is no (2, 17, 60)-net in base 9, because

3 times m-reduction [i] would yield (2, 14, 60)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 23 188827 548097 > 9¹⁴ [i]