MinT - Best Known (18−16, 18, s)-Nets in Base 9

Best Known (18−16, 18, s)-Nets in Base 9

(18−16, 18, 20)-Net over F₉ — Constructive and digital

Digital (2, 18, 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]

(18−16, 18, 37)-Net over F₉ — Upper bound on s (digital)

There is no digital (2, 18, 38)-net over F₉, because

extracting embedded orthogonal array [i] would yield linear OA(9¹⁸, 38, F₉, 16) (dual of [38, 20, 17]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(9¹⁷, 20, F₉, 16) (dual of [20, 3, 17]-code), but
    - bound for almost-MDS-codes with dimension nÂ =Â 3 [i]
  2. linear OA(9²⁰, 38, F₉, 18) (dual of [38, 18, 19]-code), but
    - discarding factors / shortening the dual code 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]

(18−16, 18, 59)-Net in Base 9 — Upper bound on s

There is no (2, 18, 60)-net in base 9, because

4 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]