MinT - Best Known (4, 37, s)-Nets in Base 9

Best Known (4, 37, s)-Nets in Base 9

(4, 37, 30)-Net over F₉ — Constructive and digital

Digital (4, 37, 30)-net over F₉, using

net from sequence [i] based on digital (4, 29)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
  - a function field by SÃ©mirat [i]

(4, 37, 49)-Net over F₉ — Upper bound on s (digital)

There is no digital (4, 37, 50)-net over F₉, because

1 times m-reduction [i] would yield digital (4, 36, 50)-net over F₉, but
- extracting embedded orthogonal array [i] would yield linear OA(9³⁶, 50, F₉, 32) (dual of [50, 14, 33]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(9³⁵, 38, F₉, 32) (dual of [38, 3, 33]-code), but
      - â€œMasâ€ bound on codes from Brouwerâ€™s database [i]
    2. linear OA(9¹⁴, 50, F₉, 12) (dual of [50, 36, 13]-code), but
      - discarding factors / shortening the dual code would yield linear OA(9¹⁴, 44, F₉, 12) (dual of [44, 30, 13]-code), but
        constructionÂ Y1 [i] would yield
        linear OA(9¹³, 17, F₉, 12) (dual of [17, 4, 13]-code), but
        â€œMPaâ€ bound on codes from Brouwerâ€™s database [i]
        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]

(4, 37, 77)-Net in Base 9 — Upper bound on s

There is no (4, 37, 78)-net in base 9, because

extracting embedded orthogonal array [i] would yield OA(9³⁷, 78, S₉, 33), but
- the linear programming bound shows that MÂ â‰¥ 199356 054373 293137 005126 556242 150856 557793 215196 056979 783877 / 900916 662343 931597 492741 > 9³⁷ [i]