MinT - Information on Result #3152593

Information on Result #3152593

There is no digital (5, 45, 77)-net over F₉, because 4 times m-reduction would yield digital (5, 41, 77)-net over F₉, but

extracting embedded orthogonal array [i] would yield linear OA(9⁴¹, 77, F₉, 36) (dual of [77, 36, 37]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(9⁴⁰, 45, F₉, 36) (dual of [45, 5, 37]-code), but
    - constructionÂ Y1 [i] would yield
      1. OA(9³⁹, 41, S₉, 36), but
        the (dual) Plotkin bound shows that MÂ â‰¥ 739 044147 071729 616580 416051 031916 488005 / 37 > 9³⁹ [i]
      2. OA(9⁵, 45, S₉, 4), but
        discarding factors would yield OA(9⁵, 44, S₉, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 60897 > 9⁵ [i]
  2. linear OA(9³⁶, 77, F₉, 32) (dual of [77, 41, 33]-code), but
    - discarding factors / shortening the dual code would yield linear OA(9³⁶, 50, F₉, 32) (dual of [50, 14, 33]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(9³⁵, 38, F₉, 32) (dual of [38, 3, 33]-code), but
        â€œMasâ€ bound on codes from Brouwerâ€™s database [i]
        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]

Mode: Bound (linear).

Optimality

Show details for fixed k and m, k and s, k and t, m and s, m and t, t and s.

Other Results with Identical Parameters

None.

Depending Results

None.