MinT - Information on Result #3148605

Information on Result #3148605

There is no digital (45, 178, 199)-net over F₄, because 1 times m-reduction would yield digital (45, 177, 199)-net over F₄, but

extracting embedded orthogonal array [i] would yield linear OA(4¹⁷⁷, 199, F₄, 132) (dual of [199, 22, 133]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(4¹⁷⁶, 187, F₄, 132) (dual of [187, 11, 133]-code), but
    - constructionÂ Y1 [i] would yield
      1. linear OA(4¹⁷⁵, 181, F₄, 132) (dual of [181, 6, 133]-code), but
        constructionÂ Y1 [i] would yield
        linear OA(4¹⁷⁴, 178, F₄, 132) (dual of [178, 4, 133]-code), but
        Griesmer bound [i]
        linear OA(4⁶, 181, F₄, 3) (dual of [181, 175, 4]-code or 181-cap in PG(5,4)), but
        discarding factors / shortening the dual code would yield linear OA(4⁶, 160, F₄, 3) (dual of [160, 154, 4]-code or 160-cap in PG(5,4)), but
        1 times Hill recurrence [i] would yield linear OA(4⁵, 42, F₄, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,4)), but
        41 is the largest size of a cap in PG(4,Â 4) [i]
      2. OA(4¹¹, 187, S₄, 6), but
        discarding factors would yield OA(4¹¹, 99, S₄, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 4 278880 > 4¹¹ [i]
  2. OA(4²², 199, S₄, 12), but
    - discarding factors would yield OA(4²², 164, S₄, 12), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 18 187369 733464 > 4²² [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.