MinT - Information on Result #599501

Information on Result #599501

There is no digital (50, 194, 226)-net over F₄, because extracting embedded orthogonal array would yield linear OA(4¹⁹⁴, 226, F₄, 144) (dual of [226, 32, 145]-code), but

constructionÂ Y1 [i] would yield
1. linear OA(4¹⁹³, 208, F₄, 144) (dual of [208, 15, 145]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁹², 200, F₄, 144) (dual of [200, 8, 145]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(4¹⁹¹, 196, F₄, 144) (dual of [196, 5, 145]-code), but
        residual code [i] would yield linear OA(4⁴⁷, 51, F₄, 36) (dual of [51, 4, 37]-code), but
        a result by Landjev, Maruta, and Hill [i]
        OA(4⁸, 200, S₄, 4), but
        discarding factors would yield OA(4⁸, 121, S₄, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 65704 > 4⁸ [i]
    2. OA(4¹⁵, 208, S₄, 8), but
      - discarding factors would yield OA(4¹⁵, 135, S₄, 8), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1082 768311 > 4¹⁵ [i]
2. OA(4³², 226, S₄, 18), but
  - discarding factors would yield OA(4³², 195, S₄, 18), but
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 18 632959 071185 877328 > 4³² [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

The following results depend on this result:

	Result		Method
1	No digital (50, 195, 226)-net over F₄	[i]	m-Reduction
2	No digital (50, 196, 226)-net over F₄	[i]
3	No digital (50, 197, 226)-net over F₄	[i]