MinT - Information on Result #599419

Information on Result #599419

There is no digital (38, 150, 165)-net over F₄, because extracting embedded orthogonal array would yield linear OA(4¹⁵⁰, 165, F₄, 112) (dual of [165, 15, 113]-code), but

constructionÂ Y1 [i] would yield
1. linear OA(4¹⁴⁹, 157, F₄, 112) (dual of [157, 8, 113]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁴⁸, 153, F₄, 112) (dual of [153, 5, 113]-code), but
      - residual code [i] would yield linear OA(4³⁶, 40, F₄, 28) (dual of [40, 4, 29]-code), but
        residual code [i] would yield linear OA(4⁸, 11, F₄, 7) (dual of [11, 3, 8]-code), but
        improvement by Maruta on the Griesmer bound [i]
    2. OA(4⁸, 157, 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, 165, 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]

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 (38, 151, 165)-net over F₄	[i]	m-Reduction
2	No digital (38, 152, 165)-net over F₄	[i]
3	No digital (38, 153, 165)-net over F₄	[i]