MinT - Information on Result #599432

Information on Result #599432

There is no digital (40, 156, 190)-net over F₄, because extracting embedded orthogonal array would yield linear OA(4¹⁵⁶, 190, F₄, 116) (dual of [190, 34, 117]-code), but

constructionÂ Y1 [i] would yield
1. linear OA(4¹⁵⁵, 170, F₄, 116) (dual of [170, 15, 117]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁵⁴, 162, F₄, 116) (dual of [162, 8, 117]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(4¹⁵³, 158, F₄, 116) (dual of [158, 5, 117]-code), but
        residual code [i] would yield linear OA(4³⁷, 41, F₄, 29) (dual of [41, 4, 30]-code), but
        1 times truncation [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]
        OA(4⁸, 162, 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, 170, 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(4³⁴, 190, S₄, 20), but
  - discarding factors would yield OA(4³⁴, 173, S₄, 20), but
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 305 996090 287239 486286 > 4³⁴ [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 (40, 157, 190)-net over F₄	[i]	m-Reduction
2	No digital (40, 158, 190)-net over F₄	[i]
3	No digital (40, 159, 190)-net over F₄	[i]