MinT - Information on Result #599460

Information on Result #599460

There is no digital (44, 172, 194)-net over F₄, because extracting embedded orthogonal array would yield linear OA(4¹⁷², 194, F₄, 128) (dual of [194, 22, 129]-code), but

constructionÂ Y1 [i] would yield
1. linear OA(4¹⁷¹, 182, F₄, 128) (dual of [182, 11, 129]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁷⁰, 176, F₄, 128) (dual of [176, 6, 129]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(4¹⁶⁹, 173, F₄, 128) (dual of [173, 4, 129]-code), but
        Griesmer bound [i]
        linear OA(4⁶, 176, F₄, 3) (dual of [176, 170, 4]-code or 176-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, 182, 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, 194, 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

The following results depend on this result:

	Result		Method
1	No digital (44, 173, 194)-net over F₄	[i]	m-Reduction
2	No digital (44, 174, 194)-net over F₄	[i]
3	No digital (44, 175, 194)-net over F₄	[i]