MinT - Information on Result #599077

Information on Result #599077

There is no digital (56, 164, 189)-net over F₃, because extracting embedded orthogonal array would yield linear OA(3¹⁶⁴, 189, F₃, 108) (dual of [189, 25, 109]-code), but

constructionÂ Y1 [i] would yield
1. linear OA(3¹⁶³, 177, F₃, 108) (dual of [177, 14, 109]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3¹⁶², 171, F₃, 108) (dual of [171, 9, 109]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(3¹⁶¹, 167, F₃, 108) (dual of [167, 6, 109]-code), but
        residual code [i] would yield linear OA(3⁵³, 58, F₃, 36) (dual of [58, 5, 37]-code), but
        residual code [i] would yield linear OA(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code), but
        â€œHNâ€ bound on codes from Brouwerâ€™s database [i]
        OA(3⁹, 171, S₃, 4), but
        discarding factors would yield OA(3⁹, 100, S₃, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 20001 > 3⁹ [i]
    2. OA(3¹⁴, 177, S₃, 6), but
      - discarding factors would yield OA(3¹⁴, 154, S₃, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 4 822665 > 3¹⁴ [i]
2. OA(3²⁵, 189, S₃, 12), but
  - discarding factors would yield OA(3²⁵, 148, S₃, 12), but
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 861032 991633 > 3²⁵ [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		This result only	Method
1	No digital (56, 165, 189)-net over F₃	[i]		m-Reduction
2	No digital (56, 166, 189)-net over F₃	[i]		m-Reduction