MinT - Information on Result #599776

Information on Result #599776

There is no digital (3, 21, 35)-net over F₇, because extracting embedded orthogonal array would yield linear OA(7²¹, 35, F₇, 18) (dual of [35, 14, 19]-code), but

constructionÂ Y1 [i] would yield
1. linear OA(7²⁰, 23, F₇, 18) (dual of [23, 3, 19]-code), but
  - â€œHi4â€ bound on codes from Brouwerâ€™s database [i]
2. linear OA(7¹⁴, 35, F₇, 12) (dual of [35, 21, 13]-code), but
  - discarding factors / shortening the dual code would yield linear OA(7¹⁴, 30, F₇, 12) (dual of [30, 16, 13]-code), but
    - constructionÂ Y1 [i] would yield
      1. linear OA(7¹³, 16, F₇, 12) (dual of [16, 3, 13]-code), but
        bound for almost-MDS-codes with dimension nÂ =Â 3 [i]
      2. linear OA(7¹⁶, 30, F₇, 14) (dual of [30, 14, 15]-code), but
        discarding factors / shortening the dual code would yield linear OA(7¹⁶, 24, F₇, 14) (dual of [24, 8, 15]-code), but
        residual code [i] would yield OA(7², 9, S₇, 2), but
        bound for OAs with strength kÂ =Â 2 [i]
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 55 > 7² [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 (3, 22, 35)-net over F₇	[i]		m-Reduction
2	No digital (3, 23, 35)-net over F₇	[i]		m-Reduction