MinT - Information on Result #599341

Information on Result #599341

There is no digital (4, 14, 30)-net over F₄, because extracting embedded orthogonal array would yield linear OA(4¹⁴, 30, F₄, 10) (dual of [30, 16, 11]-code), but

constructionÂ Y1 [i] would yield
1. linear OA(4¹³, 18, F₄, 10) (dual of [18, 5, 11]-code), but
  - â€œLizâ€ bound on codes from Brouwerâ€™s database [i]
2. linear OA(4¹⁶, 30, F₄, 12) (dual of [30, 14, 13]-code), but
  - discarding factors / shortening the dual code would yield linear OA(4¹⁶, 29, F₄, 12) (dual of [29, 13, 13]-code), but
    - constructionÂ Y1 [i] would yield
      1. linear OA(4¹⁵, 19, F₄, 12) (dual of [19, 4, 13]-code), but
        constructionÂ Y1 [i] would yield
        OA(4¹⁴, 16, S₄, 12), but
        the (dual) Plotkin bound shows that MÂ â‰¥ 4294 967296 / 13 > 4¹⁴ [i]
        linear OA(4⁴, 19, F₄, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,4)), but
        discarding factors / shortening the dual code would yield linear OA(4⁴, 18, F₄, 3) (dual of [18, 14, 4]-code or 18-cap in PG(3,4)), but
        ovoid cap is optimal [i]
      2. linear OA(4¹³, 29, F₄, 10) (dual of [29, 16, 11]-code), but
        discarding factors / shortening the dual code would yield linear OA(4¹³, 18, F₄, 10) (dual of [18, 5, 11]-code) (see above)

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 (4, 15, 30)-net over F₄	[i]		m-Reduction