MinT - Information on Result #548691

Information on Result #548691

There is no linear OA(4¹⁴, 30, F₄, 10) (dual of [30, 16, 11]-code), because constructionÂ Y1 would yield

linear OA(4¹³, 18, F₄, 10) (dual of [18, 5, 11]-code), but
- â€œLizâ€ bound on codes from Brouwerâ€™s database [i]
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).

None.

The following results depend on this result:

	Result		Method
1	No linear OOA(4¹⁵, 30, F₄, 2, 11) (dual of [(30, 2), 45, 12]-NRT-code)	[i]	m-Reduction for OOAs
2	No linear OOA(4¹⁴, 30, F₄, 2, 10) (dual of [(30, 2), 46, 11]-NRT-code)	[i]	Depth Reduction
3	No linear OOA(4¹⁴, 30, F₄, 3, 10) (dual of [(30, 3), 76, 11]-NRT-code)	[i]	Depth Reduction
4	No digital (4, 14, 30)-net over F₄	[i]	Extracting Embedded Orthogonal Array