MinT - Information on Result #548062

Information on Result #548062

There is no linear OA(4¹⁰, 31771, F₄, 3) (dual of [31771, 31761, 4]-code or 31771-cap in PG(9,4)), because removing affine subspaces would yield

linear OA(4⁷, 608, F₄, 3) (dual of [608, 601, 4]-code or 608-cap in PG(6,4)), but
- 1 times Hill recurrence [i] would yield linear OA(4⁶, 154, F₄, 3) (dual of [154, 148, 4]-code or 154-cap in PG(5,4)), but
  - constructionÂ Y1 [i] would yield
    1. 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. linear OA(4¹⁴⁸, 154, F₄, 112) (dual of [154, 6, 113]-code), but
      - discarding factors / shortening the dual code would yield linear OA(4¹⁴⁸, 153, F₄, 112) (dual of [153, 5, 113]-code), but
        residual code [i] would yield linear OA(4³⁶, 40, F₄, 28) (dual of [40, 4, 29]-code), but
        residual code [i] would yield linear OA(4⁸, 11, F₄, 7) (dual of [11, 3, 8]-code), but
        improvement by Maruta on the Griesmer bound [i]
1752-cap in AG(7,4), but
- 3 times the recursive bound from Bierbrauer and Edel [i] would yield 41-cap in AG(4,4), but
  - bound on the size of affine caps [i]
6329-cap in AG(8,4), but
- 4 times the recursive bound from Bierbrauer and Edel [i] would yield 41-cap in AG(4,4) (see above)
23085-cap in AG(9,4), but
- 5 times the recursive bound from Bierbrauer and Edel [i] would yield 41-cap in AG(4,4) (see above)

Mode: Bound (linear).

None.

The following results depend on this result:

	Result		Method
1	No linear OOA(4¹⁰, 31771, F₄, 2, 3) (dual of [(31771, 2), 63532, 4]-NRT-code)	[i]	Depth Reduction
2	No linear OOA(4¹⁰, 31771, F₄, 3, 3) (dual of [(31771, 3), 95303, 4]-NRT-code)	[i]	Depth Reduction
3	No digital (7, 10, 31771)-net over F₄	[i]	Extracting Embedded Orthogonal Array