MinT - Information on Result #599349

Information on Result #599349

There is no digital (6, 9, 8687)-net over F₄, because extracting embedded orthogonal array would yield linear OA(4⁹, 8687, F₄, 3) (dual of [8687, 8678, 4]-code or 8687-cap in PG(8,4)), but

removing affine subspaces [i] would yield
1. 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]
2. 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]
3. 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)

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

None.