MinT - Information on Result #599287

Information on Result #599287

There is no digital (82, 247, 257)-net over F₃, because extracting embedded orthogonal array would yield linear OA(3²⁴⁷, 257, F₃, 165) (dual of [257, 10, 166]-code), but

constructionÂ Y1 [i] would yield
1. linear OA(3²⁴⁶, 253, F₃, 165) (dual of [253, 7, 166]-code), but
  - residual code [i] would yield linear OA(3⁸¹, 87, F₃, 55) (dual of [87, 6, 56]-code), but
    - 1 times truncation [i] would yield linear OA(3⁸⁰, 86, F₃, 54) (dual of [86, 6, 55]-code), but
      - residual code [i] would yield linear OA(3²⁶, 31, F₃, 18) (dual of [31, 5, 19]-code), but
        residual code [i] would yield linear OA(3⁸, 12, F₃, 6) (dual of [12, 4, 7]-code), but
        residual code [i] would yield linear OA(3², 5, F₃, 2) (dual of [5, 3, 3]-code or 5-arc in PG(1,3)), but
        improvement by Maruta on the Griesmer bound [i]
2. OA(3¹⁰, 257, S₃, 4), but
  - discarding factors would yield OA(3¹⁰, 172, S₃, 4), but
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 59169 > 3¹⁰ [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		Method
1	No digital (82, 248, 257)-net over F₃	[i]	m-Reduction
2	No digital (82, 249, 257)-net over F₃	[i]
3	No digital (82, 250, 257)-net over F₃	[i]