MinT - Information on Result #599779

Information on Result #599779

There is no digital (4, 25, 53)-net over F₇, because extracting embedded orthogonal array would yield linear OA(7²⁵, 53, F₇, 21) (dual of [53, 28, 22]-code), but

constructionÂ Y1 [i] would yield
1. linear OA(7²⁴, 29, F₇, 21) (dual of [29, 5, 22]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(7²³, 25, S₇, 21), but
      - the (dual) Plotkin bound shows that MÂ â‰¥ 383 162462 761132 828802 / 11 > 7²³ [i]
    2. OA(7⁵, 29, S₇, 4), but
      - the linear programming bound shows that MÂ â‰¥ 3 384381 / 197 > 7⁵ [i]
2. linear OA(7²⁸, 53, F₇, 24) (dual of [53, 25, 25]-code), but
  - discarding factors / shortening the dual code would yield linear OA(7²⁸, 40, F₇, 24) (dual of [40, 12, 25]-code), but
    - constructionÂ Y1 [i] would yield
      1. linear OA(7²⁷, 30, F₇, 24) (dual of [30, 3, 25]-code), but
        â€œHi4â€ bound on codes from Brouwerâ€™s database [i]
      2. linear OA(7¹², 40, F₇, 10) (dual of [40, 28, 11]-code), but
        discarding factors / shortening the dual code would yield linear OA(7¹², 36, F₇, 10) (dual of [36, 24, 11]-code), but
        constructionÂ Y1 [i] would yield
        linear OA(7¹¹, 15, F₇, 10) (dual of [15, 4, 11]-code), but
        â€œBPMâ€ bound on codes from Brouwerâ€™s database [i]
        linear OA(7²⁴, 36, F₇, 21) (dual of [36, 12, 22]-code), but
        discarding factors / shortening the dual code would yield linear OA(7²⁴, 31, F₇, 21) (dual of [31, 7, 22]-code), but
        residual code [i] would yield OA(7³, 9, S₇, 3), but
        bound for OAs with index unity [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		This result only	Method
1	No digital (4, 26, 53)-net over F₇	[i]		m-Reduction
2	No digital (4, 27, 53)-net over F₇	[i]		m-Reduction