MinT - Information on Result #599909

Information on Result #599909

There is no digital (5, 78, 124)-net over F₁₆, because extracting embedded orthogonal array would yield linear OA(16⁷⁸, 124, F₁₆, 73) (dual of [124, 46, 74]-code), but

constructionÂ Y1 [i] would yield
1. OA(16⁷⁷, 81, S₁₆, 73), but
  - the linear programming bound shows that MÂ â‰¥ 20 058253 259194 796242 490750 709498 450326 191022 155255 442417 783937 598455 532112 408827 665059 378347 638784 / 37999 > 16⁷⁷ [i]
2. linear OA(16⁴⁶, 124, F₁₆, 43) (dual of [124, 78, 44]-code), but
  - discarding factors / shortening the dual code would yield linear OA(16⁴⁶, 97, F₁₆, 43) (dual of [97, 51, 44]-code), but
    - constructionÂ Y1 [i] would yield
      1. OA(16⁴⁵, 49, S₁₆, 43), but
        the linear programming bound shows that MÂ â‰¥ 79 689768 125026 220634 634045 411816 077548 174434 353547 313152 / 47 > 16⁴⁵ [i]
      2. linear OA(16⁵¹, 97, F₁₆, 48) (dual of [97, 46, 49]-code), but
        discarding factors / shortening the dual code would yield linear OA(16⁵¹, 68, F₁₆, 48) (dual of [68, 17, 49]-code), but
        residual code [i] would yield OA(16³, 19, S₁₆, 3), but
        1 times truncation [i] would yield OA(16², 18, S₁₆, 2), but
        bound for OAs with strength kÂ =Â 2 [i]
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 271 > 16² [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 (5, 79, 124)-net over F₁₆	[i]	m-Reduction
2	No digital (5, 80, 124)-net over F₁₆	[i]
3	No digital (5, 81, 124)-net over F₁₆	[i]
4	No digital (5, 82, 124)-net over F₁₆	[i]