MinT - Information on Result #548546

Information on Result #548546

There is no linear OA(2¹⁵⁷, 210, F₂, 74) (dual of [210, 53, 75]-code), because constructionÂ Y1 would yield

linear OA(2¹⁵⁶, 190, F₂, 74) (dual of [190, 34, 75]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(2¹⁵⁵, 178, F₂, 74) (dual of [178, 23, 75]-code), but
    - adding a parity check bit [i] would yield linear OA(2¹⁵⁶, 179, F₂, 75) (dual of [179, 23, 76]-code), but
      - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]
  2. OA(2³⁴, 190, S₂, 12), but
    - discarding factors would yield OA(2³⁴, 154, S₂, 12), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 17486 314616 > 2³⁴ [i]
OA(2⁵³, 210, S₂, 20), but
- discarding factors would yield OA(2⁵³, 182, S₂, 20), but
  - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 9064 436853 738748 > 2⁵³ [i]

Mode: Bound (linear).

None.

The following results depend on this result:

	Result		Method
1	No linear OA(2¹⁵⁸, 211, F₂, 75) (dual of [211, 53, 76]-code)	[i]	Truncation
2	No linear OOA(2¹⁵⁸, 210, F₂, 2, 75) (dual of [(210, 2), 262, 76]-NRT-code)	[i]	m-Reduction for OOAs
3	No linear OOA(2¹⁵⁷, 210, F₂, 2, 74) (dual of [(210, 2), 263, 75]-NRT-code)	[i]	Depth Reduction
4	No linear OOA(2¹⁵⁷, 210, F₂, 3, 74) (dual of [(210, 3), 473, 75]-NRT-code)	[i]
5	No linear OOA(2¹⁵⁷, 210, F₂, 4, 74) (dual of [(210, 4), 683, 75]-NRT-code)	[i]
6	No linear OOA(2¹⁵⁷, 210, F₂, 5, 74) (dual of [(210, 5), 893, 75]-NRT-code)	[i]
7	No linear OOA(2¹⁵⁷, 210, F₂, 6, 74) (dual of [(210, 6), 1103, 75]-NRT-code)	[i]
8	No linear OOA(2¹⁵⁷, 210, F₂, 7, 74) (dual of [(210, 7), 1313, 75]-NRT-code)	[i]
9	No linear OOA(2¹⁵⁷, 210, F₂, 8, 74) (dual of [(210, 8), 1523, 75]-NRT-code)	[i]
10	No digital (83, 157, 210)-net over F₂	[i]	Extracting Embedded Orthogonal Array
11	No linear OA(2¹⁵⁸, 244, F₂, 74) (dual of [244, 86, 75]-code)	[i]	ConstructionÂ Y1 (Bound)