MinT - Information on Result #547891

Information on Result #547891

There is no linear OA(4¹⁷², 194, F₄, 128) (dual of [194, 22, 129]-code), because constructionÂ Y1 would yield

linear OA(4¹⁷¹, 182, F₄, 128) (dual of [182, 11, 129]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(4¹⁷⁰, 176, F₄, 128) (dual of [176, 6, 129]-code), but
    - constructionÂ Y1 [i] would yield
      1. linear OA(4¹⁶⁹, 173, F₄, 128) (dual of [173, 4, 129]-code), but
        Griesmer bound [i]
      2. linear OA(4⁶, 176, F₄, 3) (dual of [176, 170, 4]-code or 176-cap in PG(5,4)), but
        discarding factors / shortening the dual code would yield linear OA(4⁶, 160, F₄, 3) (dual of [160, 154, 4]-code or 160-cap in PG(5,4)), but
        1 times Hill recurrence [i] would yield 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. OA(4¹¹, 182, S₄, 6), but
    - discarding factors would yield OA(4¹¹, 99, S₄, 6), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 4 278880 > 4¹¹ [i]
OA(4²², 194, S₄, 12), but
- discarding factors would yield OA(4²², 164, S₄, 12), but
  - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 18 187369 733464 > 4²² [i]

Mode: Bound (linear).

None.

The following results depend on this result:

	Result		Method
1	No linear OA(4¹⁷³, 195, F₄, 129) (dual of [195, 22, 130]-code)	[i]	Truncation
2	No linear OA(4¹⁷⁴, 196, F₄, 130) (dual of [196, 22, 131]-code)	[i]
3	No linear OA(4¹⁷⁵, 197, F₄, 131) (dual of [197, 22, 132]-code)	[i]
4	No linear OOA(4¹⁷³, 194, F₄, 2, 129) (dual of [(194, 2), 215, 130]-NRT-code)	[i]	m-Reduction for OOAs
5	No linear OOA(4¹⁷⁴, 194, F₄, 2, 130) (dual of [(194, 2), 214, 131]-NRT-code)	[i]
6	No linear OOA(4¹⁷⁵, 194, F₄, 2, 131) (dual of [(194, 2), 213, 132]-NRT-code)	[i]
7	No linear OOA(4¹⁷², 194, F₄, 2, 128) (dual of [(194, 2), 216, 129]-NRT-code)	[i]	Depth Reduction
8	No linear OOA(4¹⁷², 194, F₄, 3, 128) (dual of [(194, 3), 410, 129]-NRT-code)	[i]	Depth Reduction
9	No digital (44, 172, 194)-net over F₄	[i]	Extracting Embedded Orthogonal Array