MinT - Information on Result #906959

Information on Result #906959

Linear OOA(49³³, large, F₄₉, 2, 6), using generalized (u,Â u+v)-construction based on

linear OOA(49⁵, 2882402, F₄₉, 2, 2) (dual of [(2882402, 2), 5764799, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(49⁵, 5884901, F₄₉, 2, 2) (dual of [(5884901, 2), 11769797, 3]-NRT-code), using
  - appending kth column [i] based on linear OA(49⁵, 5884901, F₄₉, 2) (dual of [5884901, 5884896, 3]-code), using
    - Hamming code H(5,49) [i]
linear OOA(49⁷, 2882402, F₄₉, 2, 3) (dual of [(2882402, 2), 5764797, 4]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(49⁷, 5769603, F₄₉, 2, 3) (dual of [(5769603, 2), 11539199, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(49⁷, 5769603, F₄₉, 3) (dual of [5769603, 5769596, 4]-code or 5769603-cap in PG(6,49)), using
    - product of two caps with tangent hyperplane [i]
linear OOA(49²¹, 2882402, F₄₉, 2, 6) (dual of [(2882402, 2), 5764783, 7]-NRT-code), using
- OOA 2-folding [i] based on linear OA(49²¹, 5764804, F₄₉, 6) (dual of [5764804, 5764783, 7]-code), using
  - discarding factors / shortening the dual code based on linear OA(49²¹, 5764805, F₄₉, 6) (dual of [5764805, 5764784, 7]-code), using
    - constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
      1. linear OA(49²¹, 5764801, F₄₉, 6) (dual of [5764801, 5764780, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
      2. linear OA(49¹⁷, 5764801, F₄₉, 5) (dual of [5764801, 5764784, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
      3. linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

Mode: Constructive and linear.

Optimality

Show details for fixed k and m, k and t, m and t.

Other Results with Identical Parameters

None.

Depending Results

The following results depend on this result:

	Result		Method
1	Linear OOA(49³³, large, F₄₉, 2, 5)	[i]	Strength Reduction for OOAs
2	Linear OOA(49³³, large, F₄₉, 2, 4)	[i]
3	Linear OOA(49³³, large, F₄₉, 2, 3)	[i]
4	Linear OOA(49³⁵, large, F₄₉, 2, 6)	[i]	NRT-Code Embedding in Larger Space
5	Linear OOA(49⁴¹, large, F₄₉, 2, 6)	[i]
6	Linear OOA(49⁴³, large, F₄₉, 2, 6)	[i]
7	Linear OOA(49⁴⁵, large, F₄₉, 2, 6)	[i]