MinT - Information on Result #988552

Information on Result #988552

Linear OOA(49³⁶, 2942452, F₄₉, 3, 7) (dual of [(2942452, 3), 8827320, 8]-NRT-code), using OOA 2-folding based on linear OOA(49³⁶, 5884904, F₄₉, 2, 7) (dual of [(5884904, 2), 11769772, 8]-NRT-code), using

generalized (u,Â u+v)-construction [i] based on
1. linear OOA(49⁴, 120100, F₄₉, 2, 2) (dual of [(120100, 2), 240196, 3]-NRT-code), using
  - appending kth column [i] based on linear OA(49⁴, 120100, F₄₉, 2) (dual of [120100, 120096, 3]-code), using
    - Hamming code H(4,49) [i]
2. 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]
3. linear OOA(49²⁵, 2882402, F₄₉, 2, 7) (dual of [(2882402, 2), 5764779, 8]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(49²⁵, 5764804, F₄₉, 7) (dual of [5764804, 5764779, 8]-code), using
    - discarding factors / shortening the dual code based on linear OA(49²⁵, 5764805, F₄₉, 7) (dual of [5764805, 5764780, 8]-code), using
      - constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(49²⁵, 5764801, F₄₉, 7) (dual of [5764801, 5764776, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        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]
        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 s, k and t, m and s, m and t.

Other Results with Identical Parameters

None.

Depending Results

The following results depend on this result:

	Result		This result only	Method
1	Linear OOA(7⁷², 5884904, F₇, 3, 7) (dual of [(5884904, 3), 17654640, 8]-NRT-code)	[i]		Trace Code for OOAs