MinT - Information on Result #1590749

Information on Result #1590749

Linear OOA(2¹²², 686, F₂, 4, 22) (dual of [(686, 4), 2622, 23]-NRT-code), using embedding of OOA with Gilbertâ€“VarÅ¡amov bound based on linear OOA(2¹²², 686, F₂, 3, 22) (dual of [(686, 3), 1936, 23]-NRT-code), using

OOA 3-folding [i] based on linear OA(2¹²², 2058, F₂, 22) (dual of [2058, 1936, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2¹²², 2059, F₂, 22) (dual of [2059, 1937, 23]-code), using
  - 1 times truncation [i] based on linear OA(2¹²³, 2060, F₂, 23) (dual of [2060, 1937, 24]-code), using
    - constructionÂ X applied to C_e(22) âŠ‚ C_e(20) [i] based on
      1. linear OA(2¹²², 2048, F₂, 23) (dual of [2048, 1926, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      2. linear OA(2¹¹¹, 2048, F₂, 21) (dual of [2048, 1937, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      3. linear OA(2¹, 12, F₂, 1) (dual of [12, 11, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹, s, F₂, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
        parity-check code [i]

Mode: Linear.

Optimality

Show details for fixed k and m, n and k, k and s, k and t, n and m, m and s, m and t, n and s, n 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(2¹²³, 686, F₂, 4, 22) (dual of [(686, 4), 2621, 23]-NRT-code)	[i]		OOA Duplication
2	Linear OOA(2¹²², 343, F₂, 7, 22) (dual of [(343, 7), 2279, 23]-NRT-code)	[i]		OOA Folding
3	Linear OOA(2¹²², 343, F₂, 8, 22) (dual of [(343, 8), 2622, 23]-NRT-code)	[i]		OOA Folding