MinT - Information on Result #1661839

Information on Result #1661839

Linear OOA(2⁶⁷, 686, F₂, 6, 12) (dual of [(686, 6), 4049, 13]-NRT-code), using embedding of OOA with Gilbertâ€“VarÅ¡amov bound based on linear OOA(2⁶⁷, 686, F₂, 3, 12) (dual of [(686, 3), 1991, 13]-NRT-code), using

OOA 3-folding [i] based on linear OA(2⁶⁷, 2058, F₂, 12) (dual of [2058, 1991, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2⁶⁷, 2059, F₂, 12) (dual of [2059, 1992, 13]-code), using
  - 1 times truncation [i] based on linear OA(2⁶⁸, 2060, F₂, 13) (dual of [2060, 1992, 14]-code), using
    - constructionÂ X applied to C_e(12) âŠ‚ C_e(10) [i] based on
      1. linear OA(2⁶⁷, 2048, F₂, 13) (dual of [2048, 1981, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
      2. linear OA(2⁵⁶, 2048, F₂, 11) (dual of [2048, 1992, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [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(2⁶⁷, 685, F₂, 7, 12) (dual of [(685, 7), 4728, 13]-NRT-code)	[i]		OOA Stacking with Additional Row
2	Linear OOA(2⁶⁷, 685, F₂, 8, 12) (dual of [(685, 8), 5413, 13]-NRT-code)	[i]
3	Linear OOA(2⁶⁷, 685, F₂, 18, 12) (dual of [(685, 18), 12263, 13]-NRT-code)	[i]