MinT - Information on Result #945416

Information on Result #945416

Linear OOA(2⁷⁸, 686, F₂, 3, 14) (dual of [(686, 3), 1980, 15]-NRT-code), using OOA 3-folding based on linear OA(2⁷⁸, 2058, F₂, 14) (dual of [2058, 1980, 15]-code), using

discarding factors / shortening the dual code based on linear OA(2⁷⁸, 2059, F₂, 14) (dual of [2059, 1981, 15]-code), using
- 1 times truncation [i] based on linear OA(2⁷⁹, 2060, F₂, 15) (dual of [2060, 1981, 16]-code), using
  - constructionÂ X applied to C_e(14) âŠ‚ C_e(12) [i] based on
    1. linear OA(2⁷⁸, 2048, F₂, 15) (dual of [2048, 1970, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    2. 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]
    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: Constructive and 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		Method
1	Linear OOA(2⁷⁸, 686, F₂, 4, 14) (dual of [(686, 4), 2666, 15]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
2	Linear OOA(2⁷⁸, 686, F₂, 5, 14) (dual of [(686, 5), 3352, 15]-NRT-code)	[i]
3	Linear OOA(2⁷⁸, 686, F₂, 6, 14) (dual of [(686, 6), 4038, 15]-NRT-code)	[i]
4	Linear OOA(2⁷⁸, 686, F₂, 7, 14) (dual of [(686, 7), 4724, 15]-NRT-code)	[i]
5	Linear OOA(2⁷⁸, 686, F₂, 8, 14) (dual of [(686, 8), 5410, 15]-NRT-code)	[i]
6	Digital (64, 78, 686)-net over F₂	[i]