MinT - Information on Result #1678740

Information on Result #1678740

Linear OOA(2⁵¹, 2055, F₂, 7, 8) (dual of [(2055, 7), 14334, 9]-NRT-code), using embedding of OOA with Gilbertâ€“VarÅ¡amov bound based on linear OOA(2⁵¹, 2055, F₂, 2, 8) (dual of [(2055, 2), 4059, 9]-NRT-code), using

2¹ times duplication [i] based on linear OOA(2⁵⁰, 2055, F₂, 2, 8) (dual of [(2055, 2), 4060, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2⁵⁰, 4110, F₂, 8) (dual of [4110, 4060, 9]-code), using
  - strength reduction [i] based on linear OA(2⁵⁰, 4110, F₂, 9) (dual of [4110, 4060, 10]-code), using
    - constructionÂ X4 applied to C_e(8) âŠ‚ C_e(6) [i] based on
      1. linear OA(2⁴⁹, 4096, F₂, 9) (dual of [4096, 4047, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 2¹²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
      2. linear OA(2³⁷, 4096, F₂, 7) (dual of [4096, 4059, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 2¹²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
      3. linear OA(2¹³, 14, F₂, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,2)), using
        dual of repetition code with length 14 [i]
      4. linear OA(2¹, 14, F₂, 1) (dual of [14, 13, 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(2⁵¹, 2055, F₂, 8, 8) (dual of [(2055, 8), 16389, 9]-NRT-code)	[i]		Appending kth Column