MinT - Information on Result #1509238

Information on Result #1509238

Linear OOA(2⁴⁵, 1004, F₂, 3, 8) (dual of [(1004, 3), 2967, 9]-NRT-code), using embedding of OOA with Gilbertâ€“VarÅ¡amov bound based on linear OOA(2⁴⁵, 1004, F₂, 2, 8) (dual of [(1004, 2), 1963, 9]-NRT-code), using

discarding factors / shortening the dual code based on linear OOA(2⁴⁵, 1030, F₂, 2, 8) (dual of [(1030, 2), 2015, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2⁴⁵, 2060, F₂, 8) (dual of [2060, 2015, 9]-code), using
  - 1 times truncation [i] based on linear OA(2⁴⁶, 2061, F₂, 9) (dual of [2061, 2015, 10]-code), using
    - constructionÂ X4 applied to C_e(8) âŠ‚ C_e(6) [i] based on
      1. linear OA(2⁴⁵, 2048, F₂, 9) (dual of [2048, 2003, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
      2. linear OA(2³⁴, 2048, F₂, 7) (dual of [2048, 2014, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
      3. linear OA(2¹², 13, F₂, 11) (dual of [13, 1, 12]-code), using
        strength reduction [i] based on linear OA(2¹², 13, F₂, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,2)), using
        dual of repetition code with length 13 [i]
      4. linear OA(2¹, 13, F₂, 1) (dual of [13, 12, 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(2²²⁹, 2797205, F₂, 3, 16) (dual of [(2797205, 3), 8391386, 17]-NRT-code)	[i]		(u,Â u+v)-Construction for OOAs
2	Linear OOA(2²³⁰, 2797205, F₂, 3, 17) (dual of [(2797205, 3), 8391385, 18]-NRT-code)	[i]		(u,Â u+v)-Construction for OOAs