MinT - Information on Result #805592

Information on Result #805592

Linear OOA(2⁵¹, 517, F₂, 2, 10) (dual of [(517, 2), 983, 11]-NRT-code), using OOA 2-folding based on linear OA(2⁵¹, 1034, F₂, 10) (dual of [1034, 983, 11]-code), using

discarding factors / shortening the dual code based on linear OA(2⁵¹, 1035, F₂, 10) (dual of [1035, 984, 11]-code), using
- 1 times truncation [i] based on linear OA(2⁵², 1036, F₂, 11) (dual of [1036, 984, 12]-code), using
  - constructionÂ X4 applied to C_e(10) âŠ‚ C_e(8) [i] based on
    1. linear OA(2⁵¹, 1024, F₂, 11) (dual of [1024, 973, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    2. linear OA(2⁴¹, 1024, F₂, 9) (dual of [1024, 983, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    3. linear OA(2¹¹, 12, F₂, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,2)), using
      - dual of repetition code with length 12 [i]
    4. 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(2⁵¹, 423, F₂, 3, 10) (dual of [(423, 3), 1218, 11]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
2	Linear OOA(2⁵¹, 423, F₂, 4, 10) (dual of [(423, 4), 1641, 11]-NRT-code)	[i]
3	Linear OOA(2⁵¹, 423, F₂, 5, 10) (dual of [(423, 5), 2064, 11]-NRT-code)	[i]
4	Linear OOA(2⁵¹, 423, F₂, 6, 10) (dual of [(423, 6), 2487, 11]-NRT-code)	[i]
5	Linear OOA(2⁵¹, 423, F₂, 7, 10) (dual of [(423, 7), 2910, 11]-NRT-code)	[i]
6	Linear OOA(2⁵¹, 423, F₂, 8, 10) (dual of [(423, 8), 3333, 11]-NRT-code)	[i]
7	Digital (41, 51, 423)-net over F₂	[i]