MinT - Information on Result #1051233

Information on Result #1051233

Linear OOA(2¹¹¹, 1048581, F₂, 4, 10) (dual of [(1048581, 4), 4194213, 11]-NRT-code), using OOA 4-folding based on linear OA(2¹¹¹, 4194324, F₂, 10) (dual of [4194324, 4194213, 11]-code), using

discarding factors / shortening the dual code based on linear OA(2¹¹¹, 4194327, F₂, 10) (dual of [4194327, 4194216, 11]-code), using
- 1 times truncation [i] based on linear OA(2¹¹², 4194328, F₂, 11) (dual of [4194328, 4194216, 12]-code), using
  - constructionÂ X4 applied to C_e(10) âŠ‚ C_e(8) [i] based on
    1. linear OA(2¹¹¹, 4194304, F₂, 11) (dual of [4194304, 4194193, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 4194303Â = 2²²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    2. linear OA(2⁸⁹, 4194304, F₂, 9) (dual of [4194304, 4194215, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 4194303Â = 2²²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    3. linear OA(2²³, 24, F₂, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,2)), using
      - dual of repetition code with length 24 [i]
    4. linear OA(2¹, 24, F₂, 1) (dual of [24, 23, 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(2¹¹¹, 1048581, F₂, 5, 10) (dual of [(1048581, 5), 5242794, 11]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
2	Linear OOA(2¹¹¹, 1048581, F₂, 6, 10) (dual of [(1048581, 6), 6291375, 11]-NRT-code)	[i]
3	Linear OOA(2¹¹¹, 1048581, F₂, 7, 10) (dual of [(1048581, 7), 7339956, 11]-NRT-code)	[i]
4	Linear OOA(2¹¹¹, 1048581, F₂, 8, 10) (dual of [(1048581, 8), 8388537, 11]-NRT-code)	[i]
5	Digital (101, 111, 1048581)-net over F₂	[i]