MinT - Information on Result #2427144

Information on Result #2427144

Linear OOA(2²³³, 1046, F₂, 2, 40) (dual of [(1046, 2), 1859, 41]-NRT-code), using strength reduction based on linear OOA(2²³³, 1046, F₂, 2, 41) (dual of [(1046, 2), 1859, 42]-NRT-code), using

OOA 2-folding [i] based on linear OA(2²³³, 2092, F₂, 41) (dual of [2092, 1859, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(2²³³, 2093, F₂, 41) (dual of [2093, 1860, 42]-code), using
  - constructionÂ X applied to C_e(40) âŠ‚ C_e(34) [i] based on
    1. linear OA(2²²¹, 2048, F₂, 41) (dual of [2048, 1827, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    2. linear OA(2¹⁸⁸, 2048, F₂, 35) (dual of [2048, 1860, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    3. linear OA(2¹², 45, F₂, 5) (dual of [45, 33, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹², 48, F₂, 5) (dual of [48, 36, 6]-code), using
        adding a parity check bit [i] based on linear OA(2¹¹, 47, F₂, 4) (dual of [47, 36, 5]-code), using
        extracting embedded orthogonal array [i] based on digital (7, 11, 47)-net over F₂, using
        linear code embedding found in a computer search [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(2²³³, 1046, F₂, 3, 40) (dual of [(1046, 3), 2905, 41]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
2	Linear OOA(2²³³, 1046, F₂, 4, 40) (dual of [(1046, 4), 3951, 41]-NRT-code)	[i]
3	Linear OOA(2²³³, 1046, F₂, 5, 40) (dual of [(1046, 5), 4997, 41]-NRT-code)	[i]
4	Linear OOA(2²³³, 1046, F₂, 6, 40) (dual of [(1046, 6), 6043, 41]-NRT-code)	[i]
5	Linear OOA(2²³³, 1046, F₂, 7, 40) (dual of [(1046, 7), 7089, 41]-NRT-code)	[i]
6	Linear OOA(2²³³, 1046, F₂, 8, 40) (dual of [(1046, 8), 8135, 41]-NRT-code)	[i]
7	Digital (193, 233, 1046)-net over F₂	[i]