MinT - Information on Result #945147

Information on Result #945147

Linear OOA(2⁶¹, 1369, F₂, 3, 10) (dual of [(1369, 3), 4046, 11]-NRT-code), using OOA 3-folding based on linear OA(2⁶¹, 4107, F₂, 10) (dual of [4107, 4046, 11]-code), using

discarding factors / shortening the dual code based on linear OA(2⁶¹, 4109, F₂, 10) (dual of [4109, 4048, 11]-code), using
- 1 times truncation [i] based on linear OA(2⁶², 4110, F₂, 11) (dual of [4110, 4048, 12]-code), using
  - constructionÂ X4 applied to C_e(10) âŠ‚ C_e(8) [i] based on
    1. linear OA(2⁶¹, 4096, F₂, 11) (dual of [4096, 4035, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 2¹²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    2. 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]
    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: 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(2⁶¹, 1369, F₂, 4, 10) (dual of [(1369, 4), 5415, 11]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
2	Linear OOA(2⁶¹, 1369, F₂, 5, 10) (dual of [(1369, 5), 6784, 11]-NRT-code)	[i]
3	Linear OOA(2⁶¹, 1369, F₂, 6, 10) (dual of [(1369, 6), 8153, 11]-NRT-code)	[i]
4	Linear OOA(2⁶¹, 1369, F₂, 7, 10) (dual of [(1369, 7), 9522, 11]-NRT-code)	[i]
5	Linear OOA(2⁶¹, 1369, F₂, 8, 10) (dual of [(1369, 8), 10891, 11]-NRT-code)	[i]
6	Digital (51, 61, 1369)-net over F₂	[i]