MinT - Information on Result #1051399

Information on Result #1051399

Linear OOA(2¹⁰⁹, 65540, F₂, 4, 12) (dual of [(65540, 4), 262051, 13]-NRT-code), using OOA 4-folding based on linear OA(2¹⁰⁹, 262160, F₂, 12) (dual of [262160, 262051, 13]-code), using

discarding factors / shortening the dual code based on linear OA(2¹⁰⁹, 262163, F₂, 12) (dual of [262163, 262054, 13]-code), using
- 1 times truncation [i] based on linear OA(2¹¹⁰, 262164, F₂, 13) (dual of [262164, 262054, 14]-code), using
  - constructionÂ X4 applied to C_e(12) âŠ‚ C_e(10) [i] based on
    1. linear OA(2¹⁰⁹, 262144, F₂, 13) (dual of [262144, 262035, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 2¹⁸âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    2. linear OA(2⁹¹, 262144, F₂, 11) (dual of [262144, 262053, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 2¹⁸âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(2¹⁹, 20, F₂, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
      - dual of repetition code with length 20 [i]
    4. linear OA(2¹, 20, F₂, 1) (dual of [20, 19, 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(2¹⁰⁹, 65540, F₂, 5, 12) (dual of [(65540, 5), 327591, 13]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
2	Linear OOA(2¹⁰⁹, 65540, F₂, 6, 12) (dual of [(65540, 6), 393131, 13]-NRT-code)	[i]
3	Linear OOA(2¹⁰⁹, 65540, F₂, 7, 12) (dual of [(65540, 7), 458671, 13]-NRT-code)	[i]
4	Linear OOA(2¹⁰⁹, 65540, F₂, 8, 12) (dual of [(65540, 8), 524211, 13]-NRT-code)	[i]
5	Digital (97, 109, 65540)-net over F₂	[i]