MinT - Information on Result #675554

Information on Result #675554

Linear OA(3²⁴⁵, 6605, F₃, 44) (dual of [6605, 6360, 45]-code), using constructionÂ X applied to C_e(43) âŠ‚ C_e(37) based on

linear OA(3²³³, 6561, F₃, 44) (dual of [6561, 6328, 45]-code), using an extension C_e(43) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [i]
linear OA(3²⁰¹, 6561, F₃, 38) (dual of [6561, 6360, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
linear OA(3¹², 44, F₃, 5) (dual of [44, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹², 54, F₃, 5) (dual of [54, 42, 6]-code), using
  - (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
    1. linear OA(3¹, 13, F₃, 1) (dual of [13, 12, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
        parity-check code [i]
    2. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]
    3. linear OA(3⁸, 28, F₃, 5) (dual of [28, 20, 6]-code), using
      - dual code (with bound on d by constructionÂ Y1) [i] based on
        linear OA(3²⁰, 28, F₃, 14) (dual of [28, 8, 15]-code), using
        contraction [i] based on linear OA(3⁴⁸, 56, F₃, 29) (dual of [56, 8, 30]-code), using the BCH-code C(I) with length 56Â | 3⁶âˆ’1, defining interval IÂ =Â {0,1,…,28}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
        nonexistence of linear OA(3¹⁹, 23, F₃, 14) (dual of [23, 4, 15]-code), because
        2 times truncation [i] would yield linear OA(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code), but
        â€œHNâ€ bound on codes from Brouwerâ€™s database [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(3²⁴⁵, 4842, F₃, 2, 44) (dual of [(4842, 2), 9439, 45]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
2	Linear OOA(3²⁴⁵, 4842, F₃, 3, 44) (dual of [(4842, 3), 14281, 45]-NRT-code)	[i]
3	Linear OOA(3²⁴⁵, 4842, F₃, 4, 44) (dual of [(4842, 4), 19123, 45]-NRT-code)	[i]
4	Linear OOA(3²⁴⁵, 4842, F₃, 5, 44) (dual of [(4842, 5), 23965, 45]-NRT-code)	[i]
5	Digital (201, 245, 4842)-net over F₃	[i]
6	Linear OOA(3²⁴⁵, 1321, F₃, 5, 44) (dual of [(1321, 5), 6360, 45]-NRT-code)	[i]	OOA Folding