MinT - Information on Result #814620

Information on Result #814620

Linear OOA(3²¹¹, 88602, F₃, 2, 27) (dual of [(88602, 2), 176993, 28]-NRT-code), using OOA 2-folding based on linear OA(3²¹¹, 177204, F₃, 27) (dual of [177204, 176993, 28]-code), using

constructionÂ X applied to C([0,13]) âŠ‚ C([0,10]) [i] based on
1. linear OA(3¹⁹⁹, 177148, F₃, 27) (dual of [177148, 176949, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148Â | 3²²âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
2. linear OA(3¹⁵⁵, 177148, F₃, 21) (dual of [177148, 176993, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148Â | 3²²âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
3. linear OA(3¹², 56, F₃, 5) (dual of [56, 44, 6]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3⁴, 28, F₃, 2) (dual of [28, 24, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]
    2. 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(3²¹³, 88603, F₃, 2, 27) (dual of [(88603, 2), 176993, 28]-NRT-code)	[i]	NRT-Code Embedding in Larger Space
2	Linear OOA(3²¹², 88602, F₃, 2, 27) (dual of [(88602, 2), 176992, 28]-NRT-code)	[i]	OOA Duplication
3	Linear OOA(3²¹¹, 70004, F₃, 3, 27) (dual of [(70004, 3), 209801, 28]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
4	Linear OOA(3²¹¹, 70004, F₃, 4, 27) (dual of [(70004, 4), 279805, 28]-NRT-code)	[i]
5	Linear OOA(3²¹¹, 70004, F₃, 5, 27) (dual of [(70004, 5), 349809, 28]-NRT-code)	[i]
6	Digital (184, 211, 70004)-net over F₃	[i]