MinT - Information on Result #814611

Information on Result #814611

Linear OOA(3¹⁹³, 29551, F₃, 2, 27) (dual of [(29551, 2), 58909, 28]-NRT-code), using OOA 2-folding based on linear OA(3¹⁹³, 59102, F₃, 27) (dual of [59102, 58909, 28]-code), using

constructionÂ X applied to C([0,13]) âŠ‚ C([0,10]) [i] based on
1. linear OA(3¹⁸¹, 59050, F₃, 27) (dual of [59050, 58869, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050Â | 3²⁰âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
2. linear OA(3¹⁴¹, 59050, F₃, 21) (dual of [59050, 58909, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050Â | 3²⁰âˆ’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, 52, F₃, 5) (dual of [52, 40, 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(3¹⁹⁵, 29552, F₃, 2, 27) (dual of [(29552, 2), 58909, 28]-NRT-code)	[i]	NRT-Code Embedding in Larger Space
2	Linear OOA(3¹⁹⁴, 29551, F₃, 2, 27) (dual of [(29551, 2), 58908, 28]-NRT-code)	[i]	OOA Duplication
3	Linear OOA(3¹⁹³, 29551, F₃, 3, 27) (dual of [(29551, 3), 88460, 28]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
4	Linear OOA(3¹⁹³, 29551, F₃, 4, 27) (dual of [(29551, 4), 118011, 28]-NRT-code)	[i]
5	Linear OOA(3¹⁹³, 29551, F₃, 5, 27) (dual of [(29551, 5), 147562, 28]-NRT-code)	[i]
6	Digital (166, 193, 29551)-net over F₃	[i]