MinT - Information on Result #954574

Information on Result #954574

Linear OOA(3²¹¹, 6577, F₃, 3, 34) (dual of [(6577, 3), 19520, 35]-NRT-code), using OOA 3-folding based on linear OA(3²¹¹, 19731, F₃, 34) (dual of [19731, 19520, 35]-code), using

constructionÂ X applied to C_e(33) âŠ‚ C_e(27) [i] based on
1. linear OA(3¹⁹⁹, 19683, F₃, 34) (dual of [19683, 19484, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
2. linear OA(3¹⁶³, 19683, F₃, 28) (dual of [19683, 19520, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
3. linear OA(3¹², 48, F₃, 5) (dual of [48, 36, 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(3²¹¹, 6577, F₃, 3, 33) (dual of [(6577, 3), 19520, 34]-NRT-code)	[i]	Strength Reduction for OOAs
2	Linear OOA(3²¹⁴, 6578, F₃, 3, 34) (dual of [(6578, 3), 19520, 35]-NRT-code)	[i]	NRT-Code Embedding in Larger Space
3	Linear OOA(3²¹², 6577, F₃, 3, 34) (dual of [(6577, 3), 19519, 35]-NRT-code)	[i]	OOA Duplication
4	Linear OOA(3²¹³, 6577, F₃, 3, 34) (dual of [(6577, 3), 19518, 35]-NRT-code)	[i]	OOA Duplication