MinT - Information on Result #2267496

Information on Result #2267496

Linear OA(3²³⁴, 783, F₃, 56) (dual of [783, 549, 57]-code), using 2 times code embedding in larger space based on linear OA(3²³², 781, F₃, 56) (dual of [781, 549, 57]-code), using

constructionÂ XX applied to C₁Â =Â C([342,391]), C₂Â =Â C([336,385]), C₃Â =Â C₁Â +Â C₂Â =Â C([342,385]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([336,391]) [i] based on
1. linear OA(3¹⁹⁹, 728, F₃, 50) (dual of [728, 529, 51]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {342,343,…,391}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 51 [i]
2. linear OA(3¹⁹⁶, 728, F₃, 50) (dual of [728, 532, 51]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {336,337,…,385}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 51 [i]
3. linear OA(3²¹⁴, 728, F₃, 56) (dual of [728, 514, 57]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {336,337,…,391}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 57 [i]
4. linear OA(3¹⁷⁵, 728, F₃, 44) (dual of [728, 553, 45]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {342,343,…,385}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]
5. 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
    1. 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]
    2. 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]
6. linear OA(3⁸, 25, F₃, 5) (dual of [25, 17, 6]-code), using
  - discarding factors / shortening the dual code based on linear OA(3⁸, 26, F₃, 5) (dual of [26, 18, 6]-code), using
    - dual code (with bound on d by constructionÂ Y1) [i] based on
      1. linear OA(3¹⁸, 26, F₃, 12) (dual of [26, 8, 13]-code), using
        2 times truncation [i] based on linear OA(3²⁰, 28, F₃, 14) (dual of [28, 8, 15]-code) (see above)
      2. nonexistence of linear OA(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code) (see above)

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		This result only	Method
1	Linear OOA(3²³⁴, 261, F₃, 3, 56) (dual of [(261, 3), 549, 57]-NRT-code)	[i]		OOA Folding