MinT - Information on Result #2267643

Information on Result #2267643

Linear OA(3²³⁹, 759, F₃, 61) (dual of [759, 520, 62]-code), using 1 times code embedding in larger space based on linear OA(3²³⁸, 758, F₃, 61) (dual of [758, 520, 62]-code), using

constructionÂ XX applied to C₁Â =Â C([307,365]), C₂Â =Â C([313,367]), C₃Â =Â C₁Â +Â C₂Â =Â C([313,365]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([307,367]) [i] based on
1. linear OA(3²²³, 728, F₃, 59) (dual of [728, 505, 60]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {307,308,…,365}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 60 [i]
2. linear OA(3²¹⁴, 728, F₃, 55) (dual of [728, 514, 56]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {313,314,…,367}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 56 [i]
3. linear OA(3²²⁹, 728, F₃, 61) (dual of [728, 499, 62]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {307,308,…,367}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 62 [i]
4. linear OA(3²⁰⁸, 728, F₃, 53) (dual of [728, 520, 54]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {313,314,…,365}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 54 [i]
5. linear OA(3⁸, 23, F₃, 5) (dual of [23, 15, 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), 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(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code), because
        â€œHNâ€ bound on codes from Brouwerâ€™s database [i]
6. linear OA(3¹, 7, F₃, 1) (dual of [7, 6, 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]

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(3²³⁹, 253, F₃, 3, 61) (dual of [(253, 3), 520, 62]-NRT-code)	[i]		OOA Folding