MinT - Information on Result #705054

Information on Result #705054

Linear OA(3¹⁸⁴, 790, F₃, 43) (dual of [790, 606, 44]-code), using constructionÂ XX applied to C₁Â =Â C([722,30]), C₂Â =Â C([0,36]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,30]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([722,36]) based on

linear OA(3¹⁴², 728, F₃, 37) (dual of [728, 586, 38]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {−6,−5,…,30}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
linear OA(3¹⁴², 728, F₃, 37) (dual of [728, 586, 38]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [0,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
linear OA(3¹⁶⁶, 728, F₃, 43) (dual of [728, 562, 44]-code), using the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {−6,−5,…,36}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [i]
linear OA(3¹¹⁸, 728, F₃, 31) (dual of [728, 610, 32]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [0,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
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]
linear OA(3¹⁰, 34, F₃, 5) (dual of [34, 24, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁰, 36, F₃, 5) (dual of [36, 26, 6]-code), using
  - (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
    1. linear OA(3¹, 12, F₃, 1) (dual of [12, 11, 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, 12, F₃, 2) (dual of [12, 9, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]
    3. linear OA(3⁶, 12, F₃, 5) (dual of [12, 6, 6]-code), using
      - extended Golay code G_e(3) [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 OA(3¹⁸⁵, 791, F₃, 43) (dual of [791, 606, 44]-code)	[i]	Code Embedding in Larger Space
2	Linear OA(3¹⁸³, 789, F₃, 42) (dual of [789, 606, 43]-code)	[i]	Truncation
3	Linear OOA(3¹⁸⁴, 395, F₃, 2, 43) (dual of [(395, 2), 606, 44]-NRT-code)	[i]	OOA Folding
4	Linear OOA(3¹⁸⁴, 263, F₃, 3, 43) (dual of [(263, 3), 605, 44]-NRT-code)	[i]	OOA Folding