MinT - Information on Result #675370

Information on Result #675370

Linear OA(3²⁰⁵, 6606, F₃, 37) (dual of [6606, 6401, 38]-code), using constructionÂ X applied to C([0,18]) âŠ‚ C([0,15]) based on

linear OA(3¹⁹³, 6562, F₃, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
linear OA(3¹⁶¹, 6562, F₃, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
linear OA(3¹², 44, F₃, 5) (dual of [44, 32, 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 OA(3²⁰⁵, 6606, F₃, 36) (dual of [6606, 6401, 37]-code)	[i]	Strength Reduction
2	Linear OOA(3²⁰⁵, 4166, F₃, 2, 37) (dual of [(4166, 2), 8127, 38]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
3	Linear OOA(3²⁰⁵, 4166, F₃, 3, 37) (dual of [(4166, 3), 12293, 38]-NRT-code)	[i]
4	Linear OOA(3²⁰⁵, 4166, F₃, 4, 37) (dual of [(4166, 4), 16459, 38]-NRT-code)	[i]
5	Linear OOA(3²⁰⁵, 4166, F₃, 5, 37) (dual of [(4166, 5), 20625, 38]-NRT-code)	[i]
6	Digital (168, 205, 4166)-net over F₃	[i]
7	Linear OOA(3²⁰⁵, 3303, F₃, 2, 37) (dual of [(3303, 2), 6401, 38]-NRT-code)	[i]	OOA Folding
8	Linear OOA(3²⁰⁵, 2202, F₃, 3, 37) (dual of [(2202, 3), 6401, 38]-NRT-code)	[i]	OOA Folding