MinT - Information on Result #675058

Information on Result #675058

Linear OA(3¹⁷⁵, 19732, F₃, 27) (dual of [19732, 19557, 28]-code), using constructionÂ X applied to C([0,13]) âŠ‚ C([0,10]) based on

linear OA(3¹⁶³, 19684, F₃, 27) (dual of [19684, 19521, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
linear OA(3¹²⁷, 19684, F₃, 21) (dual of [19684, 19557, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
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(3¹⁷⁵, 10626, F₃, 2, 27) (dual of [(10626, 2), 21077, 28]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
2	Linear OOA(3¹⁷⁵, 10626, F₃, 3, 27) (dual of [(10626, 3), 31703, 28]-NRT-code)	[i]
3	Linear OOA(3¹⁷⁵, 10626, F₃, 4, 27) (dual of [(10626, 4), 42329, 28]-NRT-code)	[i]
4	Linear OOA(3¹⁷⁵, 10626, F₃, 5, 27) (dual of [(10626, 5), 52955, 28]-NRT-code)	[i]
5	Digital (148, 175, 10626)-net over F₃	[i]
6	Linear OOA(3¹⁷⁵, 9866, F₃, 2, 27) (dual of [(9866, 2), 19557, 28]-NRT-code)	[i]	OOA Folding
7	Linear OOA(3¹⁷⁵, 4933, F₃, 4, 27) (dual of [(4933, 4), 19557, 28]-NRT-code)	[i]	OOA Folding