MinT - Information on Result #815906

Information on Result #815906

Linear OOA(3²²⁹, 9866, F₃, 2, 37) (dual of [(9866, 2), 19503, 38]-NRT-code), using OOA 2-folding based on linear OA(3²²⁹, 19732, F₃, 37) (dual of [19732, 19503, 38]-code), using

constructionÂ X applied to C([0,18]) âŠ‚ C([0,15]) [i] based on
1. linear OA(3²¹⁷, 19684, F₃, 37) (dual of [19684, 19467, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
2. linear OA(3¹⁸¹, 19684, F₃, 31) (dual of [19684, 19503, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
3. 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(3²²⁹, 9866, F₃, 2, 36) (dual of [(9866, 2), 19503, 37]-NRT-code)	[i]	Strength Reduction for OOAs
2	Linear OOA(3²³¹, 9867, F₃, 2, 37) (dual of [(9867, 2), 19503, 38]-NRT-code)	[i]	NRT-Code Embedding in Larger Space
3	Linear OOA(3²³⁰, 9866, F₃, 2, 37) (dual of [(9866, 2), 19502, 38]-NRT-code)	[i]	OOA Duplication
4	Linear OOA(3²²⁹, 9866, F₃, 3, 37) (dual of [(9866, 3), 29369, 38]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
5	Linear OOA(3²²⁹, 9866, F₃, 4, 37) (dual of [(9866, 4), 39235, 38]-NRT-code)	[i]
6	Linear OOA(3²²⁹, 9866, F₃, 5, 37) (dual of [(9866, 5), 49101, 38]-NRT-code)	[i]
7	Digital (192, 229, 9866)-net over F₃	[i]