MinT - Information on Result #1523705

Information on Result #1523705

Linear OOA(3¹³⁹, 11559, F₃, 3, 21) (dual of [(11559, 3), 34538, 22]-NRT-code), using embedding of OOA with Gilbertâ€“VarÅ¡amov bound based on linear OA(3¹³⁹, 11559, F₃, 21) (dual of [11559, 11420, 22]-code), using

discarding factors / shortening the dual code based on linear OA(3¹³⁹, 19732, F₃, 21) (dual of [19732, 19593, 22]-code), using
- constructionÂ X applied to C([0,10]) âŠ‚ C([0,7]) [i] based on
  1. 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]
  2. linear OA(3⁹¹, 19684, F₃, 15) (dual of [19684, 19593, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (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
        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]
        linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [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
        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: 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(3¹³⁹, 5779, F₃, 5, 21) (dual of [(5779, 5), 28756, 22]-NRT-code)	[i]		OOA Folding