MinT - Information on Result #1523210

Information on Result #1523210

Linear OOA(3¹³⁰, 9902, F₃, 3, 20) (dual of [(9902, 3), 29576, 21]-NRT-code), using embedding of OOA with Gilbertâ€“VarÅ¡amov bound based on linear OA(3¹³⁰, 9902, F₃, 20) (dual of [9902, 9772, 21]-code), using

discarding factors / shortening the dual code based on linear OA(3¹³⁰, 19731, F₃, 20) (dual of [19731, 19601, 21]-code), using
- constructionÂ X applied to C_e(19) âŠ‚ C_e(13) [i] based on
  1. linear OA(3¹¹⁸, 19683, F₃, 20) (dual of [19683, 19565, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
  2. linear OA(3⁸², 19683, F₃, 14) (dual of [19683, 19601, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [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(3¹³⁰, 4951, F₃, 5, 20) (dual of [(4951, 5), 24625, 21]-NRT-code)	[i]		OOA Folding