MinT - Information on Result #678487

Information on Result #678487

Linear OA(4²²⁵, 262212, F₄, 31) (dual of [262212, 261987, 32]-code), using constructionÂ X applied to C_e(30) âŠ‚ C_e(22) based on

linear OA(4²⁰⁸, 262144, F₄, 31) (dual of [262144, 261936, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
linear OA(4¹⁵⁴, 262144, F₄, 23) (dual of [262144, 261990, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
linear OA(4¹⁷, 68, F₄, 7) (dual of [68, 51, 8]-code), using
- generalized (u,Â u+v)-construction [i] based on
  1. linear OA(4¹, 17, F₄, 1) (dual of [17, 16, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
      - parity-check code [i]
  2. linear OA(4³, 17, F₄, 2) (dual of [17, 14, 3]-code), using
    - discarding factors / shortening the dual code based on linear OA(4³, 21, F₄, 2) (dual of [21, 18, 3]-code), using
      - Hamming code H(3,4) [i]
  3. linear OA(4⁴, 17, F₄, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
    - ovoid in PG(3,Â 4) [i]
  4. linear OA(4⁹, 17, F₄, 7) (dual of [17, 8, 8]-code), using
    - dual code (with bound on d by constructionÂ Y1) [i] based on
      1. linear OA(4⁸, 17, F₄, 6) (dual of [17, 9, 7]-code), using
        the BCH-code C(I) with length 17Â | 4⁴âˆ’1, defining interval IÂ =Â {1,4,7,…,16}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
      2. nonexistence of linear OA(4⁷, 10, F₄, 6) (dual of [10, 3, 7]-code), because
        bound for almost-MDS-codes with dimension nÂ =Â 3 [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(4²²⁵, 173915, F₄, 2, 31) (dual of [(173915, 2), 347605, 32]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
2	Linear OOA(4²²⁵, 173915, F₄, 3, 31) (dual of [(173915, 3), 521520, 32]-NRT-code)	[i]
3	Digital (194, 225, 173915)-net over F₄	[i]