MinT - Information on Result #678321

Information on Result #678321

Linear OA(4²⁵², 262213, F₄, 35) (dual of [262213, 261961, 36]-code), using constructionÂ X applied to C([0,17]) âŠ‚ C([0,13]) based on

linear OA(4²³⁵, 262145, F₄, 35) (dual of [262145, 261910, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 4¹⁸âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
linear OA(4¹⁸¹, 262145, F₄, 27) (dual of [262145, 261964, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 4¹⁸âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [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(4²⁵², 166521, F₄, 2, 35) (dual of [(166521, 2), 332790, 36]-NRT-code)	[i]	Embedding of OOA with Gilbertâ€“VarÅ¡amov Bound
2	Linear OOA(4²⁵², 166521, F₄, 3, 35) (dual of [(166521, 3), 499311, 36]-NRT-code)	[i]
3	Digital (217, 252, 166521)-net over F₄	[i]