MinT - Information on Result #2697393

Information on Result #2697393

Linear OOA(7¹¹⁰, large, F₇, 3, 8), using 7 times NRT-code embedding in larger space based on linear OOA(7⁸⁹, large, F₇, 3, 8), using

generalized (u,Â u+v)-construction [i] based on
1. linear OOA(7⁹, 2796201, F₇, 3, 2) (dual of [(2796201, 3), 8388594, 3]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(7⁹, 6725601, F₇, 3, 2) (dual of [(6725601, 3), 20176794, 3]-NRT-code), using
    - appending 1 arbitrary column [i] based on linear OOA(7⁹, 6725601, F₇, 2, 2) (dual of [(6725601, 2), 13451193, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(7⁹, 6725601, F₇, 2) (dual of [6725601, 6725592, 3]-code), using
        Hamming code H(9,7) [i]
2. linear OOA(7²⁵, 2796201, F₇, 3, 4) (dual of [(2796201, 3), 8388578, 5]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(7²⁵, 2882404, F₇, 3, 4) (dual of [(2882404, 3), 8647187, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(7²⁵, 5764808, F₇, 4) (dual of [5764808, 5764783, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(7²⁵, 5764809, F₇, 4) (dual of [5764809, 5764784, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(7²⁵, 5764801, F₇, 4) (dual of [5764801, 5764776, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 7⁸âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(7¹⁷, 5764801, F₇, 3) (dual of [5764801, 5764784, 4]-code or 5764801-cap in PG(16,7)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 7⁸âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(7⁰, 8, F₇, 0) (dual of [8, 8, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(7⁰, s, F₇, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
3. linear OOA(7⁵⁵, 2796201, F₇, 3, 8) (dual of [(2796201, 3), 8388548, 9]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(7⁵⁵, large, F₇, 8) (dual of [large, large−55, 9]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 20176803Â | 7⁹âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

Mode: Constructive and linear.

Optimality

Show details for fixed k and m, k and t, m and t.

Other Results with Identical Parameters

None.

Depending Results

None.