MinT - Information on Result #1022006

Information on Result #1022006

Linear OOA(8¹³⁸, 5592922, F₈, 3, 15) (dual of [(5592922, 3), 16778628, 16]-NRT-code), using (u,Â u+v)-construction based on

linear OOA(8²⁴, 520, F₈, 3, 7) (dual of [(520, 3), 1536, 8]-NRT-code), using
- generalized (u,Â u+v)-construction [i] based on
  1. linear OOA(8⁰, 65, F₈, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(8⁰, s, F₈, 3, 0) for arbitrarily large s, using
      - OOA with only one run / dual of NRT-code without redundancy [i]
  2. linear OOA(8¹, 65, F₈, 3, 1) (dual of [(65, 3), 194, 2]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(8¹, s, F₈, 3, 1) for arbitrarily large s, using
      - appending 2 arbitrary columns [i] based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
        parity-check code [i]
  3. linear OOA(8¹, 65, F₈, 3, 1) (dual of [(65, 3), 194, 2]-NRT-code) (see above)
  4. linear OOA(8¹, 65, F₈, 3, 1) (dual of [(65, 3), 194, 2]-NRT-code) (see above)
  5. linear OOA(8¹, 65, F₈, 3, 1) (dual of [(65, 3), 194, 2]-NRT-code) (see above)
  6. linear OOA(8³, 65, F₈, 3, 2) (dual of [(65, 3), 192, 3]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(8³, 73, F₈, 3, 2) (dual of [(73, 3), 216, 3]-NRT-code), using
      - appending 1 arbitrary column [i] based on linear OOA(8³, 73, F₈, 2, 2) (dual of [(73, 2), 143, 3]-NRT-code), using
        appending kth column [i] based on linear OA(8³, 73, F₈, 2) (dual of [73, 70, 3]-code), using
        Hamming code H(3,8) [i]
  7. linear OOA(8⁴, 65, F₈, 3, 3) (dual of [(65, 3), 191, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(8⁴, 65, F₈, 2, 3) (dual of [(65, 2), 126, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8⁴, 65, F₈, 3) (dual of [65, 61, 4]-code or 65-cap in PG(3,8)), using
        ovoid in PG(3,Â 8) [i]
  8. linear OOA(8¹³, 65, F₈, 3, 7) (dual of [(65, 3), 182, 8]-NRT-code), using
    - extracting embedded OOA [i] based on digital (6, 13, 65)-net over F₈, using
      - base reduction for projective spaces (embedding PG(6,64) in PG(12,8)) for nets [i] based on digital (0, 7, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
linear OOA(8¹¹⁴, 5592402, F₈, 3, 15) (dual of [(5592402, 3), 16777092, 16]-NRT-code), using
- trace code [i] based on linear OOA(64⁵⁷, 2796201, F₆₄, 3, 15) (dual of [(2796201, 3), 8388546, 16]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(64⁵⁷, large, F₆₄, 15) (dual of [large, large−57, 16]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

Mode: Constructive and linear.

Optimality

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

Other Results with Identical Parameters

None.

Depending Results

None.