MinT - Information on Result #1189842

Information on Result #1189842

Digital (61, 68, 7689204)-net over F₇, using net defined by OOA based on linear OOA(7⁶⁸, 7689204, F₇, 9, 7) (dual of [(7689204, 9), 69202768, 8]-NRT-code), using

OOA stacking with additional row [i] based on linear OOA(7⁶⁸, 7689205, F₇, 3, 7) (dual of [(7689205, 3), 23067547, 8]-NRT-code), using
- generalized (u,Â u+v)-construction [i] based on
  1. linear OOA(7⁵, 2801, F₇, 3, 2) (dual of [(2801, 3), 8398, 3]-NRT-code), using
    - appending 1 arbitrary column [i] based on linear OOA(7⁵, 2801, F₇, 2, 2) (dual of [(2801, 2), 5597, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(7⁵, 2801, F₇, 2) (dual of [2801, 2796, 3]-code), using
        Hamming code H(5,7) [i]
  2. linear OOA(7¹³, 3843202, F₇, 3, 3) (dual of [(3843202, 3), 11529593, 4]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(7¹³, 6120100, F₇, 3, 3) (dual of [(6120100, 3), 18360287, 4]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OA(7¹³, 6120101, F₇, 3) (dual of [6120101, 6120088, 4]-code or 6120101-cap in PG(12,7)), using
        product of projective cap with cap in AG(6,Â 7) [i] based on linear OA(7⁷, 2499, F₇, 3) (dual of [2499, 2492, 4]-code or 2499-cap in PG(6,7)), using
        product of two caps with tangent hyperplane [i]
  3. linear OOA(7⁵⁰, 3843202, F₇, 3, 7) (dual of [(3843202, 3), 11529556, 8]-NRT-code), using
    - trace code [i] based on linear OOA(49²⁵, 1921601, F₄₉, 3, 7) (dual of [(1921601, 3), 5764778, 8]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(49²⁵, 5764803, F₄₉, 7) (dual of [5764803, 5764778, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(49²⁵, 5764805, F₄₉, 7) (dual of [5764805, 5764780, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(49²⁵, 5764801, F₄₉, 7) (dual of [5764801, 5764776, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(49²¹, 5764801, F₄₉, 6) (dual of [5764801, 5764780, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [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, t and s.

Other Results with Identical Parameters

None.

Depending Results

None.