MinT - Information on Result #1229859

Information on Result #1229859

Digital (146, 159, 5608851)-net over F₈, using generalized (u,Â u+v)-construction based on

digital (1, 4, 65)-net over F₈, using
- net defined by OOA [i] based on 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]
digital (12, 16, 16386)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹⁶, 16386, F₈, 4, 4) (dual of [(16386, 4), 65528, 5]-NRT-code), using
  - appending kth column [i] based on linear OOA(8¹⁶, 16386, F₈, 3, 4) (dual of [(16386, 3), 49142, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(8¹⁶, 32772, F₈, 4) (dual of [32772, 32756, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹⁶, 32773, F₈, 4) (dual of [32773, 32757, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(8¹⁶, 32768, F₈, 4) (dual of [32768, 32752, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(8¹¹, 32768, F₈, 3) (dual of [32768, 32757, 4]-code or 32768-cap in PG(10,8)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(8⁰, 5, F₈, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
digital (35, 41, 2796200)-net over F₈, using
- s-reduction based on digital (35, 41, 2796201)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁴¹, 2796201, F₈, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(8⁴¹, large, F₈, 6) (dual of [large, large−41, 7]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
digital (85, 98, 2796200)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁹⁸, 2796200, F₈, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
  - OOA 3-folding and stacking with additional row [i] based on linear OOA(8⁹⁸, 8388601, F₈, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(8⁹⁸, 8388602, F₈, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
      - trace code [i] based on linear OOA(64⁴⁹, 4194301, F₆₄, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁴⁹, 8388602, F₆₄, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (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, t and s.

Other Results with Identical Parameters

None.

Depending Results

None.