MinT - Information on Result #1232883

Information on Result #1232883

Digital (86, 99, 2806082)-net over F₂₇, using generalized (u,Â u+v)-construction based on

digital (8, 12, 9882)-net over F₂₇, using
- net defined by OOA [i] based on linear OOA(27¹², 9882, F₂₇, 4, 4) (dual of [(9882, 4), 39516, 5]-NRT-code), using
  - OA 2-folding and stacking [i] based on linear OA(27¹², 19764, F₂₇, 4) (dual of [19764, 19752, 5]-code), using
    - generalized (u,Â u+v)-construction [i] based on
      1. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 0) (dual of [s, s, 1]-code) for arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
      2. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      3. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      4. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      5. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      6. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      7. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      8. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      9. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      10. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      11. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      12. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      13. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      14. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      15. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      16. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      17. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      18. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      19. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      20. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      21. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      22. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      23. linear OA(27⁰, 732, F₂₇, 0) (dual of [732, 732, 1]-code) (see above)
      24. linear OA(27¹, 732, F₂₇, 1) (dual of [732, 731, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
        parity-check code [i]
      25. linear OA(27¹, 732, F₂₇, 1) (dual of [732, 731, 2]-code) (see above)
      26. linear OA(27³, 732, F₂₇, 2) (dual of [732, 729, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(27³, 757, F₂₇, 2) (dual of [757, 754, 3]-code), using
        Hamming code H(3,27) [i]
      27. linear OA(27⁷, 732, F₂₇, 4) (dual of [732, 725, 5]-code), using
        constructionÂ XX applied to C₁Â =Â C([727,1]), C₂Â =Â C([0,2]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,1]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([727,2]) [i] based on
        linear OA(27⁵, 728, F₂₇, 3) (dual of [728, 723, 4]-code or 728-cap in PG(4,27)), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,1}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(27⁵, 728, F₂₇, 3) (dual of [728, 723, 4]-code or 728-cap in PG(4,27)), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(27⁷, 728, F₂₇, 4) (dual of [728, 721, 5]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,1,2}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(27³, 728, F₂₇, 2) (dual of [728, 725, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(27⁰, 2, F₂₇, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 0) (dual of [s, s, 1]-code) for arbitrarily large s (see above)
        linear OA(27⁰, 2, F₂₇, 0) (dual of [2, 2, 1]-code) (see above)
digital (20, 26, 1398100)-net over F₂₇, using
- s-reduction based on digital (20, 26, 2796201)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27²⁶, 2796201, F₂₇, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(27²⁶, large, F₂₇, 6) (dual of [large, large−26, 7]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
digital (48, 61, 1398100)-net over F₂₇, using
- net defined by OOA [i] based on linear OOA(27⁶¹, 1398100, F₂₇, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
  - OOA 6-folding and stacking with additional row [i] based on linear OA(27⁶¹, 8388601, F₂₇, 13) (dual of [8388601, 8388540, 14]-code), using
    - discarding factors / shortening the dual code based on linear OA(27⁶¹, large, F₂₇, 13) (dual of [large, large−61, 14]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’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.