MinT - Information on Result #1224630

Information on Result #1224630

Digital (44, 53, 2113582)-net over F₃₂, using (u,Â u+v)-construction based on

digital (8, 12, 16432)-net over F₃₂, using
- net defined by OOA [i] based on linear OOA(32¹², 16432, F₃₂, 4, 4) (dual of [(16432, 4), 65716, 5]-NRT-code), using
  - OA 2-folding and stacking [i] based on linear OA(32¹², 32864, F₃₂, 4) (dual of [32864, 32852, 5]-code), using
    - generalized (u,Â u+v)-construction [i] based on
      1. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 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(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      3. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      4. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      5. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      6. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      7. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      8. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      9. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      10. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      11. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      12. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      13. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      14. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      15. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      16. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      17. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      18. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      19. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      20. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      21. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      22. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      23. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      24. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      25. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      26. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      27. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      28. linear OA(32⁰, 1027, F₃₂, 0) (dual of [1027, 1027, 1]-code) (see above)
      29. linear OA(32¹, 1027, F₃₂, 1) (dual of [1027, 1026, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(32¹, s, F₃₂, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
        parity-check code [i]
      30. linear OA(32¹, 1027, F₃₂, 1) (dual of [1027, 1026, 2]-code) (see above)
      31. linear OA(32³, 1027, F₃₂, 2) (dual of [1027, 1024, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(32³, 1057, F₃₂, 2) (dual of [1057, 1054, 3]-code), using
        Hamming code H(3,32) [i]
      32. linear OA(32⁷, 1027, F₃₂, 4) (dual of [1027, 1020, 5]-code), using
        constructionÂ XX applied to C₁Â =Â C([1022,1]), 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([1022,2]) [i] based on
        linear OA(32⁵, 1023, F₃₂, 3) (dual of [1023, 1018, 4]-code or 1023-cap in PG(4,32)), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,1}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(32⁵, 1023, F₃₂, 3) (dual of [1023, 1018, 4]-code or 1023-cap in PG(4,32)), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(32⁷, 1023, F₃₂, 4) (dual of [1023, 1016, 5]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,1,2}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(32³, 1023, F₃₂, 2) (dual of [1023, 1020, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) for arbitrarily large s (see above)
        linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 1]-code) (see above)
digital (32, 41, 2097150)-net over F₃₂, using
- net defined by OOA [i] based on linear OOA(32⁴¹, 2097150, F₃₂, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
  - OOA 4-folding and stacking with additional row [i] based on linear OA(32⁴¹, 8388601, F₃₂, 9) (dual of [8388601, 8388560, 10]-code), using
    - discarding factors / shortening the dual code based on linear OA(32⁴¹, large, F₃₂, 9) (dual of [large, large−41, 10]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (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

The following results depend on this result:

	Result		This result only	Method
1	Digital (60, 69, 4227164)-net over F₃₂	[i]		(u,Â u+v)-Construction for Nets