MinT - Information on Result #1224523

Information on Result #1224523

Digital (18, 22, large)-net over F₃₂, using (u,Â u+v)-construction based on

digital (0, 2, 33)-net over F₃₂, using
- kÂ =Â 2 construction [i]
digital (16, 20, 8388602)-net over F₃₂, using
- net defined by OOA [i] based on linear OOA(32²⁰, 8388602, F₃₂, 6, 4) (dual of [(8388602, 6), 50331592, 5]-NRT-code), using
  - OOA stacking with additional row [i] based on linear OOA(32²⁰, large, F₃₂, 2, 4), using
    - generalized (u,Â u+v)-construction [i] based on
      1. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(32⁰, s, F₃₂, 2, 0) for arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
      2. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      3. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      4. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      5. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      6. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      7. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      8. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      9. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      10. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      11. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      12. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      13. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      14. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      15. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      16. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      17. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      18. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      19. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      20. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      21. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      22. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      23. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      24. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      25. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      26. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      27. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      28. linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
      29. linear OOA(32¹, 524290, F₃₂, 2, 1) (dual of [(524290, 2), 1048579, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(32¹, s, F₃₂, 2, 1) for arbitrarily large s, using
        appending 1 arbitrary column [i] 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 OOA(32¹, 524290, F₃₂, 2, 1) (dual of [(524290, 2), 1048579, 2]-NRT-code) (see above)
      31. linear OOA(32⁵, 524290, F₃₂, 2, 2) (dual of [(524290, 2), 1048575, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(32⁵, 1082401, F₃₂, 2, 2) (dual of [(1082401, 2), 2164797, 3]-NRT-code), using
        appending kth column [i] based on linear OA(32⁵, 1082401, F₃₂, 2) (dual of [1082401, 1082396, 3]-code), using
        Hamming code H(5,32) [i]
      32. linear OOA(32¹³, 524290, F₃₂, 2, 4) (dual of [(524290, 2), 1048567, 5]-NRT-code), using
        OOA 2-folding [i] based on linear OA(32¹³, 1048580, F₃₂, 4) (dual of [1048580, 1048567, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(32¹³, 1048576, F₃₂, 4) (dual of [1048576, 1048563, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(32⁹, 1048576, F₃₂, 3) (dual of [1048576, 1048567, 4]-code or 1048576-cap in PG(8,32)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(32⁰, 4, F₃₂, 0) (dual of [4, 4, 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]

Mode: Constructive and linear.

Optimality

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

Other Results with Identical Parameters

None.

Depending Results

The following results depend on this result:

	Result		Method
1	Digital (19, 22, large)-net over F₃₂	[i]	t-Expansion
2	Digital (19, 23, large)-net over F₃₂	[i]	Net Duplication
3	Digital (18, 21, large)-net over F₃₂	[i]	m-Reduction