MinT - Information on Result #1221051

Information on Result #1221051

Digital (56, 67, 1693969)-net over F₂₅, using (u,Â u+v)-construction based on

digital (11, 16, 16249)-net over F₂₅, using
- net defined by OOA [i] based on linear OOA(25¹⁶, 16249, F₂₅, 6, 5) (dual of [(16249, 6), 97478, 6]-NRT-code), using
  - OOA stacking with additional row [i] based on linear OOA(25¹⁶, 16250, F₂₅, 2, 5) (dual of [(16250, 2), 32484, 6]-NRT-code), using
    - generalized (u,Â u+v)-construction [i] based on
      1. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(25⁰, s, F₂₅, 2, 0) for arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
      2. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      3. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      4. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      5. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      6. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      7. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      8. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      9. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      10. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      11. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      12. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      13. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      14. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      15. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      16. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      17. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      18. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      19. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      20. linear OOA(25⁰, 650, F₂₅, 2, 0) (dual of [(650, 2), 1300, 1]-NRT-code) (see above)
      21. linear OOA(25¹, 650, F₂₅, 2, 1) (dual of [(650, 2), 1299, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(25¹, s, F₂₅, 2, 1) for arbitrarily large s, using
        appending 1 arbitrary column [i] based on linear OA(25¹, s, F₂₅, 1) (dual of [s, s−1, 2]-code) for arbitrarily large s, using
        parity-check code [i]
      22. linear OOA(25¹, 650, F₂₅, 2, 1) (dual of [(650, 2), 1299, 2]-NRT-code) (see above)
      23. linear OOA(25¹, 650, F₂₅, 2, 1) (dual of [(650, 2), 1299, 2]-NRT-code) (see above)
      24. linear OOA(25³, 650, F₂₅, 2, 2) (dual of [(650, 2), 1297, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(25³, 651, F₂₅, 2, 2) (dual of [(651, 2), 1299, 3]-NRT-code), using
        appending kth column [i] based on linear OA(25³, 651, F₂₅, 2) (dual of [651, 648, 3]-code), using
        Hamming code H(3,25) [i]
      25. linear OOA(25¹⁰, 650, F₂₅, 2, 5) (dual of [(650, 2), 1290, 6]-NRT-code), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(25⁰, s, F₂₅, 2, 0) for arbitrarily large s (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25⁰, 26, F₂₅, 2, 0) (dual of [(26, 2), 52, 1]-NRT-code) (see above)
        linear OOA(25¹, 26, F₂₅, 2, 1) (dual of [(26, 2), 51, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(25¹, s, F₂₅, 2, 1) for arbitrarily large s (see above)
        linear OOA(25¹, 26, F₂₅, 2, 1) (dual of [(26, 2), 51, 2]-NRT-code) (see above)
        linear OOA(25¹, 26, F₂₅, 2, 1) (dual of [(26, 2), 51, 2]-NRT-code) (see above)
        linear OOA(25², 26, F₂₅, 2, 2) (dual of [(26, 2), 50, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(2;50,25) [i]
        linear OOA(25⁵, 26, F₂₅, 2, 5) (dual of [(26, 2), 47, 6]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(2;47,25) [i]
digital (40, 51, 1677720)-net over F₂₅, using
- net defined by OOA [i] based on linear OOA(25⁵¹, 1677720, F₂₅, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
  - OOA 5-folding and stacking with additional row [i] based on linear OA(25⁵¹, 8388601, F₂₅, 11) (dual of [8388601, 8388550, 12]-code), using
    - discarding factors / shortening the dual code based on linear OA(25⁵¹, large, F₂₅, 11) (dual of [large, large−51, 12]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (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 (77, 88, 3387938)-net over F₂₅	[i]		(u,Â u+v)-Construction for Nets