MinT - Best Known (20−4, 20, s)-Nets in Base 49

Best Known (20−4, 20, s)-Nets in Base 49

(20−4, 20, large)-Net over F₄₉ — Constructive and digital

Digital (16, 20, large)-net over F₄₉, using

49¹ times duplication [i] based on digital (15, 19, large)-net over F₄₉, using
- net defined by OOA [i] based on linear OOA(49¹⁹, large, F₄₉, 4, 4), using
  - appending kth column [i] based on linear OOA(49¹⁹, large, F₄₉, 3, 4), using
    - generalized (u,Â u+v)-construction [i] based on
      1. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(49⁰, s, F₄₉, 3, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
      2. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      3. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      4. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      5. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      6. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      7. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      8. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      9. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      10. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      11. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      12. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      13. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      14. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      15. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      16. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      17. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      18. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      19. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      20. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      21. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      22. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      23. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      24. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      25. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      26. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      27. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      28. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      29. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      30. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      31. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      32. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      33. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      34. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      35. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      36. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      37. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      38. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      39. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      40. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      41. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      42. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      43. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      44. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      45. linear OOA(49⁰, 120100, F₄₉, 3, 0) (dual of [(120100, 3), 360300, 1]-NRT-code) (see above)
      46. linear OOA(49¹, 120100, F₄₉, 3, 1) (dual of [(120100, 3), 360299, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(49¹, s, F₄₉, 3, 1) with arbitrarily large s, using
        appending 2 arbitrary columns [i] based on linear OA(49¹, s, F₄₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
      47. linear OOA(49¹, 120100, F₄₉, 3, 1) (dual of [(120100, 3), 360299, 2]-NRT-code) (see above)
      48. linear OOA(49⁴, 120100, F₄₉, 3, 2) (dual of [(120100, 3), 360296, 3]-NRT-code), using
        appending 1 arbitrary column [i] based on linear OOA(49⁴, 120100, F₄₉, 2, 2) (dual of [(120100, 2), 240196, 3]-NRT-code), using
        appending kth column [i] based on linear OA(49⁴, 120100, F₄₉, 2) (dual of [120100, 120096, 3]-code), using
        Hamming code H(4,49) [i]
      49. linear OOA(49¹³, 2882402, F₄₉, 3, 4) (dual of [(2882402, 3), 8647193, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(49¹³, 5764804, F₄₉, 4) (dual of [5764804, 5764791, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(49¹³, 5764805, F₄₉, 4) (dual of [5764805, 5764792, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(49¹³, 5764801, F₄₉, 4) (dual of [5764801, 5764788, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(49⁹, 5764801, F₄₉, 3) (dual of [5764801, 5764792, 4]-code or 5764801-cap in PG(8,49)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(20−4, 20, large)-Net in Base 49 — Upper bound on s

There is no (16, 20, large)-net in base 49, because

2 times m-reduction [i] would yield (16, 18, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]

Best Known (20−4, 20, s)-Nets in Base 49

(20−4, 20, large)-Net over F49 — Constructive and digital

(20−4, 20, large)-Net in Base 49 — Upper bound on s

(20−4, 20, large)-Net over F₄₉ — Constructive and digital