MinT - Best Known (51−3, 51, s)-Nets in Base 49

Best Known (51−3, 51, s)-Nets in Base 49

(51−3, 51, large)-Net over F₄₉ — Constructive and digital

Digital (48, 51, large)-net over F₄₉, using

t-expansion [i] based on digital (46, 51, large)-net over F₄₉, using
- 3 times m-reduction [i] based on digital (46, 54, large)-net over F₄₉, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 120100)-net over F₄₉, using
      - s-reduction based on digital (0, 0, s)-net over F₄₉ with arbitrarily large s, using
        arbitrary set of 49⁰ points [i]
    2. digital (0, 0, 120100)-net over F₄₉ (see above)
    3. digital (0, 0, 120100)-net over F₄₉ (see above)
    4. digital (0, 0, 120100)-net over F₄₉ (see above)
    5. digital (0, 0, 120100)-net over F₄₉ (see above)
    6. digital (0, 0, 120100)-net over F₄₉ (see above)
    7. digital (0, 0, 120100)-net over F₄₉ (see above)
    8. digital (0, 0, 120100)-net over F₄₉ (see above)
    9. digital (0, 0, 120100)-net over F₄₉ (see above)
    10. digital (0, 0, 120100)-net over F₄₉ (see above)
    11. digital (0, 0, 120100)-net over F₄₉ (see above)
    12. digital (0, 0, 120100)-net over F₄₉ (see above)
    13. digital (0, 0, 120100)-net over F₄₉ (see above)
    14. digital (0, 0, 120100)-net over F₄₉ (see above)
    15. digital (0, 0, 120100)-net over F₄₉ (see above)
    16. digital (0, 0, 120100)-net over F₄₉ (see above)
    17. digital (0, 0, 120100)-net over F₄₉ (see above)
    18. digital (0, 0, 120100)-net over F₄₉ (see above)
    19. digital (0, 0, 120100)-net over F₄₉ (see above)
    20. digital (0, 0, 120100)-net over F₄₉ (see above)
    21. digital (0, 0, 120100)-net over F₄₉ (see above)
    22. digital (0, 0, 120100)-net over F₄₉ (see above)
    23. digital (0, 0, 120100)-net over F₄₉ (see above)
    24. digital (0, 0, 120100)-net over F₄₉ (see above)
    25. digital (0, 0, 120100)-net over F₄₉ (see above)
    26. digital (0, 0, 120100)-net over F₄₉ (see above)
    27. digital (0, 0, 120100)-net over F₄₉ (see above)
    28. digital (0, 0, 120100)-net over F₄₉ (see above)
    29. digital (0, 0, 120100)-net over F₄₉ (see above)
    30. digital (0, 0, 120100)-net over F₄₉ (see above)
    31. digital (0, 0, 120100)-net over F₄₉ (see above)
    32. digital (0, 0, 120100)-net over F₄₉ (see above)
    33. digital (0, 0, 120100)-net over F₄₉ (see above)
    34. digital (0, 0, 120100)-net over F₄₉ (see above)
    35. digital (0, 0, 120100)-net over F₄₉ (see above)
    36. digital (0, 0, 120100)-net over F₄₉ (see above)
    37. digital (0, 0, 120100)-net over F₄₉ (see above)
    38. digital (0, 0, 120100)-net over F₄₉ (see above)
    39. digital (0, 0, 120100)-net over F₄₉ (see above)
    40. digital (0, 0, 120100)-net over F₄₉ (see above)
    41. digital (0, 0, 120100)-net over F₄₉ (see above)
    42. digital (0, 1, 120100)-net over F₄₉, using
      - s-reduction based on digital (0, 1, s)-net over F₄₉ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    43. digital (0, 1, 120100)-net over F₄₉ (see above)
    44. digital (0, 1, 120100)-net over F₄₉ (see above)
    45. digital (0, 1, 120100)-net over F₄₉ (see above)
    46. digital (2, 4, 120100)-net over F₄₉, using
      - kÂ =Â 2 construction [i]
    47. digital (2, 4, 120100)-net over F₄₉ (see above)
    48. digital (9, 13, 1441201)-net over F₄₉, using
      - s-reduction based on digital (9, 13, 2882402)-net over F₄₉, using
        net defined by OOA [i] based on linear OOA(49¹³, 2882402, F₄₉, 4, 4) (dual of [(2882402, 4), 11529595, 5]-NRT-code), using
        appending kth column [i] based on 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]
    49. digital (21, 29, 1441201)-net over F₄₉, using
      - net defined by OOA [i] based on linear OOA(49²⁹, 1441201, F₄₉, 8, 8) (dual of [(1441201, 8), 11529579, 9]-NRT-code), using
        OA 4-folding and stacking [i] based on linear OA(49²⁹, 5764804, F₄₉, 8) (dual of [5764804, 5764775, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(49²⁹, 5764805, F₄₉, 8) (dual of [5764805, 5764776, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(49²⁹, 5764801, F₄₉, 8) (dual of [5764801, 5764772, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(49²⁵, 5764801, F₄₉, 7) (dual of [5764801, 5764776, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code) (see above)

(51−3, 51, large)-Net in Base 49 — Upper bound on s

There is no (48, 51, large)-net in base 49, because

1 times m-reduction [i] would yield (48, 50, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]

Best Known (51−3, 51, s)-Nets in Base 49

(51−3, 51, large)-Net over F49 — Constructive and digital

(51−3, 51, large)-Net in Base 49 — Upper bound on s

(51−3, 51, large)-Net over F₄₉ — Constructive and digital