MinT - Best Known (33, 33+4, s)-Nets in Base 49

Best Known (33, 33+4, s)-Nets in Base 49

(33, 33+4, large)-Net over F₄₉ — Constructive and digital

Digital (33, 37, large)-net over F₄₉, using

3 times m-reduction [i] based on digital (33, 40, 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, 0, 120100)-net over F₄₉ (see above)
  43. 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]
  44. digital (0, 1, 120100)-net over F₄₉ (see above)
  45. digital (0, 1, 120100)-net over F₄₉ (see above)
  46. digital (0, 1, 120100)-net over F₄₉ (see above)
  47. digital (2, 4, 120100)-net over F₄₉, using
    - kÂ =Â 2 construction [i]
  48. digital (4, 7, 1921601)-net over F₄₉, using
    - s-reduction based on digital (4, 7, 5769603)-net over F₄₉, using
      - net defined by OOA [i] based on linear OOA(49⁷, 5769603, F₄₉, 3, 3) (dual of [(5769603, 3), 17308802, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(49⁷, 5769603, F₄₉, 2, 3) (dual of [(5769603, 2), 11539199, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(49⁷, 5769603, F₄₉, 3) (dual of [5769603, 5769596, 4]-code or 5769603-cap in PG(6,49)), using
        product of two caps with tangent hyperplane [i]
  49. digital (18, 25, 1921601)-net over F₄₉, using
    - net defined by OOA [i] based on linear OOA(49²⁵, 1921601, F₄₉, 7, 7) (dual of [(1921601, 7), 13451182, 8]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OA(49²⁵, 5764804, F₄₉, 7) (dual of [5764804, 5764779, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(49²⁵, 5764805, F₄₉, 7) (dual of [5764805, 5764780, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        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(49²¹, 5764801, F₄₉, 6) (dual of [5764801, 5764780, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [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]

(33, 33+4, large)-Net in Base 49 — Upper bound on s

There is no (33, 37, large)-net in base 49, because

2 times m-reduction [i] would yield (33, 35, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]

Best Known (33, 33+4, s)-Nets in Base 49

(33, 33+4, large)-Net over F49 — Constructive and digital

(33, 33+4, large)-Net in Base 49 — Upper bound on s

(33, 33+4, large)-Net over F₄₉ — Constructive and digital