MinT - Best Known (37, 44, s)-Nets in Base 27

Best Known (37, 44, s)-Nets in Base 27

(37, 44, 5612840)-Net over F₂₇ — Constructive and digital

Digital (37, 44, 5612840)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (2, 4, 20440)-net over F₂₇, using
  - kÂ =Â 2 construction [i]
2. digital (6, 9, 2796200)-net over F₂₇, using
  - s-reduction based on digital (6, 9, large)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁹, large, F₂₇, 3, 3), using
      - appending kth column [i] based on linear OOA(27⁹, large, F₂₇, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(27⁹, large, F₂₇, 3) (dual of [large, large−9, 4]-code), using
        product of projective cap with (hyper)oval [i] based on linear OA(27⁷, 532899, F₂₇, 3) (dual of [532899, 532892, 4]-code or 532899-cap in PG(6,27)), using
        product of two caps with tangent hyperplane [i]
3. digital (24, 31, 2796200)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27³¹, 2796200, F₂₇, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(27³¹, 8388601, F₂₇, 7) (dual of [8388601, 8388570, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(27³¹, large, F₂₇, 7) (dual of [large, large−31, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(37, 44, large)-Net in Base 27 — Constructive

(37, 44, large)-net in base 27, using

base change [i] based on digital (26, 33, large)-net over F₈₁, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 0, 177147)-net over F₈₁, using
    - s-reduction based on digital (0, 0, s)-net over F₈₁ with arbitrarily large s, using
      - arbitrary set of 81⁰ points [i]
  2. digital (0, 0, 177147)-net over F₈₁ (see above)
  3. digital (0, 0, 177147)-net over F₈₁ (see above)
  4. digital (0, 0, 177147)-net over F₈₁ (see above)
  5. digital (0, 0, 177147)-net over F₈₁ (see above)
  6. digital (0, 0, 177147)-net over F₈₁ (see above)
  7. digital (0, 0, 177147)-net over F₈₁ (see above)
  8. digital (0, 0, 177147)-net over F₈₁ (see above)
  9. digital (0, 0, 177147)-net over F₈₁ (see above)
  10. digital (0, 0, 177147)-net over F₈₁ (see above)
  11. digital (0, 0, 177147)-net over F₈₁ (see above)
  12. digital (0, 0, 177147)-net over F₈₁ (see above)
  13. digital (0, 0, 177147)-net over F₈₁ (see above)
  14. digital (0, 0, 177147)-net over F₈₁ (see above)
  15. digital (0, 0, 177147)-net over F₈₁ (see above)
  16. digital (0, 0, 177147)-net over F₈₁ (see above)
  17. digital (0, 0, 177147)-net over F₈₁ (see above)
  18. digital (0, 0, 177147)-net over F₈₁ (see above)
  19. digital (0, 0, 177147)-net over F₈₁ (see above)
  20. digital (0, 0, 177147)-net over F₈₁ (see above)
  21. digital (0, 0, 177147)-net over F₈₁ (see above)
  22. digital (0, 0, 177147)-net over F₈₁ (see above)
  23. digital (0, 0, 177147)-net over F₈₁ (see above)
  24. digital (0, 0, 177147)-net over F₈₁ (see above)
  25. digital (0, 0, 177147)-net over F₈₁ (see above)
  26. digital (0, 0, 177147)-net over F₈₁ (see above)
  27. digital (0, 0, 177147)-net over F₈₁ (see above)
  28. digital (0, 0, 177147)-net over F₈₁ (see above)
  29. digital (0, 0, 177147)-net over F₈₁ (see above)
  30. digital (0, 0, 177147)-net over F₈₁ (see above)
  31. digital (0, 0, 177147)-net over F₈₁ (see above)
  32. digital (0, 0, 177147)-net over F₈₁ (see above)
  33. digital (0, 0, 177147)-net over F₈₁ (see above)
  34. digital (0, 0, 177147)-net over F₈₁ (see above)
  35. digital (0, 0, 177147)-net over F₈₁ (see above)
  36. digital (0, 0, 177147)-net over F₈₁ (see above)
  37. digital (0, 0, 177147)-net over F₈₁ (see above)
  38. digital (0, 0, 177147)-net over F₈₁ (see above)
  39. digital (0, 0, 177147)-net over F₈₁ (see above)
  40. digital (0, 0, 177147)-net over F₈₁ (see above)
  41. digital (0, 0, 177147)-net over F₈₁ (see above)
  42. digital (0, 0, 177147)-net over F₈₁ (see above)
  43. digital (0, 0, 177147)-net over F₈₁ (see above)
  44. digital (0, 0, 177147)-net over F₈₁ (see above)
  45. digital (0, 0, 177147)-net over F₈₁ (see above)
  46. digital (0, 0, 177147)-net over F₈₁ (see above)
  47. digital (0, 0, 177147)-net over F₈₁ (see above)
  48. digital (0, 0, 177147)-net over F₈₁ (see above)
  49. digital (0, 0, 177147)-net over F₈₁ (see above)
  50. digital (0, 0, 177147)-net over F₈₁ (see above)
  51. digital (0, 0, 177147)-net over F₈₁ (see above)
  52. digital (0, 0, 177147)-net over F₈₁ (see above)
  53. digital (0, 0, 177147)-net over F₈₁ (see above)
  54. digital (0, 0, 177147)-net over F₈₁ (see above)
  55. digital (0, 0, 177147)-net over F₈₁ (see above)
  56. digital (0, 0, 177147)-net over F₈₁ (see above)
  57. digital (0, 0, 177147)-net over F₈₁ (see above)
  58. digital (0, 0, 177147)-net over F₈₁ (see above)
  59. digital (0, 0, 177147)-net over F₈₁ (see above)
  60. digital (0, 0, 177147)-net over F₈₁ (see above)
  61. digital (0, 0, 177147)-net over F₈₁ (see above)
  62. digital (0, 0, 177147)-net over F₈₁ (see above)
  63. digital (0, 0, 177147)-net over F₈₁ (see above)
  64. digital (0, 0, 177147)-net over F₈₁ (see above)
  65. digital (0, 0, 177147)-net over F₈₁ (see above)
  66. digital (0, 0, 177147)-net over F₈₁ (see above)
  67. digital (0, 0, 177147)-net over F₈₁ (see above)
  68. digital (0, 0, 177147)-net over F₈₁ (see above)
  69. digital (0, 0, 177147)-net over F₈₁ (see above)
  70. digital (0, 0, 177147)-net over F₈₁ (see above)
  71. digital (0, 0, 177147)-net over F₈₁ (see above)
  72. digital (0, 0, 177147)-net over F₈₁ (see above)
  73. digital (0, 0, 177147)-net over F₈₁ (see above)
  74. digital (0, 0, 177147)-net over F₈₁ (see above)
  75. digital (0, 1, 177147)-net over F₈₁, using
    - s-reduction based on digital (0, 1, s)-net over F₈₁ with arbitrarily large s, using
      - construction for strength kÂ =Â 1 [i]
  76. digital (0, 1, 177147)-net over F₈₁ (see above)
  77. digital (0, 1, 177147)-net over F₈₁ (see above)
  78. digital (0, 1, 177147)-net over F₈₁ (see above)
  79. digital (2, 4, 177147)-net over F₈₁, using
    - s-reduction based on digital (2, 4, 538084)-net over F₈₁, using
      - kÂ =Â 2 construction [i]
  80. digital (3, 6, 177147)-net over F₈₁, using
    - s-reduction based on digital (3, 6, 538248)-net over F₈₁, using
      - net defined by OOA [i] based on linear OOA(81⁶, 538248, F₈₁, 3, 3) (dual of [(538248, 3), 1614738, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(81⁶, 538248, F₈₁, 2, 3) (dual of [(538248, 2), 1076490, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(81⁶, 538248, F₈₁, 3) (dual of [538248, 538242, 4]-code or 538248-cap in PG(5,81)), using
        large caps in PG(5,Â b) [i]
  81. digital (12, 19, 177147)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81¹⁹, 177147, F₈₁, 7, 7) (dual of [(177147, 7), 1240010, 8]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OA(81¹⁹, 531442, F₈₁, 7) (dual of [531442, 531423, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 531442Â | 81⁶âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(37, 44, large)-Net over F₂₇ — Digital

Digital (37, 44, large)-net over F₂₇, using

t-expansion [i] based on digital (36, 44, large)-net over F₂₇, using
- 2 times m-reduction [i] based on digital (36, 46, large)-net over F₂₇, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁴⁶, large, F₂₇, 10) (dual of [large, large−46, 11]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(37, 44, large)-Net in Base 27 — Upper bound on s

There is no (37, 44, large)-net in base 27, because

5 times m-reduction [i] would yield (37, 39, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (37, 44, s)-Nets in Base 27

(37, 44, 5612840)-Net over F27 — Constructive and digital

(37, 44, large)-Net in Base 27 — Constructive

(37, 44, large)-Net over F27 — Digital

(37, 44, large)-Net in Base 27 — Upper bound on s

(37, 44, 5612840)-Net over F₂₇ — Constructive and digital

(37, 44, large)-Net over F₂₇ — Digital