MinT - Best Known (92, 109, s)-Nets in Base 27

Best Known (92, 109, s)-Nets in Base 27

(92, 109, 1053847)-Net over F₂₇ — Constructive and digital

Digital (92, 109, 1053847)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (20, 28, 5272)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 6, 351)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27⁶, 351, F₂₇, 4, 4) (dual of [(351, 4), 1398, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(27⁶, 702, F₂₇, 4) (dual of [702, 696, 5]-code), using
        1 times truncation [i] based on linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
        a code by Danev and Olsson [i]
    2. digital (14, 22, 4921)-net over F₂₇, using
      - net defined by OOA [i] based on linear OOA(27²², 4921, F₂₇, 8, 8) (dual of [(4921, 8), 39346, 9]-NRT-code), using
        OA 4-folding and stacking [i] based on linear OA(27²², 19684, F₂₇, 8) (dual of [19684, 19662, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(27²², 19686, F₂₇, 8) (dual of [19686, 19664, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(27²², 19683, F₂₇, 8) (dual of [19683, 19661, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(27¹⁹, 19683, F₂₇, 7) (dual of [19683, 19664, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 27³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(27⁰, 3, F₂₇, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (64, 81, 1048575)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁸¹, 1048575, F₂₇, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(27⁸¹, 8388601, F₂₇, 17) (dual of [8388601, 8388520, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁸¹, large, F₂₇, 17) (dual of [large, large−81, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(92, 109, 1055235)-Net in Base 27 — Constructive

(92, 109, 1055235)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (20, 28, 6660)-net in base 27, using
  - base change [i] based on digital (13, 21, 6660)-net over F₈₁, using
    - generalized (u,Â u+v)-construction [i] based on
      1. digital (0, 0, 82)-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, 82)-net over F₈₁ (see above)
      3. digital (0, 0, 82)-net over F₈₁ (see above)
      4. digital (0, 0, 82)-net over F₈₁ (see above)
      5. digital (0, 0, 82)-net over F₈₁ (see above)
      6. digital (0, 0, 82)-net over F₈₁ (see above)
      7. digital (0, 0, 82)-net over F₈₁ (see above)
      8. digital (0, 0, 82)-net over F₈₁ (see above)
      9. digital (0, 0, 82)-net over F₈₁ (see above)
      10. digital (0, 0, 82)-net over F₈₁ (see above)
      11. digital (0, 0, 82)-net over F₈₁ (see above)
      12. digital (0, 0, 82)-net over F₈₁ (see above)
      13. digital (0, 0, 82)-net over F₈₁ (see above)
      14. digital (0, 0, 82)-net over F₈₁ (see above)
      15. digital (0, 0, 82)-net over F₈₁ (see above)
      16. digital (0, 0, 82)-net over F₈₁ (see above)
      17. digital (0, 0, 82)-net over F₈₁ (see above)
      18. digital (0, 0, 82)-net over F₈₁ (see above)
      19. digital (0, 0, 82)-net over F₈₁ (see above)
      20. digital (0, 0, 82)-net over F₈₁ (see above)
      21. digital (0, 0, 82)-net over F₈₁ (see above)
      22. digital (0, 0, 82)-net over F₈₁ (see above)
      23. digital (0, 0, 82)-net over F₈₁ (see above)
      24. digital (0, 0, 82)-net over F₈₁ (see above)
      25. digital (0, 0, 82)-net over F₈₁ (see above)
      26. digital (0, 0, 82)-net over F₈₁ (see above)
      27. digital (0, 0, 82)-net over F₈₁ (see above)
      28. digital (0, 0, 82)-net over F₈₁ (see above)
      29. digital (0, 0, 82)-net over F₈₁ (see above)
      30. digital (0, 0, 82)-net over F₈₁ (see above)
      31. digital (0, 0, 82)-net over F₈₁ (see above)
      32. digital (0, 0, 82)-net over F₈₁ (see above)
      33. digital (0, 0, 82)-net over F₈₁ (see above)
      34. digital (0, 0, 82)-net over F₈₁ (see above)
      35. digital (0, 0, 82)-net over F₈₁ (see above)
      36. digital (0, 0, 82)-net over F₈₁ (see above)
      37. digital (0, 0, 82)-net over F₈₁ (see above)
      38. digital (0, 0, 82)-net over F₈₁ (see above)
      39. digital (0, 0, 82)-net over F₈₁ (see above)
      40. digital (0, 0, 82)-net over F₈₁ (see above)
      41. digital (0, 0, 82)-net over F₈₁ (see above)
      42. digital (0, 0, 82)-net over F₈₁ (see above)
      43. digital (0, 0, 82)-net over F₈₁ (see above)
      44. digital (0, 0, 82)-net over F₈₁ (see above)
      45. digital (0, 0, 82)-net over F₈₁ (see above)
      46. digital (0, 0, 82)-net over F₈₁ (see above)
      47. digital (0, 0, 82)-net over F₈₁ (see above)
      48. digital (0, 0, 82)-net over F₈₁ (see above)
      49. digital (0, 0, 82)-net over F₈₁ (see above)
      50. digital (0, 0, 82)-net over F₈₁ (see above)
      51. digital (0, 0, 82)-net over F₈₁ (see above)
      52. digital (0, 0, 82)-net over F₈₁ (see above)
      53. digital (0, 0, 82)-net over F₈₁ (see above)
      54. digital (0, 0, 82)-net over F₈₁ (see above)
      55. digital (0, 0, 82)-net over F₈₁ (see above)
      56. digital (0, 0, 82)-net over F₈₁ (see above)
      57. digital (0, 0, 82)-net over F₈₁ (see above)
      58. digital (0, 0, 82)-net over F₈₁ (see above)
      59. digital (0, 0, 82)-net over F₈₁ (see above)
      60. digital (0, 0, 82)-net over F₈₁ (see above)
      61. digital (0, 0, 82)-net over F₈₁ (see above)
      62. digital (0, 0, 82)-net over F₈₁ (see above)
      63. digital (0, 0, 82)-net over F₈₁ (see above)
      64. digital (0, 0, 82)-net over F₈₁ (see above)
      65. digital (0, 0, 82)-net over F₈₁ (see above)
      66. digital (0, 0, 82)-net over F₈₁ (see above)
      67. digital (0, 0, 82)-net over F₈₁ (see above)
      68. digital (0, 0, 82)-net over F₈₁ (see above)
      69. digital (0, 0, 82)-net over F₈₁ (see above)
      70. digital (0, 0, 82)-net over F₈₁ (see above)
      71. digital (0, 0, 82)-net over F₈₁ (see above)
      72. digital (0, 0, 82)-net over F₈₁ (see above)
      73. digital (0, 0, 82)-net over F₈₁ (see above)
      74. digital (0, 1, 82)-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]
      75. digital (0, 1, 82)-net over F₈₁ (see above)
      76. digital (0, 1, 82)-net over F₈₁ (see above)
      77. digital (0, 1, 82)-net over F₈₁ (see above)
      78. digital (0, 2, 82)-net over F₈₁, using
        kÂ =Â 2 construction [i]
      79. digital (0, 2, 82)-net over F₈₁ (see above)
      80. digital (0, 4, 82)-net over F₈₁, using
        net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
        Niederreiter sequence [i]
      81. digital (1, 9, 100)-net over F₈₁, using
        net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
        a function field by SÃ©mirat [i]
2. digital (64, 81, 1048575)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁸¹, 1048575, F₂₇, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(27⁸¹, 8388601, F₂₇, 17) (dual of [8388601, 8388520, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁸¹, large, F₂₇, 17) (dual of [large, large−81, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(92, 109, large)-Net over F₂₇ — Digital

Digital (92, 109, large)-net over F₂₇, using

27³ times duplication [i] based on digital (89, 106, large)-net over F₂₇, using
- t-expansion [i] based on digital (84, 106, large)-net over F₂₇, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27¹⁰⁶, large, F₂₇, 22) (dual of [large, large−106, 23]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(92, 109, large)-Net in Base 27 — Upper bound on s

There is no (92, 109, large)-net in base 27, because

15 times m-reduction [i] would yield (92, 94, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (92, 109, s)-Nets in Base 27

(92, 109, 1053847)-Net over F27 — Constructive and digital

(92, 109, 1055235)-Net in Base 27 — Constructive

(92, 109, large)-Net over F27 — Digital

(92, 109, large)-Net in Base 27 — Upper bound on s

(92, 109, 1053847)-Net over F₂₇ — Constructive and digital

(92, 109, large)-Net over F₂₇ — Digital