MinT - Best Known (72, 72+10, s)-Nets in Base 27

Best Known (72, 72+10, s)-Nets in Base 27

(72, 72+10, 6710880)-Net over F₂₇ — Constructive and digital

Digital (72, 82, 6710880)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 6, 1677720)-net over F₂₇, using
  - s-reduction based on digital (4, 6, large)-net over F₂₇, using
    - kÂ =Â 2 construction [i]
2. digital (6, 9, 1677720)-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 (16, 21, 1677720)-net over F₂₇, using
  - s-reduction based on digital (16, 21, 4194301)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27²¹, 4194301, F₂₇, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(27²¹, large, F₂₇, 5) (dual of [large, large−21, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
4. digital (36, 46, 1677720)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁴⁶, 1677720, F₂₇, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(27⁴⁶, 8388600, F₂₇, 10) (dual of [8388600, 8388554, 11]-code), using
      - discarding factors / shortening the dual code 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]

(72, 72+10, large)-Net in Base 27 — Constructive

(72, 82, large)-net in base 27, using

27² times duplication [i] based on (70, 80, large)-net in base 27, using
- base change [i] based on digital (50, 60, large)-net over F₈₁, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 106288)-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, 106288)-net over F₈₁ (see above)
    3. digital (0, 0, 106288)-net over F₈₁ (see above)
    4. digital (0, 0, 106288)-net over F₈₁ (see above)
    5. digital (0, 0, 106288)-net over F₈₁ (see above)
    6. digital (0, 0, 106288)-net over F₈₁ (see above)
    7. digital (0, 0, 106288)-net over F₈₁ (see above)
    8. digital (0, 0, 106288)-net over F₈₁ (see above)
    9. digital (0, 0, 106288)-net over F₈₁ (see above)
    10. digital (0, 0, 106288)-net over F₈₁ (see above)
    11. digital (0, 0, 106288)-net over F₈₁ (see above)
    12. digital (0, 0, 106288)-net over F₈₁ (see above)
    13. digital (0, 0, 106288)-net over F₈₁ (see above)
    14. digital (0, 0, 106288)-net over F₈₁ (see above)
    15. digital (0, 0, 106288)-net over F₈₁ (see above)
    16. digital (0, 0, 106288)-net over F₈₁ (see above)
    17. digital (0, 0, 106288)-net over F₈₁ (see above)
    18. digital (0, 0, 106288)-net over F₈₁ (see above)
    19. digital (0, 0, 106288)-net over F₈₁ (see above)
    20. digital (0, 0, 106288)-net over F₈₁ (see above)
    21. digital (0, 0, 106288)-net over F₈₁ (see above)
    22. digital (0, 0, 106288)-net over F₈₁ (see above)
    23. digital (0, 0, 106288)-net over F₈₁ (see above)
    24. digital (0, 0, 106288)-net over F₈₁ (see above)
    25. digital (0, 0, 106288)-net over F₈₁ (see above)
    26. digital (0, 0, 106288)-net over F₈₁ (see above)
    27. digital (0, 0, 106288)-net over F₈₁ (see above)
    28. digital (0, 0, 106288)-net over F₈₁ (see above)
    29. digital (0, 0, 106288)-net over F₈₁ (see above)
    30. digital (0, 0, 106288)-net over F₈₁ (see above)
    31. digital (0, 0, 106288)-net over F₈₁ (see above)
    32. digital (0, 0, 106288)-net over F₈₁ (see above)
    33. digital (0, 0, 106288)-net over F₈₁ (see above)
    34. digital (0, 0, 106288)-net over F₈₁ (see above)
    35. digital (0, 0, 106288)-net over F₈₁ (see above)
    36. digital (0, 0, 106288)-net over F₈₁ (see above)
    37. digital (0, 0, 106288)-net over F₈₁ (see above)
    38. digital (0, 0, 106288)-net over F₈₁ (see above)
    39. digital (0, 0, 106288)-net over F₈₁ (see above)
    40. digital (0, 0, 106288)-net over F₈₁ (see above)
    41. digital (0, 0, 106288)-net over F₈₁ (see above)
    42. digital (0, 0, 106288)-net over F₈₁ (see above)
    43. digital (0, 0, 106288)-net over F₈₁ (see above)
    44. digital (0, 0, 106288)-net over F₈₁ (see above)
    45. digital (0, 0, 106288)-net over F₈₁ (see above)
    46. digital (0, 0, 106288)-net over F₈₁ (see above)
    47. digital (0, 0, 106288)-net over F₈₁ (see above)
    48. digital (0, 0, 106288)-net over F₈₁ (see above)
    49. digital (0, 0, 106288)-net over F₈₁ (see above)
    50. digital (0, 0, 106288)-net over F₈₁ (see above)
    51. digital (0, 0, 106288)-net over F₈₁ (see above)
    52. digital (0, 0, 106288)-net over F₈₁ (see above)
    53. digital (0, 0, 106288)-net over F₈₁ (see above)
    54. digital (0, 0, 106288)-net over F₈₁ (see above)
    55. digital (0, 0, 106288)-net over F₈₁ (see above)
    56. digital (0, 0, 106288)-net over F₈₁ (see above)
    57. digital (0, 0, 106288)-net over F₈₁ (see above)
    58. digital (0, 0, 106288)-net over F₈₁ (see above)
    59. digital (0, 0, 106288)-net over F₈₁ (see above)
    60. digital (0, 0, 106288)-net over F₈₁ (see above)
    61. digital (0, 0, 106288)-net over F₈₁ (see above)
    62. digital (0, 0, 106288)-net over F₈₁ (see above)
    63. digital (0, 0, 106288)-net over F₈₁ (see above)
    64. digital (0, 0, 106288)-net over F₈₁ (see above)
    65. digital (0, 0, 106288)-net over F₈₁ (see above)
    66. digital (0, 0, 106288)-net over F₈₁ (see above)
    67. digital (0, 0, 106288)-net over F₈₁ (see above)
    68. digital (0, 0, 106288)-net over F₈₁ (see above)
    69. digital (0, 0, 106288)-net over F₈₁ (see above)
    70. digital (0, 0, 106288)-net over F₈₁ (see above)
    71. digital (0, 0, 106288)-net over F₈₁ (see above)
    72. digital (0, 1, 106288)-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]
    73. digital (0, 1, 106288)-net over F₈₁ (see above)
    74. digital (0, 1, 106288)-net over F₈₁ (see above)
    75. digital (0, 1, 106288)-net over F₈₁ (see above)
    76. digital (0, 1, 106288)-net over F₈₁ (see above)
    77. digital (2, 4, 106288)-net over F₈₁, using
      - s-reduction based on digital (2, 4, 538084)-net over F₈₁, using
        kÂ =Â 2 construction [i]
    78. digital (2, 4, 106288)-net over F₈₁ (see above)
    79. digital (3, 6, 106288)-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]
    80. digital (8, 13, 106288)-net over F₈₁, using
      - s-reduction based on digital (8, 13, 265721)-net over F₈₁, using
        net defined by OOA [i] based on linear OOA(81¹³, 265721, F₈₁, 5, 5) (dual of [(265721, 5), 1328592, 6]-NRT-code), using
        appending kth column [i] based on linear OOA(81¹³, 265721, F₈₁, 4, 5) (dual of [(265721, 4), 1062871, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(81¹³, 531443, F₈₁, 5) (dual of [531443, 531430, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹³, 531444, F₈₁, 5) (dual of [531444, 531431, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(81¹³, 531441, F₈₁, 5) (dual of [531441, 531428, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(81¹⁰, 531441, F₈₁, 4) (dual of [531441, 531431, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁰, s, F₈₁, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
    81. digital (18, 28, 106288)-net over F₈₁, using
      - net defined by OOA [i] based on linear OOA(81²⁸, 106288, F₈₁, 10, 10) (dual of [(106288, 10), 1062852, 11]-NRT-code), using
        OA 5-folding and stacking [i] based on linear OA(81²⁸, 531440, F₈₁, 10) (dual of [531440, 531412, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(81²⁸, 531441, F₈₁, 10) (dual of [531441, 531413, 11]-code), using
        an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(72, 72+10, large)-Net over F₂₇ — Digital

Digital (72, 82, large)-net over F₂₇, using

9 times m-reduction [i] based on digital (72, 91, large)-net over F₂₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁹¹, large, F₂₇, 19) (dual of [large, large−91, 20]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(72, 72+10, large)-Net in Base 27 — Upper bound on s

There is no (72, 82, large)-net in base 27, because

8 times m-reduction [i] would yield (72, 74, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (72, 72+10, s)-Nets in Base 27

(72, 72+10, 6710880)-Net over F27 — Constructive and digital

(72, 72+10, large)-Net in Base 27 — Constructive

(72, 72+10, large)-Net over F27 — Digital

(72, 72+10, large)-Net in Base 27 — Upper bound on s

(72, 72+10, 6710880)-Net over F₂₇ — Constructive and digital

(72, 72+10, large)-Net over F₂₇ — Digital