MinT - Best Known (69−9, 69, s)-Nets in Base 27

Best Known (69−9, 69, s)-Nets in Base 27

(69−9, 69, 6292207)-Net over F₂₇ — Constructive and digital

Digital (60, 69, 6292207)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 3, 757)-net over F₂₇, using
  - kÂ =Â 2 construction [i]
2. digital (6, 9, 2097150)-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 (12, 16, 2097150)-net over F₂₇, using
  - s-reduction based on digital (12, 16, 4194301)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27¹⁶, 4194301, F₂₇, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(27¹⁶, 8388602, F₂₇, 4) (dual of [8388602, 8388586, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹⁶, large, F₂₇, 4) (dual of [large, large−16, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
4. digital (32, 41, 2097150)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁴¹, 2097150, F₂₇, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(27⁴¹, 8388601, F₂₇, 9) (dual of [8388601, 8388560, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁴¹, large, F₂₇, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(69−9, 69, large)-Net in Base 27 — Constructive

(60, 69, large)-net in base 27, using

27¹ times duplication [i] based on (59, 68, large)-net in base 27, using
- base change [i] based on digital (42, 51, large)-net over F₈₁, using
  - 81¹ times duplication [i] based on digital (41, 50, large)-net over F₈₁, using
    - generalized (u,Â u+v)-construction [i] based on
      1. digital (0, 0, 132860)-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, 132860)-net over F₈₁ (see above)
      3. digital (0, 0, 132860)-net over F₈₁ (see above)
      4. digital (0, 0, 132860)-net over F₈₁ (see above)
      5. digital (0, 0, 132860)-net over F₈₁ (see above)
      6. digital (0, 0, 132860)-net over F₈₁ (see above)
      7. digital (0, 0, 132860)-net over F₈₁ (see above)
      8. digital (0, 0, 132860)-net over F₈₁ (see above)
      9. digital (0, 0, 132860)-net over F₈₁ (see above)
      10. digital (0, 0, 132860)-net over F₈₁ (see above)
      11. digital (0, 0, 132860)-net over F₈₁ (see above)
      12. digital (0, 0, 132860)-net over F₈₁ (see above)
      13. digital (0, 0, 132860)-net over F₈₁ (see above)
      14. digital (0, 0, 132860)-net over F₈₁ (see above)
      15. digital (0, 0, 132860)-net over F₈₁ (see above)
      16. digital (0, 0, 132860)-net over F₈₁ (see above)
      17. digital (0, 0, 132860)-net over F₈₁ (see above)
      18. digital (0, 0, 132860)-net over F₈₁ (see above)
      19. digital (0, 0, 132860)-net over F₈₁ (see above)
      20. digital (0, 0, 132860)-net over F₈₁ (see above)
      21. digital (0, 0, 132860)-net over F₈₁ (see above)
      22. digital (0, 0, 132860)-net over F₈₁ (see above)
      23. digital (0, 0, 132860)-net over F₈₁ (see above)
      24. digital (0, 0, 132860)-net over F₈₁ (see above)
      25. digital (0, 0, 132860)-net over F₈₁ (see above)
      26. digital (0, 0, 132860)-net over F₈₁ (see above)
      27. digital (0, 0, 132860)-net over F₈₁ (see above)
      28. digital (0, 0, 132860)-net over F₈₁ (see above)
      29. digital (0, 0, 132860)-net over F₈₁ (see above)
      30. digital (0, 0, 132860)-net over F₈₁ (see above)
      31. digital (0, 0, 132860)-net over F₈₁ (see above)
      32. digital (0, 0, 132860)-net over F₈₁ (see above)
      33. digital (0, 0, 132860)-net over F₈₁ (see above)
      34. digital (0, 0, 132860)-net over F₈₁ (see above)
      35. digital (0, 0, 132860)-net over F₈₁ (see above)
      36. digital (0, 0, 132860)-net over F₈₁ (see above)
      37. digital (0, 0, 132860)-net over F₈₁ (see above)
      38. digital (0, 0, 132860)-net over F₈₁ (see above)
      39. digital (0, 0, 132860)-net over F₈₁ (see above)
      40. digital (0, 0, 132860)-net over F₈₁ (see above)
      41. digital (0, 0, 132860)-net over F₈₁ (see above)
      42. digital (0, 0, 132860)-net over F₈₁ (see above)
      43. digital (0, 0, 132860)-net over F₈₁ (see above)
      44. digital (0, 0, 132860)-net over F₈₁ (see above)
      45. digital (0, 0, 132860)-net over F₈₁ (see above)
      46. digital (0, 0, 132860)-net over F₈₁ (see above)
      47. digital (0, 0, 132860)-net over F₈₁ (see above)
      48. digital (0, 0, 132860)-net over F₈₁ (see above)
      49. digital (0, 0, 132860)-net over F₈₁ (see above)
      50. digital (0, 0, 132860)-net over F₈₁ (see above)
      51. digital (0, 0, 132860)-net over F₈₁ (see above)
      52. digital (0, 0, 132860)-net over F₈₁ (see above)
      53. digital (0, 0, 132860)-net over F₈₁ (see above)
      54. digital (0, 0, 132860)-net over F₈₁ (see above)
      55. digital (0, 0, 132860)-net over F₈₁ (see above)
      56. digital (0, 0, 132860)-net over F₈₁ (see above)
      57. digital (0, 0, 132860)-net over F₈₁ (see above)
      58. digital (0, 0, 132860)-net over F₈₁ (see above)
      59. digital (0, 0, 132860)-net over F₈₁ (see above)
      60. digital (0, 0, 132860)-net over F₈₁ (see above)
      61. digital (0, 0, 132860)-net over F₈₁ (see above)
      62. digital (0, 0, 132860)-net over F₈₁ (see above)
      63. digital (0, 0, 132860)-net over F₈₁ (see above)
      64. digital (0, 0, 132860)-net over F₈₁ (see above)
      65. digital (0, 0, 132860)-net over F₈₁ (see above)
      66. digital (0, 0, 132860)-net over F₈₁ (see above)
      67. digital (0, 0, 132860)-net over F₈₁ (see above)
      68. digital (0, 0, 132860)-net over F₈₁ (see above)
      69. digital (0, 0, 132860)-net over F₈₁ (see above)
      70. digital (0, 0, 132860)-net over F₈₁ (see above)
      71. digital (0, 0, 132860)-net over F₈₁ (see above)
      72. digital (0, 0, 132860)-net over F₈₁ (see above)
      73. digital (0, 1, 132860)-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]
      74. digital (0, 1, 132860)-net over F₈₁ (see above)
      75. digital (0, 1, 132860)-net over F₈₁ (see above)
      76. digital (0, 1, 132860)-net over F₈₁ (see above)
      77. digital (0, 1, 132860)-net over F₈₁ (see above)
      78. digital (2, 4, 132860)-net over F₈₁, using
        s-reduction based on digital (2, 4, 538084)-net over F₈₁, using
        kÂ =Â 2 construction [i]
      79. digital (3, 6, 132860)-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 (6, 10, 132860)-net over F₈₁, using
        s-reduction based on digital (6, 10, 265722)-net over F₈₁, using
        net defined by OOA [i] based on linear OOA(81¹⁰, 265722, F₈₁, 4, 4) (dual of [(265722, 4), 1062878, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(81¹⁰, 531444, F₈₁, 4) (dual of [531444, 531434, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        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(81⁷, 531441, F₈₁, 3) (dual of [531441, 531434, 4]-code or 531441-cap in PG(6,81)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [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 (16, 25, 132860)-net over F₈₁, using
        net defined by OOA [i] based on linear OOA(81²⁵, 132860, F₈₁, 9, 9) (dual of [(132860, 9), 1195715, 10]-NRT-code), using
        OOA 4-folding and stacking with additional row [i] based on linear OA(81²⁵, 531441, F₈₁, 9) (dual of [531441, 531416, 10]-code), using
        an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(69−9, 69, large)-Net over F₂₇ — Digital

Digital (60, 69, large)-net over F₂₇, using

7 times m-reduction [i] based on digital (60, 76, large)-net over F₂₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁷⁶, large, F₂₇, 16) (dual of [large, large−76, 17]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(69−9, 69, large)-Net in Base 27 — Upper bound on s

There is no (60, 69, large)-net in base 27, because

7 times m-reduction [i] would yield (60, 62, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (69−9, 69, s)-Nets in Base 27

(69−9, 69, 6292207)-Net over F27 — Constructive and digital

(69−9, 69, large)-Net in Base 27 — Constructive

(69−9, 69, large)-Net over F27 — Digital

(69−9, 69, large)-Net in Base 27 — Upper bound on s

(69−9, 69, 6292207)-Net over F₂₇ — Constructive and digital

(69−9, 69, large)-Net over F₂₇ — Digital