MinT - Best Known (96, 96+12, s)-Nets in Base 27

Best Known (96, 96+12, s)-Nets in Base 27

(96, 96+12, 5592400)-Net over F₂₇ — Constructive and digital

Digital (96, 108, 5592400)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (20, 26, 2796201)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27²⁶, 2796201, F₂₇, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(27²⁶, large, F₂₇, 6) (dual of [large, large−26, 7]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
2. digital (70, 82, 2796200)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁸², 2796200, F₂₇, 14, 12) (dual of [(2796200, 14), 39146718, 13]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(27⁸², 8388601, F₂₇, 2, 12) (dual of [(8388601, 2), 16777120, 13]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(27⁸², 8388602, F₂₇, 2, 12) (dual of [(8388602, 2), 16777122, 13]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(27²⁶, 4194301, F₂₇, 2, 6) (dual of [(4194301, 2), 8388576, 7]-NRT-code), using
        OOA 2-folding [i] based on linear OA(27²⁶, 8388602, F₂₇, 6) (dual of [8388602, 8388576, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(27²⁶, large, F₂₇, 6) (dual of [large, large−26, 7]-code) (see above)
        linear OOA(27⁵⁶, 4194301, F₂₇, 2, 12) (dual of [(4194301, 2), 8388546, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(27⁵⁶, 8388602, F₂₇, 12) (dual of [8388602, 8388546, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁵⁶, large, F₂₇, 12) (dual of [large, large−56, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(96, 96+12, 7174494)-Net in Base 27 — Constructive

(96, 108, 7174494)-net in base 27, using

base change [i] based on digital (69, 81, 7174494)-net over F₈₁, using
- 81¹ times duplication [i] based on digital (68, 80, 7174494)-net over F₈₁, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 88574)-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, 88574)-net over F₈₁ (see above)
    3. digital (0, 0, 88574)-net over F₈₁ (see above)
    4. digital (0, 0, 88574)-net over F₈₁ (see above)
    5. digital (0, 0, 88574)-net over F₈₁ (see above)
    6. digital (0, 0, 88574)-net over F₈₁ (see above)
    7. digital (0, 0, 88574)-net over F₈₁ (see above)
    8. digital (0, 0, 88574)-net over F₈₁ (see above)
    9. digital (0, 0, 88574)-net over F₈₁ (see above)
    10. digital (0, 0, 88574)-net over F₈₁ (see above)
    11. digital (0, 0, 88574)-net over F₈₁ (see above)
    12. digital (0, 0, 88574)-net over F₈₁ (see above)
    13. digital (0, 0, 88574)-net over F₈₁ (see above)
    14. digital (0, 0, 88574)-net over F₈₁ (see above)
    15. digital (0, 0, 88574)-net over F₈₁ (see above)
    16. digital (0, 0, 88574)-net over F₈₁ (see above)
    17. digital (0, 0, 88574)-net over F₈₁ (see above)
    18. digital (0, 0, 88574)-net over F₈₁ (see above)
    19. digital (0, 0, 88574)-net over F₈₁ (see above)
    20. digital (0, 0, 88574)-net over F₈₁ (see above)
    21. digital (0, 0, 88574)-net over F₈₁ (see above)
    22. digital (0, 0, 88574)-net over F₈₁ (see above)
    23. digital (0, 0, 88574)-net over F₈₁ (see above)
    24. digital (0, 0, 88574)-net over F₈₁ (see above)
    25. digital (0, 0, 88574)-net over F₈₁ (see above)
    26. digital (0, 0, 88574)-net over F₈₁ (see above)
    27. digital (0, 0, 88574)-net over F₈₁ (see above)
    28. digital (0, 0, 88574)-net over F₈₁ (see above)
    29. digital (0, 0, 88574)-net over F₈₁ (see above)
    30. digital (0, 0, 88574)-net over F₈₁ (see above)
    31. digital (0, 0, 88574)-net over F₈₁ (see above)
    32. digital (0, 0, 88574)-net over F₈₁ (see above)
    33. digital (0, 0, 88574)-net over F₈₁ (see above)
    34. digital (0, 0, 88574)-net over F₈₁ (see above)
    35. digital (0, 0, 88574)-net over F₈₁ (see above)
    36. digital (0, 0, 88574)-net over F₈₁ (see above)
    37. digital (0, 0, 88574)-net over F₈₁ (see above)
    38. digital (0, 0, 88574)-net over F₈₁ (see above)
    39. digital (0, 0, 88574)-net over F₈₁ (see above)
    40. digital (0, 0, 88574)-net over F₈₁ (see above)
    41. digital (0, 0, 88574)-net over F₈₁ (see above)
    42. digital (0, 0, 88574)-net over F₈₁ (see above)
    43. digital (0, 0, 88574)-net over F₈₁ (see above)
    44. digital (0, 0, 88574)-net over F₈₁ (see above)
    45. digital (0, 0, 88574)-net over F₈₁ (see above)
    46. digital (0, 0, 88574)-net over F₈₁ (see above)
    47. digital (0, 0, 88574)-net over F₈₁ (see above)
    48. digital (0, 0, 88574)-net over F₈₁ (see above)
    49. digital (0, 0, 88574)-net over F₈₁ (see above)
    50. digital (0, 0, 88574)-net over F₈₁ (see above)
    51. digital (0, 0, 88574)-net over F₈₁ (see above)
    52. digital (0, 0, 88574)-net over F₈₁ (see above)
    53. digital (0, 0, 88574)-net over F₈₁ (see above)
    54. digital (0, 0, 88574)-net over F₈₁ (see above)
    55. digital (0, 0, 88574)-net over F₈₁ (see above)
    56. digital (0, 0, 88574)-net over F₈₁ (see above)
    57. digital (0, 0, 88574)-net over F₈₁ (see above)
    58. digital (0, 0, 88574)-net over F₈₁ (see above)
    59. digital (0, 0, 88574)-net over F₈₁ (see above)
    60. digital (0, 0, 88574)-net over F₈₁ (see above)
    61. digital (0, 0, 88574)-net over F₈₁ (see above)
    62. digital (0, 0, 88574)-net over F₈₁ (see above)
    63. digital (0, 0, 88574)-net over F₈₁ (see above)
    64. digital (0, 0, 88574)-net over F₈₁ (see above)
    65. digital (0, 0, 88574)-net over F₈₁ (see above)
    66. digital (0, 0, 88574)-net over F₈₁ (see above)
    67. digital (0, 0, 88574)-net over F₈₁ (see above)
    68. digital (0, 0, 88574)-net over F₈₁ (see above)
    69. digital (0, 0, 88574)-net over F₈₁ (see above)
    70. digital (0, 1, 88574)-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]
    71. digital (0, 1, 88574)-net over F₈₁ (see above)
    72. digital (0, 1, 88574)-net over F₈₁ (see above)
    73. digital (0, 1, 88574)-net over F₈₁ (see above)
    74. digital (0, 1, 88574)-net over F₈₁ (see above)
    75. digital (0, 1, 88574)-net over F₈₁ (see above)
    76. digital (2, 4, 88574)-net over F₈₁, using
      - s-reduction based on digital (2, 4, 538084)-net over F₈₁, using
        kÂ =Â 2 construction [i]
    77. digital (2, 4, 88574)-net over F₈₁ (see above)
    78. digital (3, 6, 88574)-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]
    79. digital (6, 10, 88574)-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]
    80. digital (10, 16, 88574)-net over F₈₁, using
      - s-reduction based on digital (10, 16, 177148)-net over F₈₁, using
        net defined by OOA [i] based on linear OOA(81¹⁶, 177148, F₈₁, 6, 6) (dual of [(177148, 6), 1062872, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(81¹⁶, 531444, F₈₁, 6) (dual of [531444, 531428, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(81¹⁶, 531441, F₈₁, 6) (dual of [531441, 531425, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        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(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code) (see above)
    81. digital (22, 34, 88574)-net over F₈₁, using
      - net defined by OOA [i] based on linear OOA(81³⁴, 88574, F₈₁, 12, 12) (dual of [(88574, 12), 1062854, 13]-NRT-code), using
        OA 6-folding and stacking [i] based on linear OA(81³⁴, 531444, F₈₁, 12) (dual of [531444, 531410, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(81³⁴, 531441, F₈₁, 12) (dual of [531441, 531407, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(81³¹, 531441, F₈₁, 11) (dual of [531441, 531410, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(81⁰, 3, F₈₁, 0) (dual of [3, 3, 1]-code) (see above)

(96, 96+12, large)-Net over F₂₇ — Digital

Digital (96, 108, large)-net over F₂₇, using

27² times duplication [i] based on digital (94, 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]

(96, 96+12, large)-Net in Base 27 — Upper bound on s

There is no (96, 108, large)-net in base 27, because

10 times m-reduction [i] would yield (96, 98, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (96, 96+12, s)-Nets in Base 27

(96, 96+12, 5592400)-Net over F27 — Constructive and digital

(96, 96+12, 7174494)-Net in Base 27 — Constructive

(96, 96+12, large)-Net over F27 — Digital

(96, 96+12, large)-Net in Base 27 — Upper bound on s

(96, 96+12, 5592400)-Net over F₂₇ — Constructive and digital

(96, 96+12, large)-Net over F₂₇ — Digital