MinT - Best Known (80−10, 80, s)-Nets in Base 27

Best Known (80−10, 80, s)-Nets in Base 27

(80−10, 80, 5152454)-Net over F₂₇ — Constructive and digital

Digital (70, 80, 5152454)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 5, 551881)-net over F₂₇, using
  - kÂ =Â 2 construction [i]
2. digital (5, 8, 1245133)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁸, 1245133, F₂₇, 3, 3) (dual of [(1245133, 3), 3735391, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(27⁸, 1245133, F₂₇, 2, 3) (dual of [(1245133, 2), 2490258, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(27⁸, 1245133, F₂₇, 3) (dual of [1245133, 1245125, 4]-code or 1245133-cap in PG(7,27)), using
        product of projective cap with 27²-ovoid cap in AG(3,Â 27) [i] based on linear OA(27⁵, 1708, F₂₇, 3) (dual of [1708, 1703, 4]-code or 1708-cap in PG(4,27)), using
        large caps in PG(4,Â b) [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]

(80−10, 80, large)-Net in Base 27 — Constructive

(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]

(80−10, 80, large)-Net over F₂₇ — Digital

Digital (70, 80, large)-net over F₂₇, using

t-expansion [i] based on digital (68, 80, large)-net over F₂₇, using
- 6 times m-reduction [i] based on digital (68, 86, large)-net over F₂₇, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁸⁶, large, F₂₇, 18) (dual of [large, large−86, 19]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(80−10, 80, large)-Net in Base 27 — Upper bound on s

There is no (70, 80, large)-net in base 27, because

8 times m-reduction [i] would yield (70, 72, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (80−10, 80, s)-Nets in Base 27

(80−10, 80, 5152454)-Net over F27 — Constructive and digital

(80−10, 80, large)-Net in Base 27 — Constructive

(80−10, 80, large)-Net over F27 — Digital

(80−10, 80, large)-Net in Base 27 — Upper bound on s

(80−10, 80, 5152454)-Net over F₂₇ — Constructive and digital

(80−10, 80, large)-Net over F₂₇ — Digital