MinT - Best Known (170−17, 170, s)-Nets in Base 8

Best Known (170−17, 170, s)-Nets in Base 8

(170−17, 170, 2105470)-Net over F₈ — Constructive and digital

Digital (153, 170, 2105470)-net over F₈, using

trace code for nets [i] based on digital (68, 85, 1052735)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (12, 20, 4160)-net over F₆₄, using
    - generalized (u,Â u+v)-construction [i] based on
      1. digital (0, 0, 65)-net over F₆₄, using
        s-reduction based on digital (0, 0, s)-net over F₆₄ with arbitrarily large s, using
        arbitrary set of 64⁰ points [i]
      2. digital (0, 0, 65)-net over F₆₄ (see above)
      3. digital (0, 0, 65)-net over F₆₄ (see above)
      4. digital (0, 0, 65)-net over F₆₄ (see above)
      5. digital (0, 0, 65)-net over F₆₄ (see above)
      6. digital (0, 0, 65)-net over F₆₄ (see above)
      7. digital (0, 0, 65)-net over F₆₄ (see above)
      8. digital (0, 0, 65)-net over F₆₄ (see above)
      9. digital (0, 0, 65)-net over F₆₄ (see above)
      10. digital (0, 0, 65)-net over F₆₄ (see above)
      11. digital (0, 0, 65)-net over F₆₄ (see above)
      12. digital (0, 0, 65)-net over F₆₄ (see above)
      13. digital (0, 0, 65)-net over F₆₄ (see above)
      14. digital (0, 0, 65)-net over F₆₄ (see above)
      15. digital (0, 0, 65)-net over F₆₄ (see above)
      16. digital (0, 0, 65)-net over F₆₄ (see above)
      17. digital (0, 0, 65)-net over F₆₄ (see above)
      18. digital (0, 0, 65)-net over F₆₄ (see above)
      19. digital (0, 0, 65)-net over F₆₄ (see above)
      20. digital (0, 0, 65)-net over F₆₄ (see above)
      21. digital (0, 0, 65)-net over F₆₄ (see above)
      22. digital (0, 0, 65)-net over F₆₄ (see above)
      23. digital (0, 0, 65)-net over F₆₄ (see above)
      24. digital (0, 0, 65)-net over F₆₄ (see above)
      25. digital (0, 0, 65)-net over F₆₄ (see above)
      26. digital (0, 0, 65)-net over F₆₄ (see above)
      27. digital (0, 0, 65)-net over F₆₄ (see above)
      28. digital (0, 0, 65)-net over F₆₄ (see above)
      29. digital (0, 0, 65)-net over F₆₄ (see above)
      30. digital (0, 0, 65)-net over F₆₄ (see above)
      31. digital (0, 0, 65)-net over F₆₄ (see above)
      32. digital (0, 0, 65)-net over F₆₄ (see above)
      33. digital (0, 0, 65)-net over F₆₄ (see above)
      34. digital (0, 0, 65)-net over F₆₄ (see above)
      35. digital (0, 0, 65)-net over F₆₄ (see above)
      36. digital (0, 0, 65)-net over F₆₄ (see above)
      37. digital (0, 0, 65)-net over F₆₄ (see above)
      38. digital (0, 0, 65)-net over F₆₄ (see above)
      39. digital (0, 0, 65)-net over F₆₄ (see above)
      40. digital (0, 0, 65)-net over F₆₄ (see above)
      41. digital (0, 0, 65)-net over F₆₄ (see above)
      42. digital (0, 0, 65)-net over F₆₄ (see above)
      43. digital (0, 0, 65)-net over F₆₄ (see above)
      44. digital (0, 0, 65)-net over F₆₄ (see above)
      45. digital (0, 0, 65)-net over F₆₄ (see above)
      46. digital (0, 0, 65)-net over F₆₄ (see above)
      47. digital (0, 0, 65)-net over F₆₄ (see above)
      48. digital (0, 0, 65)-net over F₆₄ (see above)
      49. digital (0, 0, 65)-net over F₆₄ (see above)
      50. digital (0, 0, 65)-net over F₆₄ (see above)
      51. digital (0, 0, 65)-net over F₆₄ (see above)
      52. digital (0, 0, 65)-net over F₆₄ (see above)
      53. digital (0, 0, 65)-net over F₆₄ (see above)
      54. digital (0, 0, 65)-net over F₆₄ (see above)
      55. digital (0, 0, 65)-net over F₆₄ (see above)
      56. digital (0, 0, 65)-net over F₆₄ (see above)
      57. digital (0, 1, 65)-net over F₆₄, using
        s-reduction based on digital (0, 1, s)-net over F₆₄ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
      58. digital (0, 1, 65)-net over F₆₄ (see above)
      59. digital (0, 1, 65)-net over F₆₄ (see above)
      60. digital (0, 1, 65)-net over F₆₄ (see above)
      61. digital (0, 2, 65)-net over F₆₄, using
        kÂ =Â 2 construction [i]
      62. digital (0, 2, 65)-net over F₆₄ (see above)
      63. digital (0, 4, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
      64. digital (0, 8, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
  2. digital (48, 65, 1048575)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁶⁵, 1048575, F₆₄, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
      - OOA 8-folding and stacking with additional row [i] based on linear OA(64⁶⁵, 8388601, F₆₄, 17) (dual of [8388601, 8388536, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶⁵, large, F₆₄, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(170−17, 170, 2129919)-Net in Base 8 — Constructive

(153, 170, 2129919)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (32, 40, 32769)-net in base 8, using
  - base change [i] based on digital (22, 30, 32769)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16³⁰, 32769, F₁₆, 8, 8) (dual of [(32769, 8), 262122, 9]-NRT-code), using
      - OA 4-folding and stacking [i] based on linear OA(16³⁰, 131076, F₁₆, 8) (dual of [131076, 131046, 9]-code), using
        trace code [i] based on linear OA(256¹⁵, 65538, F₂₅₆, 8) (dual of [65538, 65523, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(256¹⁵, 65536, F₂₅₆, 8) (dual of [65536, 65521, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(256¹³, 65536, F₂₅₆, 7) (dual of [65536, 65523, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 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 (113, 130, 2097150)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹³⁰, 2097150, F₈, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(8¹³⁰, 8388601, F₈, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8¹³⁰, 8388602, F₈, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
        trace code [i] based on linear OOA(64⁶⁵, 4194301, F₆₄, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁶⁵, 8388602, F₆₄, 17) (dual of [8388602, 8388537, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶⁵, large, F₆₄, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(170−17, 170, large)-Net over F₈ — Digital

Digital (153, 170, large)-net over F₈, using

8² times duplication [i] based on digital (151, 168, large)-net over F₈, using
- t-expansion [i] based on digital (144, 168, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶⁸, large, F₈, 24) (dual of [large, large−168, 25]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(170−17, 170, large)-Net in Base 8 — Upper bound on s

There is no (153, 170, large)-net in base 8, because

15 times m-reduction [i] would yield (153, 155, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (170−17, 170, s)-Nets in Base 8

(170−17, 170, 2105470)-Net over F8 — Constructive and digital

(170−17, 170, 2129919)-Net in Base 8 — Constructive

(170−17, 170, large)-Net over F8 — Digital

(170−17, 170, large)-Net in Base 8 — Upper bound on s

(170−17, 170, 2105470)-Net over F₈ — Constructive and digital

(170−17, 170, large)-Net over F₈ — Digital