MinT - Best Known (82, 109, s)-Nets in Base 8

Best Known (82, 109, s)-Nets in Base 8

(82, 109, 708)-Net over F₈ — Constructive and digital

Digital (82, 109, 708)-net over F₈, using

8¹ times duplication [i] based on digital (81, 108, 708)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 40, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 20, 177)-net over F₆₄, using
      - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]
  2. digital (41, 68, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 34, 177)-net over F₆₄, using
      - net from sequence [i] based on digital (7, 176)-sequence over F₆₄ (see above)

(82, 109, 1028)-Net in Base 8 — Constructive

(82, 109, 1028)-net in base 8, using

8¹ times duplication [i] based on (81, 108, 1028)-net in base 8, using
- base change [i] based on digital (54, 81, 1028)-net over F₁₆, using
  - 16¹ times duplication [i] based on digital (53, 80, 1028)-net over F₁₆, using
    - (u,Â u+v)-construction [i] based on
      1. digital (13, 26, 514)-net over F₁₆, using
        trace code for nets [i] based on digital (0, 13, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
      2. digital (27, 54, 514)-net over F₁₆, using
        trace code for nets [i] based on digital (0, 27, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(82, 109, 9222)-Net over F₈ — Digital

Digital (82, 109, 9222)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 109, large)-Net in Base 8 — Upper bound on s

There is no (82, 109, large)-net in base 8, because

25 times m-reduction [i] would yield (82, 84, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]