MinT - Best Known (69, 108, s)-Nets in Base 8

Best Known (69, 108, s)-Nets in Base 8

(69, 108, 354)-Net over F₈ — Constructive and digital

Digital (69, 108, 354)-net over F₈, using

16 times m-reduction [i] based on digital (69, 124, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 62, 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]

(69, 108, 516)-Net in Base 8 — Constructive

(69, 108, 516)-net in base 8, using

base change [i] based on digital (42, 81, 516)-net over F₁₆, using
- 1 times m-reduction [i] based on digital (42, 82, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 41, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(69, 108, 810)-Net over F₈ — Digital

Digital (69, 108, 810)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 108, 138007)-Net in Base 8 — Upper bound on s

There is no (69, 108, 138008)-net in base 8, because

1 times m-reduction [i] would yield (69, 107, 138008)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 272179 357116 208085 291009 000013 280059 655853 314873 864788 391312 191939 964883 228878 821034 256811 790649 > 8¹⁰⁷ [i]