MinT - Best Known (109, 168, s)-Nets in Base 8

Best Known (109, 168, s)-Nets in Base 8

(109, 168, 388)-Net over F₈ — Constructive and digital

Digital (109, 168, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 36, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (73, 132, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 66, 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]

(109, 168, 576)-Net in Base 8 — Constructive

(109, 168, 576)-net in base 8, using

t-expansion [i] based on (108, 168, 576)-net in base 8, using
- 4 times m-reduction [i] based on (108, 172, 576)-net in base 8, using
  - trace code for nets [i] based on (22, 86, 288)-net in base 64, using
    - 5 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
      - base change [i] based on digital (9, 78, 288)-net over F₁₂₈, using
        net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(109, 168, 1353)-Net over F₈ — Digital

Digital (109, 168, 1353)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 168, 264573)-Net in Base 8 — Upper bound on s

There is no (109, 168, 264574)-net in base 8, because

1 times m-reduction [i] would yield (109, 167, 264574)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 546899 656540 870638 169025 272307 245889 908857 497143 766720 766674 559716 872616 757982 207159 356679 588078 128318 722183 539679 064570 104059 257953 508929 614461 409161 > 8¹⁶⁷ [i]