MinT - Best Known (109, 109+57, s)-Nets in Base 8

Best Known (109, 109+57, s)-Nets in Base 8

(109, 109+57, 400)-Net over F₈ — Constructive and digital

Digital (109, 166, 400)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 38, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
2. digital (71, 128, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 64, 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, 109+57, 576)-Net in Base 8 — Constructive

(109, 166, 576)-net in base 8, using

t-expansion [i] based on (108, 166, 576)-net in base 8, using
- 6 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, 109+57, 1502)-Net over F₈ — Digital

Digital (109, 166, 1502)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+57, 338582)-Net in Base 8 — Upper bound on s

There is no (109, 166, 338583)-net in base 8, because

1 times m-reduction [i] would yield (109, 165, 338583)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102295 599149 675197 880610 824042 353889 822923 248310 145138 577455 703358 101163 342794 349601 328558 242078 520061 337177 436341 688290 352487 414283 328144 571785 720464 > 8¹⁶⁵ [i]