MinT - Best Known (109, 109+61, s)-Nets in Base 8

Best Known (109, 109+61, s)-Nets in Base 8

(109, 109+61, 379)-Net over F₈ — Constructive and digital

Digital (109, 170, 379)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 34, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (75, 136, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 68, 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+61, 576)-Net in Base 8 — Constructive

(109, 170, 576)-net in base 8, using

t-expansion [i] based on (108, 170, 576)-net in base 8, using
- 2 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+61, 1226)-Net over F₈ — Digital

Digital (109, 170, 1226)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+61, 210406)-Net in Base 8 — Upper bound on s

There is no (109, 170, 210407)-net in base 8, because

1 times m-reduction [i] would yield (109, 169, 210407)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 419 034535 690183 811134 117887 211713 225254 196003 750869 015380 566669 691980 129065 495648 938928 773188 333132 419230 641601 545041 332459 573541 180246 408514 150599 455020 > 8¹⁶⁹ [i]