MinT - Best Known (109, 172, s)-Nets in Base 8

Best Known (109, 172, s)-Nets in Base 8

(109, 172, 368)-Net over F₈ — Constructive and digital

Digital (109, 172, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 32, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (77, 140, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 70, 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, 172, 576)-Net in Base 8 — Constructive

(109, 172, 576)-net in base 8, using

t-expansion [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, 172, 1111)-Net over F₈ — Digital

Digital (109, 172, 1111)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 172, 169998)-Net in Base 8 — Upper bound on s

There is no (109, 172, 169999)-net in base 8, because

1 times m-reduction [i] would yield (109, 171, 169999)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 26815 960038 192996 335705 297769 828083 396012 948854 639431 282137 904931 839014 723866 946571 000007 427036 341413 289572 686693 054910 363822 844673 432194 814304 800447 581184 > 8¹⁷¹ [i]