MinT - Best Known (108, 108+63, s)-Nets in Base 8

Best Known (108, 108+63, s)-Nets in Base 8

(108, 108+63, 363)-Net over F₈ — Constructive and digital

Digital (108, 171, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 31, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [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]

(108, 108+63, 576)-Net in Base 8 — Constructive

(108, 171, 576)-net in base 8, using

1 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]

(108, 108+63, 1072)-Net over F₈ — Digital

Digital (108, 171, 1072)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(108, 108+63, 158968)-Net in Base 8 — Upper bound on s

There is no (108, 171, 158969)-net in base 8, because

1 times m-reduction [i] would yield (108, 170, 158969)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3352 342876 927252 210288 276374 370124 373063 602049 990291 473286 929194 525206 733941 566443 074061 582938 486547 823893 827080 210566 824793 767760 847324 533002 158992 025184 > 8¹⁷⁰ [i]