MinT - Best Known (92, 92+39, s)-Nets in Base 8

Best Known (92, 92+39, s)-Nets in Base 8

(92, 92+39, 484)-Net over F₈ — Constructive and digital

Digital (92, 131, 484)-net over F₈, using

8¹ times duplication [i] based on digital (91, 130, 484)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (19, 38, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 19, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
  2. digital (53, 92, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 46, 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]

(92, 92+39, 585)-Net in Base 8 — Constructive

(92, 131, 585)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (0, 19, 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. (73, 112, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 56, 288)-net in base 64, using
    - base change [i] based on digital (9, 48, 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]

(92, 92+39, 2805)-Net over F₈ — Digital

Digital (92, 131, 2805)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(92, 92+39, 1710610)-Net in Base 8 — Upper bound on s

There is no (92, 131, 1710611)-net in base 8, because

1 times m-reduction [i] would yield (92, 130, 1710611)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2521 730049 298895 828577 122232 768858 716063 616703 957445 776161 827670 339028 256273 375749 517520 738382 154037 818458 995523 615896 > 8¹³⁰ [i]