MinT - Best Known (85, 85+63, s)-Nets in Base 9

Best Known (85, 85+63, s)-Nets in Base 9

(85, 85+63, 344)-Net over F₉ — Constructive and digital

Digital (85, 148, 344)-net over F₉, using

t-expansion [i] based on digital (82, 148, 344)-net over F₉, using
- 2 times m-reduction [i] based on digital (82, 150, 344)-net over F₉, using
  - trace code for nets [i] based on digital (7, 75, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(85, 85+63, 557)-Net over F₉ — Digital

Digital (85, 148, 557)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+63, 51968)-Net in Base 9 — Upper bound on s

There is no (85, 148, 51969)-net in base 9, because

1 times m-reduction [i] would yield (85, 147, 51969)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 187 815465 289149 924969 420649 952984 804141 923233 133468 961370 377607 197097 195899 280331 965230 869813 307428 466418 894194 194823 171887 893026 700405 861177 > 9¹⁴⁷ [i]