MinT - Best Known (65, 65+81, s)-Nets in Base 8

Best Known (65, 65+81, s)-Nets in Base 8

(65, 65+81, 113)-Net over F₈ — Constructive and digital

Digital (65, 146, 113)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 51, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]
2. digital (14, 95, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(65, 65+81, 157)-Net over F₈ — Digital

Digital (65, 146, 157)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+81, 4205)-Net in Base 8 — Upper bound on s

There is no (65, 146, 4206)-net in base 8, because

1 times m-reduction [i] would yield (65, 145, 4206)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 89266 724690 654083 534894 013555 263726 336893 328405 846750 847293 743463 571469 971295 942319 653588 949035 017440 634435 871331 349427 487464 788969 > 8¹⁴⁵ [i]