MinT - Best Known (45, 45+49, s)-Nets in Base 8

Best Known (45, 45+49, s)-Nets in Base 8

(45, 45+49, 99)-Net over F₈ — Constructive and digital

Digital (45, 94, 99)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 31, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (14, 63, 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]

(45, 45+49, 144)-Net over F₈ — Digital

Digital (45, 94, 144)-net over F₈, using

net from sequence [i] based on digital (45, 143)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
  - Drinfeld modules of rank 1 [i]

(45, 45+49, 4408)-Net in Base 8 — Upper bound on s

There is no (45, 94, 4409)-net in base 8, because

1 times m-reduction [i] would yield (45, 93, 4409)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 975001 161498 367003 825188 376475 584178 400966 790196 882014 870194 265924 677960 459679 387531 > 8⁹³ [i]