MinT - Best Known (31, 31+33, s)-Nets in Base 8

Best Known (31, 31+33, s)-Nets in Base 8

(31, 31+33, 79)-Net over F₈ — Constructive and digital

Digital (31, 64, 79)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (14, 47, 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]

(31, 31+33, 109)-Net over F₈ — Digital

Digital (31, 64, 109)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 31+33, 3484)-Net in Base 8 — Upper bound on s

There is no (31, 64, 3485)-net in base 8, because

1 times m-reduction [i] would yield (31, 63, 3485)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 786 431141 046798 444709 562784 211078 709156 198952 258833 590131 > 8⁶³ [i]