MinT - Best Known (70−37, 70, s)-Nets in Base 8

Best Known (70−37, 70, s)-Nets in Base 8

(70−37, 70, 79)-Net over F₈ — Constructive and digital

Digital (33, 70, 79)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 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, 51, 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]

(70−37, 70, 106)-Net over F₈ — Digital

Digital (33, 70, 106)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(70−37, 70, 3114)-Net in Base 8 — Upper bound on s

There is no (33, 70, 3115)-net in base 8, because

1 times m-reduction [i] would yield (33, 69, 3115)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 206 715548 256031 377013 817190 144822 448954 376627 926237 709200 931066 > 8⁶⁹ [i]