MinT - Best Known (41, 41+17, s)-Nets in Base 9

Best Known (41, 41+17, s)-Nets in Base 9

(41, 41+17, 464)-Net over F₉ — Constructive and digital

Digital (41, 58, 464)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (12, 20, 232)-net over F₉, using
  - trace code for nets [i] based on digital (2, 10, 116)-net over F₈₁, using
    - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]
2. digital (21, 38, 232)-net over F₉, using
  - trace code for nets [i] based on digital (2, 19, 116)-net over F₈₁, using
    - net from sequence [i] based on digital (2, 115)-sequence over F₈₁ (see above)

(41, 41+17, 2455)-Net over F₉ — Digital

Digital (41, 58, 2455)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(41, 41+17, 2961947)-Net in Base 9 — Upper bound on s

There is no (41, 58, 2961948)-net in base 9, because

1 times m-reduction [i] would yield (41, 57, 2961948)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 2 465040 671035 817820 095777 718704 857724 847799 814863 864065 > 9⁵⁷ [i]