MinT - Best Known (31, 31+13, s)-Nets in Base 9

Best Known (31, 31+13, s)-Nets in Base 9

(31, 31+13, 432)-Net over F₉ — Constructive and digital

Digital (31, 44, 432)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 14, 200)-net over F₉, using
  - trace code for nets [i] based on digital (1, 7, 100)-net over F₈₁, using
    - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
        a function field by SÃ©mirat [i]
2. digital (17, 30, 232)-net over F₉, using
  - trace code for nets [i] based on digital (2, 15, 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]

(31, 31+13, 2091)-Net over F₉ — Digital

Digital (31, 44, 2091)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 31+13, 2581484)-Net in Base 9 — Upper bound on s

There is no (31, 44, 2581485)-net in base 9, because

1 times m-reduction [i] would yield (31, 43, 2581485)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 107752 744681 670770 781490 337271 155394 745777 > 9⁴³ [i]