MinT - Best Known (42−13, 42, s)-Nets in Base 9

Best Known (42−13, 42, s)-Nets in Base 9

(42−13, 42, 400)-Net over F₉ — Constructive and digital

Digital (29, 42, 400)-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 (15, 28, 200)-net over F₉, using
  - trace code for nets [i] based on digital (1, 14, 100)-net over F₈₁, using
    - net from sequence [i] based on digital (1, 99)-sequence over F₈₁ (see above)

(42−13, 42, 1452)-Net over F₉ — Digital

Digital (29, 42, 1452)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(42−13, 42, 1241046)-Net in Base 9 — Upper bound on s

There is no (29, 42, 1241047)-net in base 9, because

1 times m-reduction [i] would yield (29, 41, 1241047)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1330 281721 824728 383270 408768 299682 757457 > 9⁴¹ [i]