MinT - Best Known (64−19, 64, s)-Nets in Base 9

Best Known (64−19, 64, s)-Nets in Base 9

(64−19, 64, 464)-Net over F₉ — Constructive and digital

Digital (45, 64, 464)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (13, 22, 232)-net over F₉, using
  - trace code for nets [i] based on digital (2, 11, 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 (23, 42, 232)-net over F₉, using
  - trace code for nets [i] based on digital (2, 21, 116)-net over F₈₁, using
    - net from sequence [i] based on digital (2, 115)-sequence over F₈₁ (see above)

(64−19, 64, 2342)-Net over F₉ — Digital

Digital (45, 64, 2342)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(64−19, 64, 2479465)-Net in Base 9 — Upper bound on s

There is no (45, 64, 2479466)-net in base 9, because

1 times m-reduction [i] would yield (45, 63, 2479466)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 310021 264516 006503 694215 565902 202419 890752 972913 012081 747153 > 9⁶³ [i]