MinT - Best Known (68−22, 68, s)-Nets in Base 9

Best Known (68−22, 68, s)-Nets in Base 9

(68−22, 68, 364)-Net over F₉ — Constructive and digital

Digital (46, 68, 364)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (11, 22, 164)-net over F₉, using
  - trace code for nets [i] based on digital (0, 11, 82)-net over F₈₁, using
    - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
      - Niederreiter sequence [i]
2. digital (24, 46, 200)-net over F₉, using
  - trace code for nets [i] based on digital (1, 23, 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]

(68−22, 68, 1345)-Net over F₉ — Digital

Digital (46, 68, 1345)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−22, 68, 486264)-Net in Base 9 — Upper bound on s

There is no (46, 68, 486265)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 77357 048945 170425 427842 688881 352601 072607 785140 190575 070390 355033 > 9⁶⁸ [i]