MinT - Best Known (232−74, 232, s)-Nets in Base 3

Best Known (232−74, 232, s)-Nets in Base 3

(232−74, 232, 172)-Net over F₃ — Constructive and digital

Digital (158, 232, 172)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 50, 24)-net over F₃, using
  - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (108, 182, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 91, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(232−74, 232, 416)-Net over F₃ — Digital

Digital (158, 232, 416)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(232−74, 232, 7150)-Net in Base 3 — Upper bound on s

There is no (158, 232, 7151)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 492 698342 138631 611249 913137 605194 478033 915959 270793 983827 442096 238510 808995 766850 481463 818794 555344 987319 603735 > 3²³² [i]