MinT - Best Known (227−72, 227, s)-Nets in Base 3

Best Known (227−72, 227, s)-Nets in Base 3

(227−72, 227, 172)-Net over F₃ — Constructive and digital

Digital (155, 227, 172)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 49, 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 (106, 178, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 89, 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]

(227−72, 227, 415)-Net over F₃ — Digital

Digital (155, 227, 415)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−72, 227, 7246)-Net in Base 3 — Upper bound on s

There is no (155, 227, 7247)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 034640 492771 376564 394749 589580 982497 959902 483077 960553 127665 050167 601543 148127 143549 205860 574710 996413 111225 > 3²²⁷ [i]