MinT - Best Known (156, 233, s)-Nets in Base 3

Best Known (156, 233, s)-Nets in Base 3

(156, 233, 164)-Net over F₃ — Constructive and digital

Digital (156, 233, 164)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 45, 16)-net over F₃, using
  - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (111, 188, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 94, 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]

(156, 233, 373)-Net over F₃ — Digital

Digital (156, 233, 373)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(156, 233, 6110)-Net in Base 3 — Upper bound on s

There is no (156, 233, 6111)-net in base 3, because

1 times m-reduction [i] would yield (156, 232, 6111)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 493 847178 439494 682727 129229 874261 212865 571979 731800 889807 632981 037146 257151 884290 134158 578216 278978 521446 218869 > 3²³² [i]