MinT - Best Known (164, 164+78, s)-Nets in Base 3

Best Known (164, 164+78, s)-Nets in Base 3

(164, 164+78, 172)-Net over F₃ — Constructive and digital

Digital (164, 242, 172)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 52, 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 (112, 190, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 95, 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]

(164, 164+78, 417)-Net over F₃ — Digital

Digital (164, 242, 417)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(164, 164+78, 6992)-Net in Base 3 — Upper bound on s

There is no (164, 242, 6993)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 190705 814400 143041 841631 511657 953146 117890 270999 274281 959429 208626 477756 146382 476472 217490 282540 049836 738680 830283 > 3²⁴² [i]