MinT - Best Known (165, 248, s)-Nets in Base 3

Best Known (165, 248, s)-Nets in Base 3

(165, 248, 164)-Net over F₃ — Constructive and digital

Digital (165, 248, 164)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 48, 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 (117, 200, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 100, 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]

(165, 248, 379)-Net over F₃ — Digital

Digital (165, 248, 379)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 248, 6002)-Net in Base 3 — Upper bound on s

There is no (165, 248, 6003)-net in base 3, because

1 times m-reduction [i] would yield (165, 247, 6003)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7084 314019 364815 522345 187007 204853 556304 526343 762149 192078 798498 476585 581571 444316 878132 544931 846007 676785 105258 159447 > 3²⁴⁷ [i]