MinT - Best Known (213−69, 213, s)-Nets in Base 3

Best Known (213−69, 213, s)-Nets in Base 3

(213−69, 213, 164)-Net over F₃ — Constructive and digital

Digital (144, 213, 164)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 41, 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 (103, 172, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 86, 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]

(213−69, 213, 368)-Net over F₃ — Digital

Digital (144, 213, 368)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−69, 213, 6355)-Net in Base 3 — Upper bound on s

There is no (144, 213, 6356)-net in base 3, because

1 times m-reduction [i] would yield (144, 212, 6356)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 141435 978266 265531 757626 378668 772691 977745 490916 393138 124781 954697 660847 367599 672547 737229 278690 949145 > 3²¹² [i]