MinT - Best Known (210−67, 210, s)-Nets in Base 3

Best Known (210−67, 210, s)-Nets in Base 3

(210−67, 210, 167)-Net over F₃ — Constructive and digital

Digital (143, 210, 167)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 42, 19)-net over F₃, using
  - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
      - an explicitly constructive algebraic function field [i]
2. digital (101, 168, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 84, 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]

(210−67, 210, 381)-Net over F₃ — Digital

Digital (143, 210, 381)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−67, 210, 6887)-Net in Base 3 — Upper bound on s

There is no (143, 210, 6888)-net in base 3, because

1 times m-reduction [i] would yield (143, 209, 6888)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5244 560058 310381 268594 426932 917149 172181 611050 332396 968240 965282 344492 950261 736182 754981 713502 496721 > 3²⁰⁹ [i]