MinT - Best Known (210−51, 210, s)-Nets in Base 3

Best Known (210−51, 210, s)-Nets in Base 3

(210−51, 210, 400)-Net over F₃ — Constructive and digital

Digital (159, 210, 400)-net over F₃, using

3² times duplication [i] based on digital (157, 208, 400)-net over F₃, using
- trace code for nets [i] based on digital (1, 52, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
      - a function field by SÃ©mirat [i]

(210−51, 210, 993)-Net over F₃ — Digital

Digital (159, 210, 993)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−51, 210, 49558)-Net in Base 3 — Upper bound on s

There is no (159, 210, 49559)-net in base 3, because

1 times m-reduction [i] would yield (159, 209, 49559)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5230 372163 613307 173989 791183 048361 786702 011569 684960 802816 703286 278498 089081 051052 990539 967144 059647 > 3²⁰⁹ [i]