MinT - Best Known (164, 164+45, s)-Nets in Base 3

Best Known (164, 164+45, s)-Nets in Base 3

(164, 164+45, 688)-Net over F₃ — Constructive and digital

Digital (164, 209, 688)-net over F₃, using

3¹ times duplication [i] based on digital (163, 208, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 52, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(164, 164+45, 1615)-Net over F₃ — Digital

Digital (164, 209, 1615)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(164, 164+45, 146799)-Net in Base 3 — Upper bound on s

There is no (164, 209, 146800)-net in base 3, because

1 times m-reduction [i] would yield (164, 208, 146800)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1742 872728 509521 041132 847257 550476 733353 523253 843563 957028 551969 173745 188984 474562 104770 074671 416481 > 3²⁰⁸ [i]