MinT - Best Known (165, 165+81, s)-Nets in Base 4

Best Known (165, 165+81, s)-Nets in Base 4

(165, 165+81, 450)-Net over F₄ — Constructive and digital

Digital (165, 246, 450)-net over F₄, using

4 times m-reduction [i] based on digital (165, 250, 450)-net over F₄, using
- trace code for nets [i] based on digital (40, 125, 225)-net over F₁₆, using
  - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
      - a function field by GarcÃa and Quoos [i]

(165, 165+81, 690)-Net over F₄ — Digital

Digital (165, 246, 690)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+81, 25569)-Net in Base 4 — Upper bound on s

There is no (165, 246, 25570)-net in base 4, because

1 times m-reduction [i] would yield (165, 245, 25570)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3196 799063 652821 793221 548891 302781 158693 429690 981557 599413 281502 389613 909748 493881 079195 282357 916567 809034 706867 481959 787696 064914 803918 116530 419000 > 4²⁴⁵ [i]