MinT - Best Known (165, 165+77, s)-Nets in Base 4

Best Known (165, 165+77, s)-Nets in Base 4

(165, 165+77, 450)-Net over F₄ — Constructive and digital

Digital (165, 242, 450)-net over F₄, using

8 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+77, 774)-Net over F₄ — Digital

Digital (165, 242, 774)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+77, 32930)-Net in Base 4 — Upper bound on s

There is no (165, 242, 32931)-net in base 4, because

1 times m-reduction [i] would yield (165, 241, 32931)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 493384 048067 976482 568965 374131 411469 277013 882795 563999 971635 963678 413257 455676 480495 027786 256621 172711 095386 888697 140506 280291 308260 256285 995680 > 4²⁴¹ [i]