MinT - Best Known (165, 165+75, s)-Nets in Base 4

Best Known (165, 165+75, s)-Nets in Base 4

(165, 165+75, 450)-Net over F₄ — Constructive and digital

Digital (165, 240, 450)-net over F₄, using

10 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+75, 824)-Net over F₄ — Digital

Digital (165, 240, 824)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+75, 37794)-Net in Base 4 — Upper bound on s

There is no (165, 240, 37795)-net in base 4, because

1 times m-reduction [i] would yield (165, 239, 37795)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 781102 185683 758582 263528 973510 803152 235700 671132 955762 232022 536623 950776 860161 097778 884990 401308 259493 605459 692139 256261 325010 236898 141905 183680 > 4²³⁹ [i]