MinT - Best Known (165, 165+65, s)-Nets in Base 4

Best Known (165, 165+65, s)-Nets in Base 4

(165, 165+65, 531)-Net over F₄ — Constructive and digital

Digital (165, 230, 531)-net over F₄, using

7 times m-reduction [i] based on digital (165, 237, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 79, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(165, 165+65, 1204)-Net over F₄ — Digital

Digital (165, 230, 1204)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+65, 86720)-Net in Base 4 — Upper bound on s

There is no (165, 230, 86721)-net in base 4, because

1 times m-reduction [i] would yield (165, 229, 86721)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 744521 903781 803911 817005 737790 605640 287910 268727 260548 134930 312943 462687 060107 390927 664107 859329 318973 422093 153137 194843 272624 328861 713778 > 4²²⁹ [i]