MinT - Best Known (165, 165+79, s)-Nets in Base 4

Best Known (165, 165+79, s)-Nets in Base 4

(165, 165+79, 450)-Net over F₄ — Constructive and digital

Digital (165, 244, 450)-net over F₄, using

6 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+79, 730)-Net over F₄ — Digital

Digital (165, 244, 730)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+79, 28914)-Net in Base 4 — Upper bound on s

There is no (165, 244, 28915)-net in base 4, because

1 times m-reduction [i] would yield (165, 243, 28915)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 199 836497 153062 689081 955703 754082 204510 935543 156532 459312 197723 159161 921562 544338 789618 172986 801033 493570 846773 319297 688113 129608 795770 650552 743136 > 4²⁴³ [i]