MinT - Best Known (137−37, 137, s)-Nets in Base 5

Best Known (137−37, 137, s)-Nets in Base 5

(137−37, 137, 408)-Net over F₅ — Constructive and digital

Digital (100, 137, 408)-net over F₅, using

3 times m-reduction [i] based on digital (100, 140, 408)-net over F₅, using
- trace code for nets [i] based on digital (30, 70, 204)-net over F₂₅, using
  - net from sequence [i] based on digital (30, 203)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 30 and N(F) ≥ 204, using
      - a function field by GarcÃa and Quoos [i]

(137−37, 137, 1650)-Net over F₅ — Digital

Digital (100, 137, 1650)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−37, 137, 360712)-Net in Base 5 — Upper bound on s

There is no (100, 137, 360713)-net in base 5, because

1 times m-reduction [i] would yield (100, 136, 360713)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 114797 610547 219934 819872 926141 316526 676623 184894 454876 332058 475163 806932 379155 025855 816358 033625 > 5¹³⁶ [i]