MinT - Best Known (102−33, 102, s)-Nets in Base 5

Best Known (102−33, 102, s)-Nets in Base 5

(102−33, 102, 258)-Net over F₅ — Constructive and digital

Digital (69, 102, 258)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (0, 16, 6)-net over F₅, using
  - net from sequence [i] based on digital (0, 5)-sequence over F₅, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 0 and N(F) ≥ 6, using
      - the rational function field F₅(x) [i]
    - Niederreiter sequence [i]
2. digital (53, 86, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 43, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]

(102−33, 102, 556)-Net over F₅ — Digital

Digital (69, 102, 556)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(102−33, 102, 43915)-Net in Base 5 — Upper bound on s

There is no (69, 102, 43916)-net in base 5, because

1 times m-reduction [i] would yield (69, 101, 43916)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 39456 984652 057525 785908 571844 175880 890180 589308 500704 981364 041183 340545 > 5¹⁰¹ [i]