MinT - Best Known (109−44, 109, s)-Nets in Base 25

Best Known (109−44, 109, s)-Nets in Base 25

(109−44, 109, 356)-Net over F₂₅ — Constructive and digital

Digital (65, 109, 356)-net over F₂₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (9, 23, 104)-net over F₂₅, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by GarcÃa and Quoos [i]
2. digital (10, 32, 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]
3. digital (10, 54, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(109−44, 109, 2481)-Net over F₂₅ — Digital

Digital (65, 109, 2481)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−44, 109, 3182713)-Net in Base 25 — Upper bound on s

There is no (65, 109, 3182714)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 237 390052 605888 819134 294169 899142 785958 617445 466965 990278 592901 191345 584937 399835 110659 485949 303483 797986 451029 988776 226734 134534 915299 329901 578064 695265 > 25¹⁰⁹ [i]