MinT - Best Known (62, 62+41, s)-Nets in Base 25

Best Known (62, 62+41, s)-Nets in Base 25

(62, 62+41, 356)-Net over F₂₅ — Constructive and digital

Digital (62, 103, 356)-net over F₂₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (9, 22, 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, 30, 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, 51, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(62, 62+41, 2634)-Net over F₂₅ — Digital

Digital (62, 103, 2634)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(62, 62+41, 4662169)-Net in Base 25 — Upper bound on s

There is no (62, 103, 4662170)-net in base 25, because

1 times m-reduction [i] would yield (62, 102, 4662170)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 38893 907599 929643 390679 781067 092383 793291 457137 152305 107213 907189 621204 051896 706032 818269 025846 902114 548858 998346 457534 089291 593247 311056 590145 > 25¹⁰² [i]