MinT - Best Known (54, 54+39, s)-Nets in Base 25

Best Known (54, 54+39, s)-Nets in Base 25

(54, 54+39, 326)-Net over F₂₅ — Constructive and digital

Digital (54, 93, 326)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (10, 29, 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]
2. digital (25, 64, 200)-net over F₂₅, using
  - net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
      - a function field by GarcÃa and Quoos [i]

(54, 54+39, 1670)-Net over F₂₅ — Digital

Digital (54, 93, 1670)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+39, 1940773)-Net in Base 25 — Upper bound on s

There is no (54, 93, 1940774)-net in base 25, because

1 times m-reduction [i] would yield (54, 92, 1940774)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 407 835424 488777 712957 905237 644775 870898 842896 507927 705344 344998 157596 262472 887016 012151 134564 306512 001379 766336 441749 414026 610545 > 25⁹² [i]