MinT - Best Known (90−45, 90, s)-Nets in Base 5

Best Known (90−45, 90, s)-Nets in Base 5

(90−45, 90, 78)-Net over F₅ — Constructive and digital

Digital (45, 90, 78)-net over F₅, using

t-expansion [i] based on digital (38, 90, 78)-net over F₅, using
- net from sequence [i] based on digital (38, 77)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 38 and N(F) ≥ 78, using
    - an explicitly constructive algebraic function field [i]

(90−45, 90, 96)-Net over F₅ — Digital

Digital (45, 90, 96)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−45, 90, 1506)-Net in Base 5 — Upper bound on s

There is no (45, 90, 1507)-net in base 5, because

1 times m-reduction [i] would yield (45, 89, 1507)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 163 520705 920111 465563 049031 289793 224732 060341 495175 410828 243625 > 5⁸⁹ [i]