MinT - Best Known (45, 45+63, s)-Nets in Base 5

Best Known (45, 45+63, s)-Nets in Base 5

(45, 45+63, 78)-Net over F₅ — Constructive and digital

Digital (45, 108, 78)-net over F₅, using

t-expansion [i] based on digital (38, 108, 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]

(45, 45+63, 88)-Net over F₅ — Digital

Digital (45, 108, 88)-net over F₅, using

net from sequence [i] based on digital (45, 87)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 45 and N(F) ≥ 88, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(45, 45+63, 780)-Net in Base 5 — Upper bound on s

There is no (45, 108, 781)-net in base 5, because

1 times m-reduction [i] would yield (45, 107, 781)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 636 470398 110713 211729 104579 600263 918717 386441 620063 864841 243598 583940 287805 > 5¹⁰⁷ [i]