MinT - Best Known (55, 55+57, s)-Nets in Base 5

Best Known (55, 55+57, s)-Nets in Base 5

(55, 55+57, 82)-Net over F₅ — Constructive and digital

Digital (55, 112, 82)-net over F₅, using

t-expansion [i] based on digital (48, 112, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(55, 55+57, 108)-Net over F₅ — Digital

Digital (55, 112, 108)-net over F₅, using

net from sequence [i] based on digital (55, 107)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 55 and N(F) ≥ 108, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(55, 55+57, 1646)-Net in Base 5 — Upper bound on s

There is no (55, 112, 1647)-net in base 5, because

1 times m-reduction [i] would yield (55, 111, 1647)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 389697 105759 256767 101923 575123 434589 709222 468237 531629 293069 119863 813350 375825 > 5¹¹¹ [i]