MinT - Best Known (42, 42+70, s)-Nets in Base 5

Best Known (42, 42+70, s)-Nets in Base 5

(42, 42+70, 78)-Net over F₅ — Constructive and digital

Digital (42, 112, 78)-net over F₅, using

t-expansion [i] based on digital (38, 112, 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]

(42, 42+70, 80)-Net over F₅ — Digital

Digital (42, 112, 80)-net over F₅, using

t-expansion [i] based on digital (41, 112, 80)-net over F₅, using
- net from sequence [i] based on digital (41, 79)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 41 and N(F) ≥ 80, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(42, 42+70, 574)-Net in Base 5 — Upper bound on s

There is no (42, 112, 575)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 006169 865815 415873 324751 654538 092437 662148 350016 068308 626796 703063 759767 981173 > 5¹¹² [i]