MinT - Best Known (112−68, 112, s)-Nets in Base 5

Best Known (112−68, 112, s)-Nets in Base 5

(112−68, 112, 78)-Net over F₅ — Constructive and digital

Digital (44, 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]

(112−68, 112, 84)-Net over F₅ — Digital

Digital (44, 112, 84)-net over F₅, using

t-expansion [i] based on digital (43, 112, 84)-net over F₅, using
- net from sequence [i] based on digital (43, 83)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 43 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(112−68, 112, 654)-Net in Base 5 — Upper bound on s

There is no (44, 112, 655)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 986851 925989 319864 381726 584145 231808 081245 539168 535787 731826 749559 353970 677625 > 5¹¹² [i]