MinT - Best Known (109−66, 109, s)-Nets in Base 5

Best Known (109−66, 109, s)-Nets in Base 5

(109−66, 109, 78)-Net over F₅ — Constructive and digital

Digital (43, 109, 78)-net over F₅, using

t-expansion [i] based on digital (38, 109, 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]

(109−66, 109, 84)-Net over F₅ — Digital

Digital (43, 109, 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]

(109−66, 109, 645)-Net in Base 5 — Upper bound on s

There is no (43, 109, 646)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 15435 202870 685917 647703 796554 206976 893401 539403 804285 208627 880393 994777 817625 > 5¹⁰⁹ [i]