MinT - Best Known (109−72, 109, s)-Nets in Base 5

Best Known (109−72, 109, s)-Nets in Base 5

(109−72, 109, 72)-Net over F₅ — Constructive and digital

Digital (37, 109, 72)-net over F₅, using

t-expansion [i] based on digital (31, 109, 72)-net over F₅, using
- net from sequence [i] based on digital (31, 71)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
    - an explicitly constructive algebraic function field [i]

(109−72, 109, 76)-Net over F₅ — Digital

Digital (37, 109, 76)-net over F₅, using

t-expansion [i] based on digital (34, 109, 76)-net over F₅, using
- net from sequence [i] based on digital (34, 75)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 34 and N(F) ≥ 76, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(109−72, 109, 440)-Net in Base 5 — Upper bound on s

There is no (37, 109, 441)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 15786 183457 333828 080148 422320 721021 896091 914172 376455 679163 061654 775317 815025 > 5¹⁰⁹ [i]