MinT - Best Known (37, 37+89, s)-Nets in Base 5

Best Known (37, 37+89, s)-Nets in Base 5

(37, 37+89, 72)-Net over F₅ — Constructive and digital

Digital (37, 126, 72)-net over F₅, using

t-expansion [i] based on digital (31, 126, 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]

(37, 37+89, 76)-Net over F₅ — Digital

Digital (37, 126, 76)-net over F₅, using

t-expansion [i] based on digital (34, 126, 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]

(37, 37+89, 385)-Net in Base 5 — Upper bound on s

There is no (37, 126, 386)-net in base 5, because

1 times m-reduction [i] would yield (37, 125, 386)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2404 250627 421547 141968 781401 042601 147795 948916 239517 840742 990466 875061 900117 815874 482625 > 5¹²⁵ [i]