MinT - Best Known (72−37, 72, s)-Nets in Base 5

Best Known (72−37, 72, s)-Nets in Base 5

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

Digital (35, 72, 72)-net over F₅, using

t-expansion [i] based on digital (31, 72, 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]

(72−37, 72, 76)-Net over F₅ — Digital

Digital (35, 72, 76)-net over F₅, using

t-expansion [i] based on digital (34, 72, 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]

(72−37, 72, 1066)-Net in Base 5 — Upper bound on s

There is no (35, 72, 1067)-net in base 5, because

1 times m-reduction [i] would yield (35, 71, 1067)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 42 850717 437702 568857 643000 584384 340537 795750 420025 > 5⁷¹ [i]