MinT - Best Known (89−55, 89, s)-Nets in Base 5

Best Known (89−55, 89, s)-Nets in Base 5

(89−55, 89, 72)-Net over F₅ — Constructive and digital

Digital (34, 89, 72)-net over F₅, using

t-expansion [i] based on digital (31, 89, 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]

(89−55, 89, 76)-Net over F₅ — Digital

Digital (34, 89, 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]

(89−55, 89, 498)-Net in Base 5 — Upper bound on s

There is no (34, 89, 499)-net in base 5, because

1 times m-reduction [i] would yield (34, 88, 499)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 32 848186 583904 098336 624234 539758 838413 827336 775055 862166 954533 > 5⁸⁸ [i]