MinT - Best Known (89−35, 89, s)-Nets in Base 25

Best Known (89−35, 89, s)-Nets in Base 25

(89−35, 89, 326)-Net over F₂₅ — Constructive and digital

Digital (54, 89, 326)-net over F₂₅, using

3 times m-reduction [i] based on digital (54, 92, 326)-net over F₂₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 29, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]
  2. digital (25, 63, 200)-net over F₂₅, using
    - net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
        a function field by GarcÃa and Quoos [i]

(89−35, 89, 2591)-Net over F₂₅ — Digital

Digital (54, 89, 2591)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−35, 89, 5153811)-Net in Base 25 — Upper bound on s

There is no (54, 89, 5153812)-net in base 25, because

1 times m-reduction [i] would yield (54, 88, 5153812)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 1044 048801 678736 240798 106101 974242 410738 599729 969877 051933 118162 804649 444682 243858 997554 736425 163865 110196 490542 951805 715425 > 25⁸⁸ [i]