MinT - Best Known (69−31, 69, s)-Nets in Base 5

Best Known (69−31, 69, s)-Nets in Base 5

(69−31, 69, 104)-Net over F₅ — Constructive and digital

Digital (38, 69, 104)-net over F₅, using

1 times m-reduction [i] based on digital (38, 70, 104)-net over F₅, using
- trace code for nets [i] based on digital (3, 35, 52)-net over F₂₅, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by GarcÃa and Quoos [i]

(69−31, 69, 114)-Net over F₅ — Digital

Digital (38, 69, 114)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−31, 69, 2357)-Net in Base 5 — Upper bound on s

There is no (38, 69, 2358)-net in base 5, because

1 times m-reduction [i] would yield (38, 68, 2358)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 340517 816727 601822 684679 363415 435152 597219 645945 > 5⁶⁸ [i]