MinT - Best Known (116−25, 116, s)-Nets in Base 5

Best Known (116−25, 116, s)-Nets in Base 5

(116−25, 116, 504)-Net over F₅ — Constructive and digital

Digital (91, 116, 504)-net over F₅, using

5² times duplication [i] based on digital (89, 114, 504)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (32, 44, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 22, 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 (45, 70, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 35, 126)-net over F₂₅, using
      - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(116−25, 116, 5869)-Net over F₅ — Digital

Digital (91, 116, 5869)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(116−25, 116, 6603336)-Net in Base 5 — Upper bound on s

There is no (91, 116, 6603337)-net in base 5, because

1 times m-reduction [i] would yield (91, 115, 6603337)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 240 741622 926948 781113 953923 572378 562850 789489 206281 283457 895502 348603 016602 941265 > 5¹¹⁵ [i]