MinT - Best Known (140−31, 140, s)-Nets in Base 5

Best Known (140−31, 140, s)-Nets in Base 5

(140−31, 140, 504)-Net over F₅ — Constructive and digital

Digital (109, 140, 504)-net over F₅, using

4 times m-reduction [i] based on digital (109, 144, 504)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (37, 54, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 27, 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 (55, 90, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 45, 126)-net over F₂₅, using
      - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(140−31, 140, 5518)-Net over F₅ — Digital

Digital (109, 140, 5518)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(140−31, 140, 4817566)-Net in Base 5 — Upper bound on s

There is no (109, 140, 4817567)-net in base 5, because

1 times m-reduction [i] would yield (109, 139, 4817567)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 14 349317 480678 972805 241268 933593 130667 704873 342345 900849 044351 477915 153168 141952 869185 337685 959749 > 5¹³⁹ [i]