MinT - Best Known (129−29, 129, s)-Nets in Base 5

Best Known (129−29, 129, s)-Nets in Base 5

(129−29, 129, 504)-Net over F₅ — Constructive and digital

Digital (100, 129, 504)-net over F₅, using

1 times m-reduction [i] based on digital (100, 130, 504)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (35, 50, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 25, 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 (50, 80, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 40, 126)-net over F₂₅, using
      - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(129−29, 129, 4704)-Net over F₅ — Digital

Digital (100, 129, 4704)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(129−29, 129, 3715176)-Net in Base 5 — Upper bound on s

There is no (100, 129, 3715177)-net in base 5, because

1 times m-reduction [i] would yield (100, 128, 3715177)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 293873 905121 734508 615791 097560 464040 564323 465689 279379 870367 235066 791317 720707 017850 882025 > 5¹²⁸ [i]