MinT - Best Known (125−27, 125, s)-Nets in Base 5

Best Known (125−27, 125, s)-Nets in Base 5

(125−27, 125, 504)-Net over F₅ — Constructive and digital

Digital (98, 125, 504)-net over F₅, using

t-expansion [i] based on digital (97, 125, 504)-net over F₅, using
- 1 times m-reduction [i] based on digital (97, 126, 504)-net over F₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (34, 48, 252)-net over F₅, using
      - trace code for nets [i] based on digital (10, 24, 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 (49, 78, 252)-net over F₅, using
      - trace code for nets [i] based on digital (10, 39, 126)-net over F₂₅, using
        net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(125−27, 125, 6062)-Net over F₅ — Digital

Digital (98, 125, 6062)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−27, 125, 6583306)-Net in Base 5 — Upper bound on s

There is no (98, 125, 6583307)-net in base 5, because

1 times m-reduction [i] would yield (98, 124, 6583307)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 470 198141 755306 519796 220535 844617 959046 498180 384645 056405 519832 288083 907471 536644 041725 > 5¹²⁴ [i]