MinT - Best Known (69, 69+25, s)-Nets in Base 5

Best Known (69, 69+25, s)-Nets in Base 5

(69, 69+25, 304)-Net over F₅ — Constructive and digital

Digital (69, 94, 304)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (12, 24, 52)-net over F₅, using
  - trace code for nets [i] based on digital (0, 12, 26)-net over F₂₅, using
    - net from sequence [i] based on digital (0, 25)-sequence over F₂₅, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using
        the rational function field F₂₅(x) [i]
      - Niederreiter sequence [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, 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]

(69, 69+25, 1351)-Net over F₅ — Digital

Digital (69, 94, 1351)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+25, 345388)-Net in Base 5 — Upper bound on s

There is no (69, 94, 345389)-net in base 5, because

1 times m-reduction [i] would yield (69, 93, 345389)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 100974 848056 109383 639123 229301 792832 100141 817936 959120 204217 823953 > 5⁹³ [i]