MinT - Best Known (42, 57, s)-Nets in Base 5

Best Known (42, 57, s)-Nets in Base 5

(42, 57, 258)-Net over F₅ — Constructive and digital

Digital (42, 57, 258)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 6)-net over F₅, using
  - net from sequence [i] based on digital (0, 5)-sequence over F₅, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 0 and N(F) ≥ 6, using
      - the rational function field F₅(x) [i]
    - Niederreiter sequence [i]
2. 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]

(42, 57, 1067)-Net over F₅ — Digital

Digital (42, 57, 1067)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 57, 330074)-Net in Base 5 — Upper bound on s

There is no (42, 57, 330075)-net in base 5, because

1 times m-reduction [i] would yield (42, 56, 330075)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1387 783036 137639 441759 699836 604975 879061 > 5⁵⁶ [i]