MinT - Best Known (137−31, 137, s)-Nets in Base 5

Best Known (137−31, 137, s)-Nets in Base 5

(137−31, 137, 504)-Net over F₅ — Constructive and digital

Digital (106, 137, 504)-net over F₅, using

t-expansion [i] based on digital (105, 137, 504)-net over F₅, using
- 1 times m-reduction [i] based on digital (105, 138, 504)-net over F₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (36, 52, 252)-net over F₅, using
      - trace code for nets [i] based on digital (10, 26, 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 (53, 86, 252)-net over F₅, using
      - trace code for nets [i] based on digital (10, 43, 126)-net over F₂₅, using
        net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(137−31, 137, 4700)-Net over F₅ — Digital

Digital (106, 137, 4700)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−31, 137, 3491671)-Net in Base 5 — Upper bound on s

There is no (106, 137, 3491672)-net in base 5, because

1 times m-reduction [i] would yield (106, 136, 3491672)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 114794 721624 125952 556773 408604 870152 505223 471784 620688 498269 187916 755115 325062 304364 398991 523681 > 5¹³⁶ [i]