MinT - Best Known (62−27, 62, s)-Nets in Base 25

Best Known (62−27, 62, s)-Nets in Base 25

(62−27, 62, 252)-Net over F₂₅ — Constructive and digital

Digital (35, 62, 252)-net over F₂₅, using

3 times m-reduction [i] based on digital (35, 65, 252)-net over F₂₅, using
- (u,Â u+v)-construction [i] based on
  1. 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 (10, 40, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(62−27, 62, 961)-Net over F₂₅ — Digital

Digital (35, 62, 961)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(62−27, 62, 856557)-Net in Base 25 — Upper bound on s

There is no (35, 62, 856558)-net in base 25, because

1 times m-reduction [i] would yield (35, 61, 856558)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 18 808191 013692 482295 857763 385062 299886 412415 080604 475991 393509 880275 875686 162223 873425 > 25⁶¹ [i]