MinT - Best Known (69−30, 69, s)-Nets in Base 25

Best Known (69−30, 69, s)-Nets in Base 25

(69−30, 69, 252)-Net over F₂₅ — Constructive and digital

Digital (39, 69, 252)-net over F₂₅, using

8 times m-reduction [i] based on digital (39, 77, 252)-net over F₂₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 29, 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, 48, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(69−30, 69, 1045)-Net over F₂₅ — Digital

Digital (39, 69, 1045)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−30, 69, 721231)-Net in Base 25 — Upper bound on s

There is no (39, 69, 721232)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 2 869873 128892 543255 662816 743598 104047 699120 644361 926753 754323 166821 722230 979012 040121 808565 346945 > 25⁶⁹ [i]