MinT - Best Known (69−29, 69, s)-Nets in Base 25

Best Known (69−29, 69, s)-Nets in Base 25

(69−29, 69, 252)-Net over F₂₅ — Constructive and digital

Digital (40, 69, 252)-net over F₂₅, using

11 times m-reduction [i] based on digital (40, 80, 252)-net over F₂₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 30, 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, 50, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(69−29, 69, 1325)-Net over F₂₅ — Digital

Digital (40, 69, 1325)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−29, 69, 1553240)-Net in Base 25 — Upper bound on s

There is no (40, 69, 1553241)-net in base 25, because

1 times m-reduction [i] would yield (40, 68, 1553241)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 114794 373224 895412 657788 349161 328781 155932 305555 824222 364778 965700 054071 660590 667441 423971 538065 > 25⁶⁸ [i]