MinT - Best Known (68−29, 68, s)-Nets in Base 25

Best Known (68−29, 68, s)-Nets in Base 25

(68−29, 68, 252)-Net over F₂₅ — Constructive and digital

Digital (39, 68, 252)-net over F₂₅, using

9 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)

(68−29, 68, 1183)-Net over F₂₅ — Digital

Digital (39, 68, 1183)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−29, 68, 1234199)-Net in Base 25 — Upper bound on s

There is no (39, 68, 1234200)-net in base 25, because

1 times m-reduction [i] would yield (39, 67, 1234200)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 4591 784692 775347 296108 482992 404299 717071 970470 776063 555648 421908 627194 839144 138197 666780 455041 > 25⁶⁷ [i]