MinT - Best Known (68−53, 68, s)-Nets in Base 25

Best Known (68−53, 68, s)-Nets in Base 25

(68−53, 68, 126)-Net over F₂₅ — Constructive and digital

Digital (15, 68, 126)-net over F₂₅, using

t-expansion [i] based on digital (10, 68, 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]

(68−53, 68, 140)-Net over F₂₅ — Digital

Digital (15, 68, 140)-net over F₂₅, using

net from sequence [i] based on digital (15, 139)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 15 and N(F) ≥ 140, using
  - an algebraic function field by Teo [i]

(68−53, 68, 1746)-Net in Base 25 — Upper bound on s

There is no (15, 68, 1747)-net in base 25, because

1 times m-reduction [i] would yield (15, 67, 1747)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 4642 916857 674027 395817 934793 652685 023062 670858 916348 604422 870486 550984 472553 547485 477106 820049 > 25⁶⁷ [i]