MinT - Best Known (70−55, 70, s)-Nets in Base 25

Best Known (70−55, 70, s)-Nets in Base 25

(70−55, 70, 126)-Net over F₂₅ — Constructive and digital

Digital (15, 70, 126)-net over F₂₅, using

t-expansion [i] based on digital (10, 70, 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]

(70−55, 70, 140)-Net over F₂₅ — Digital

Digital (15, 70, 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]

(70−55, 70, 1686)-Net in Base 25 — Upper bound on s

There is no (15, 70, 1687)-net in base 25, because

1 times m-reduction [i] would yield (15, 69, 1687)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 2 873522 118145 097133 948442 460880 656731 834454 968227 487700 821243 013682 576874 892738 905138 641491 147065 > 25⁶⁹ [i]