MinT - Best Known (51−29, 51, s)-Nets in Base 25

Best Known (51−29, 51, s)-Nets in Base 25

Digital (22, 51, 148)-net over F₂₅, using

t-expansion [i] based on digital (19, 51, 148)-net over F₂₅, using
- net from sequence [i] based on digital (19, 147)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 19 and N(F) ≥ 148, using
    - a function field by GarcÃa and Quoos [i]

Digital (22, 51, 172)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(25⁵¹, 172, F₂₅, 2, 29) (dual of [(172, 2), 293, 30]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(25⁴⁹, 171, F₂₅, 2, 29) (dual of [(171, 2), 293, 30]-NRT-code), using
  - extended algebraic-geometric NRT-code AG_e(2;F,312P) [i] based on function field F/F₂₅ with g(F) = 20 and N(F) ≥ 171, using
    - an algebraic function field by Teo [i]

There is no (22, 51, 24761)-net in base 25, because

1 times m-reduction [i] would yield (22, 50, 24761)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 7889 373466 589941 714974 375001 359894 018432 761997 780368 216632 625073 876625 > 25⁵⁰ [i]