MinT - Best Known (81−65, 81, s)-Nets in Base 25

Best Known (81−65, 81, s)-Nets in Base 25

(81−65, 81, 126)-Net over F₂₅ — Constructive and digital

Digital (16, 81, 126)-net over F₂₅, using

t-expansion [i] based on digital (10, 81, 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]

(81−65, 81, 150)-Net over F₂₅ — Digital

Digital (16, 81, 150)-net over F₂₅, using

net from sequence [i] based on digital (16, 149)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 16 and N(F) ≥ 150, using
  - an algebraic function field by Teo [i]

(81−65, 81, 1648)-Net in Base 25 — Upper bound on s

There is no (16, 81, 1649)-net in base 25, because

1 times m-reduction [i] would yield (16, 80, 1649)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 6894 089662 033537 985353 166291 086026 317894 475171 287623 378544 548903 088117 336131 876356 505357 521637 013013 410962 984705 > 25⁸⁰ [i]