MinT - Best Known (18, 81, s)-Nets in Base 25

Best Known (18, 81, s)-Nets in Base 25

(18, 81, 126)-Net over F₂₅ — Constructive and digital

Digital (18, 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]

(18, 81, 153)-Net over F₂₅ — Digital

Digital (18, 81, 153)-net over F₂₅, using

net from sequence [i] based on digital (18, 152)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 18 and N(F) ≥ 153, using
  - an algebraic function field by Teo [i]

(18, 81, 2079)-Net in Base 25 — Upper bound on s

There is no (18, 81, 2080)-net in base 25, because

1 times m-reduction [i] would yield (18, 80, 2080)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 6860 339259 824700 607443 317851 775696 668549 285542 872130 619532 388515 307755 635625 198344 942402 606514 758072 467648 463105 > 25⁸⁰ [i]