MinT - Best Known (82−65, 82, s)-Nets in Base 25

Best Known (82−65, 82, s)-Nets in Base 25

(82−65, 82, 126)-Net over F₂₅ — Constructive and digital

Digital (17, 82, 126)-net over F₂₅, using

t-expansion [i] based on digital (10, 82, 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]

(82−65, 82, 150)-Net over F₂₅ — Digital

Digital (17, 82, 150)-net over F₂₅, using

t-expansion [i] based on digital (16, 82, 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]

(82−65, 82, 1824)-Net in Base 25 — Upper bound on s

There is no (17, 82, 1825)-net in base 25, because

1 times m-reduction [i] would yield (17, 81, 1825)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 171528 461865 429232 832474 310957 962352 860844 047339 489962 963798 886923 738253 406534 389990 667387 535331 268396 546701 836033 > 25⁸¹ [i]