MinT - Best Known (18, 69, s)-Nets in Base 25

Best Known (18, 69, s)-Nets in Base 25

(18, 69, 126)-Net over F₂₅ — Constructive and digital

Digital (18, 69, 126)-net over F₂₅, using

t-expansion [i] based on digital (10, 69, 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, 69, 153)-Net over F₂₅ — Digital

Digital (18, 69, 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, 69, 2677)-Net in Base 25 — Upper bound on s

There is no (18, 69, 2678)-net in base 25, because

1 times m-reduction [i] would yield (18, 68, 2678)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 115501 460343 505366 368047 772197 865662 720039 457553 012128 246968 981312 188538 552356 852218 655951 842193 > 25⁶⁸ [i]