MinT - Best Known (81−23, 81, s)-Nets in Base 5

Best Known (81−23, 81, s)-Nets in Base 5

(81−23, 81, 270)-Net over F₅ — Constructive and digital

Digital (58, 81, 270)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 15, 18)-net over F₅, using
  - net from sequence [i] based on digital (4, 17)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, using
      - an explicitly constructive algebraic function field [i]
2. digital (43, 66, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 33, 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]

(81−23, 81, 859)-Net over F₅ — Digital

Digital (58, 81, 859)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−23, 81, 148713)-Net in Base 5 — Upper bound on s

There is no (58, 81, 148714)-net in base 5, because

1 times m-reduction [i] would yield (58, 80, 148714)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 82 722478 860144 337008 330433 322159 560888 515237 841993 323337 > 5⁸⁰ [i]