MinT - Best Known (81−33, 81, s)-Nets in Base 5

Best Known (81−33, 81, s)-Nets in Base 5

(81−33, 81, 132)-Net over F₅ — Constructive and digital

Digital (48, 81, 132)-net over F₅, using

7 times m-reduction [i] based on digital (48, 88, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 44, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(81−33, 81, 181)-Net over F₅ — Digital

Digital (48, 81, 181)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−33, 81, 5301)-Net in Base 5 — Upper bound on s

There is no (48, 81, 5302)-net in base 5, because

1 times m-reduction [i] would yield (48, 80, 5302)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 82 874208 527234 696574 503113 205809 259335 275055 447638 681345 > 5⁸⁰ [i]