MinT - Best Known (89−37, 89, s)-Nets in Base 5

Best Known (89−37, 89, s)-Nets in Base 5

(89−37, 89, 132)-Net over F₅ — Constructive and digital

Digital (52, 89, 132)-net over F₅, using

7 times m-reduction [i] based on digital (52, 96, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 48, 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]

(89−37, 89, 181)-Net over F₅ — Digital

Digital (52, 89, 181)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−37, 89, 4921)-Net in Base 5 — Upper bound on s

There is no (52, 89, 4922)-net in base 5, because

1 times m-reduction [i] would yield (52, 88, 4922)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 32 351022 191514 393829 090551 882583 319462 031794 317098 296096 531905 > 5⁸⁸ [i]