MinT - Best Known (85−31, 85, s)-Nets in Base 5

Best Known (85−31, 85, s)-Nets in Base 5

(85−31, 85, 252)-Net over F₅ — Constructive and digital

Digital (54, 85, 252)-net over F₅, using

3 times m-reduction [i] based on digital (54, 88, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 44, 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]

(85−31, 85, 292)-Net over F₅ — Digital

Digital (54, 85, 292)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(85−31, 85, 13169)-Net in Base 5 — Upper bound on s

There is no (54, 85, 13170)-net in base 5, because

1 times m-reduction [i] would yield (54, 84, 13170)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 51701 187866 968828 671382 163989 581289 225496 978160 567139 387369 > 5⁸⁴ [i]