MinT - Best Known (82−33, 82, s)-Nets in Base 5

Best Known (82−33, 82, s)-Nets in Base 5

(82−33, 82, 132)-Net over F₅ — Constructive and digital

Digital (49, 82, 132)-net over F₅, using

8 times m-reduction [i] based on digital (49, 90, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 45, 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]

(82−33, 82, 192)-Net over F₅ — Digital

Digital (49, 82, 192)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−33, 82, 5863)-Net in Base 5 — Upper bound on s

There is no (49, 82, 5864)-net in base 5, because

1 times m-reduction [i] would yield (49, 81, 5864)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 414 045319 086997 628603 026032 903273 181370 756062 036611 590145 > 5⁸¹ [i]