MinT - Best Known (86−31, 86, s)-Nets in Base 5

Best Known (86−31, 86, s)-Nets in Base 5

(86−31, 86, 252)-Net over F₅ — Constructive and digital

Digital (55, 86, 252)-net over F₅, using

4 times m-reduction [i] based on digital (55, 90, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 45, 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]

(86−31, 86, 309)-Net over F₅ — Digital

Digital (55, 86, 309)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−31, 86, 14662)-Net in Base 5 — Upper bound on s

There is no (55, 86, 14663)-net in base 5, because

1 times m-reduction [i] would yield (55, 85, 14663)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 258532 840943 363314 904025 177104 810905 552961 371352 289366 353765 > 5⁸⁵ [i]