MinT - Best Known (139−57, 139, s)-Nets in Base 5

Best Known (139−57, 139, s)-Nets in Base 5

(139−57, 139, 252)-Net over F₅ — Constructive and digital

Digital (82, 139, 252)-net over F₅, using

5 times m-reduction [i] based on digital (82, 144, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 72, 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]

(139−57, 139, 274)-Net over F₅ — Digital

Digital (82, 139, 274)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−57, 139, 7847)-Net in Base 5 — Upper bound on s

There is no (82, 139, 7848)-net in base 5, because

1 times m-reduction [i] would yield (82, 138, 7848)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 873981 319201 589237 337057 492675 485852 270711 924775 968127 615002 394574 870586 882303 487787 169020 570625 > 5¹³⁸ [i]