MinT - Best Known (77, 77+65, s)-Nets in Base 5

Best Known (77, 77+65, s)-Nets in Base 5

(77, 77+65, 132)-Net over F₅ — Constructive and digital

Digital (77, 142, 132)-net over F₅, using

4 times m-reduction [i] based on digital (77, 146, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 73, 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]

(77, 77+65, 188)-Net over F₅ — Digital

Digital (77, 142, 188)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+65, 3819)-Net in Base 5 — Upper bound on s

There is no (77, 142, 3820)-net in base 5, because

1 times m-reduction [i] would yield (77, 141, 3820)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 360 431676 720807 551279 277816 134729 875187 519166 723806 761337 309239 126521 041778 670175 702471 124929 884161 > 5¹⁴¹ [i]