MinT - Best Known (68, 68+59, s)-Nets in Base 5

Best Known (68, 68+59, s)-Nets in Base 5

(68, 68+59, 132)-Net over F₅ — Constructive and digital

Digital (68, 127, 132)-net over F₅, using

1 times m-reduction [i] based on digital (68, 128, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 64, 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]

(68, 68+59, 162)-Net over F₅ — Digital

Digital (68, 127, 162)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+59, 3155)-Net in Base 5 — Upper bound on s

There is no (68, 127, 3156)-net in base 5, because

1 times m-reduction [i] would yield (68, 126, 3156)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 11816 528249 007489 362693 151258 098372 646045 174657 553975 510777 980660 944043 250492 780930 632145 > 5¹²⁶ [i]