MinT - Best Known (68, 68+33, s)-Nets in Base 5

Best Known (68, 68+33, s)-Nets in Base 5

(68, 68+33, 252)-Net over F₅ — Constructive and digital

Digital (68, 101, 252)-net over F₅, using

15 times m-reduction [i] based on digital (68, 116, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 58, 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]

(68, 68+33, 530)-Net over F₅ — Digital

Digital (68, 101, 530)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+33, 39711)-Net in Base 5 — Upper bound on s

There is no (68, 101, 39712)-net in base 5, because

1 times m-reduction [i] would yield (68, 100, 39712)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 7890 420784 928704 142768 284983 947474 059625 134040 795101 903311 727141 576705 > 5¹⁰⁰ [i]