MinT - Best Known (109−37, 109, s)-Nets in Base 5

Best Known (109−37, 109, s)-Nets in Base 5

(109−37, 109, 252)-Net over F₅ — Constructive and digital

Digital (72, 109, 252)-net over F₅, using

15 times m-reduction [i] based on digital (72, 124, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 62, 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]

(109−37, 109, 477)-Net over F₅ — Digital

Digital (72, 109, 477)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−37, 109, 29491)-Net in Base 5 — Upper bound on s

There is no (72, 109, 29492)-net in base 5, because

1 times m-reduction [i] would yield (72, 108, 29492)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3082 190057 098700 300790 968871 941118 684182 057199 600431 788850 683743 907105 420225 > 5¹⁰⁸ [i]