MinT - Best Known (142−101, 142, s)-Nets in Base 5

Best Known (142−101, 142, s)-Nets in Base 5

(142−101, 142, 78)-Net over F₅ — Constructive and digital

Digital (41, 142, 78)-net over F₅, using

t-expansion [i] based on digital (38, 142, 78)-net over F₅, using
- net from sequence [i] based on digital (38, 77)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 38 and N(F) ≥ 78, using
    - an explicitly constructive algebraic function field [i]

(142−101, 142, 80)-Net over F₅ — Digital

Digital (41, 142, 80)-net over F₅, using

net from sequence [i] based on digital (41, 79)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 41 and N(F) ≥ 80, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(142−101, 142, 419)-Net in Base 5 — Upper bound on s

There is no (41, 142, 420)-net in base 5, because

1 times m-reduction [i] would yield (41, 141, 420)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 368 884432 896416 151060 222383 900155 438902 321769 612823 401744 704958 514691 153963 821037 299782 218635 658945 > 5¹⁴¹ [i]