MinT - Best Known (69, 69+73, s)-Nets in Base 5

Best Known (69, 69+73, s)-Nets in Base 5

(69, 69+73, 84)-Net over F₅ — Constructive and digital

Digital (69, 142, 84)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 38, 12)-net over F₅, using
  - net from sequence [i] based on digital (2, 11)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 2 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 104, 72)-net over F₅, using
  - net from sequence [i] based on digital (31, 71)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
      - an explicitly constructive algebraic function field [i]

(69, 69+73, 132)-Net over F₅ — Digital

Digital (69, 142, 132)-net over F₅, using

t-expansion [i] based on digital (67, 142, 132)-net over F₅, using
- net from sequence [i] based on digital (67, 131)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 67 and N(F) ≥ 132, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(69, 69+73, 1924)-Net in Base 5 — Upper bound on s

There is no (69, 142, 1925)-net in base 5, because

1 times m-reduction [i] would yield (69, 141, 1925)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 359 641364 800979 492388 561138 332476 282019 884758 056019 609539 087497 540159 023664 943862 546124 506253 656689 > 5¹⁴¹ [i]