MinT - Best Known (70, 143, s)-Nets in Base 5

Best Known (70, 143, s)-Nets in Base 5

(70, 143, 88)-Net over F₅ — Constructive and digital

Digital (70, 143, 88)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 39, 16)-net over F₅, using
  - net from sequence [i] based on digital (3, 15)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 3 and N(F) ≥ 16, 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]

(70, 143, 132)-Net over F₅ — Digital

Digital (70, 143, 132)-net over F₅, using

t-expansion [i] based on digital (67, 143, 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]

(70, 143, 2014)-Net in Base 5 — Upper bound on s

There is no (70, 143, 2015)-net in base 5, because

1 times m-reduction [i] would yield (70, 142, 2015)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1823 338914 966071 543966 468442 712051 736640 202956 874160 521423 637462 255653 056230 223201 595502 668042 431345 > 5¹⁴² [i]