MinT - Best Known (69, 69+71, s)-Nets in Base 5

Best Known (69, 69+71, s)-Nets in Base 5

(69, 69+71, 88)-Net over F₅ — Constructive and digital

Digital (69, 140, 88)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 38, 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, 102, 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+71, 132)-Net over F₅ — Digital

Digital (69, 140, 132)-net over F₅, using

t-expansion [i] based on digital (67, 140, 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+71, 2049)-Net in Base 5 — Upper bound on s

There is no (69, 140, 2050)-net in base 5, because

1 times m-reduction [i] would yield (69, 139, 2050)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 14 393383 646299 481428 222678 748590 752311 417911 450002 721629 067502 708169 912359 557000 660755 008232 720809 > 5¹³⁹ [i]