MinT - Best Known (140−69, 140, s)-Nets in Base 5

Best Known (140−69, 140, s)-Nets in Base 5

(140−69, 140, 93)-Net over F₅ — Constructive and digital

Digital (71, 140, 93)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (6, 40, 21)-net over F₅, using
  - net from sequence [i] based on digital (6, 20)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 6 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 100, 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]

(140−69, 140, 144)-Net over F₅ — Digital

Digital (71, 140, 144)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(140−69, 140, 2412)-Net in Base 5 — Upper bound on s

There is no (71, 140, 2413)-net in base 5, because

1 times m-reduction [i] would yield (71, 139, 2413)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 14 403520 996898 839978 552206 583048 336963 153039 864939 842344 327480 890929 643990 378889 251622 896119 003545 > 5¹³⁹ [i]