MinT - Best Known (138−69, 138, s)-Nets in Base 5

Best Known (138−69, 138, s)-Nets in Base 5

(138−69, 138, 90)-Net over F₅ — Constructive and digital

Digital (69, 138, 90)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 38, 18)-net over F₅, using
  - net from sequence [i] based on digital (4, 17)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, 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]

(138−69, 138, 135)-Net over F₅ — Digital

Digital (69, 138, 135)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−69, 138, 2192)-Net in Base 5 — Upper bound on s

There is no (69, 138, 2193)-net in base 5, because

1 times m-reduction [i] would yield (69, 137, 2193)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 577963 728671 845951 331926 430159 895300 554195 548165 463155 344785 418525 929622 466707 899763 646680 001945 > 5¹³⁷ [i]