MinT - Best Known (69, 69+65, s)-Nets in Base 5

Best Known (69, 69+65, s)-Nets in Base 5

(69, 69+65, 93)-Net over F₅ — Constructive and digital

Digital (69, 134, 93)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (6, 38, 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, 96, 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+65, 147)-Net over F₅ — Digital

Digital (69, 134, 147)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+65, 2546)-Net in Base 5 — Upper bound on s

There is no (69, 134, 2547)-net in base 5, because

1 times m-reduction [i] would yield (69, 133, 2547)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 924 480262 434931 838600 898539 125051 428584 129588 945203 630503 994264 091931 868478 366087 614026 990465 > 5¹³³ [i]