MinT - Best Known (69−32, 69, s)-Nets in Base 3

Best Known (69−32, 69, s)-Nets in Base 3

(69−32, 69, 42)-Net over F₃ — Constructive and digital

Digital (37, 69, 42)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (6, 22, 14)-net over F₃, using
  - net from sequence [i] based on digital (6, 13)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 6 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (15, 47, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]

(69−32, 69, 53)-Net over F₃ — Digital

Digital (37, 69, 53)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−32, 69, 372)-Net in Base 3 — Upper bound on s

There is no (37, 69, 373)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 834 787145 482746 331867 694145 031617 > 3⁶⁹ [i]