MinT - Best Known (65−30, 65, s)-Nets in Base 3

Best Known (65−30, 65, s)-Nets in Base 3

(65−30, 65, 41)-Net over F₃ — Constructive and digital

Digital (35, 65, 41)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (5, 20, 13)-net over F₃, using
  - net from sequence [i] based on digital (5, 12)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 4, N(F)Â =Â 12, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
2. digital (15, 45, 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]

(65−30, 65, 52)-Net over F₃ — Digital

Digital (35, 65, 52)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(65−30, 65, 360)-Net in Base 3 — Upper bound on s

There is no (35, 65, 361)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10 330168 518651 935532 410640 954891 > 3⁶⁵ [i]