MinT - Best Known (165−82, 165, s)-Nets in Base 3

Best Known (165−82, 165, s)-Nets in Base 3

(165−82, 165, 65)-Net over F₃ — Constructive and digital

Digital (83, 165, 65)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 56, 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]
2. digital (27, 109, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(165−82, 165, 86)-Net over F₃ — Digital

Digital (83, 165, 86)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−82, 165, 631)-Net in Base 3 — Upper bound on s

There is no (83, 165, 632)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5 333878 124725 481942 229599 319388 370246 478535 441435 604883 166489 321914 389625 563505 > 3¹⁶⁵ [i]