MinT - Best Known (222−100, 222, s)-Nets in Base 3

Best Known (222−100, 222, s)-Nets in Base 3

(222−100, 222, 85)-Net over F₃ — Constructive and digital

Digital (122, 222, 85)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 77, 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]
2. digital (45, 145, 48)-net over F₃, using
  - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
      - an explicitly constructive algebraic function field [i]

(222−100, 222, 147)-Net over F₃ — Digital

Digital (122, 222, 147)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−100, 222, 1230)-Net in Base 3 — Upper bound on s

There is no (122, 222, 1231)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8397 282180 122819 202689 106034 498500 244787 583986 067486 504480 346931 571952 169720 104385 978447 956073 174026 530621 > 3²²² [i]