MinT - Best Known (73, 139, s)-Nets in Base 3

Best Known (73, 139, s)-Nets in Base 3

(73, 139, 61)-Net over F₃ — Constructive and digital

Digital (73, 139, 61)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 46, 24)-net over F₃, using
  - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (27, 93, 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]

(73, 139, 85)-Net over F₃ — Digital

Digital (73, 139, 85)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(73, 139, 641)-Net in Base 3 — Upper bound on s

There is no (73, 139, 642)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 178473 159053 187655 043319 556197 862005 562031 808116 556525 018554 850757 > 3¹³⁹ [i]