MinT - Best Known (213−94, 213, s)-Nets in Base 3

Best Known (213−94, 213, s)-Nets in Base 3

(213−94, 213, 85)-Net over F₃ — Constructive and digital

Digital (119, 213, 85)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 74, 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, 139, 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]

(213−94, 213, 150)-Net over F₃ — Digital

Digital (119, 213, 150)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−94, 213, 1288)-Net in Base 3 — Upper bound on s

There is no (119, 213, 1289)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 427432 735546 670445 431581 830748 948000 340434 775478 118216 341804 616839 930040 712854 648507 137965 696201 257995 > 3²¹³ [i]