MinT - Best Known (139, 139+64, s)-Nets in Base 3

Best Known (139, 139+64, s)-Nets in Base 3

(139, 139+64, 167)-Net over F₃ — Constructive and digital

Digital (139, 203, 167)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 41, 19)-net over F₃, using
  - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
      - an explicitly constructive algebraic function field [i]
2. digital (98, 162, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 81, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(139, 139+64, 385)-Net over F₃ — Digital

Digital (139, 203, 385)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+64, 6769)-Net in Base 3 — Upper bound on s

There is no (139, 203, 6770)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7 178180 186432 847042 038375 849826 035893 226133 997457 372022 532941 322172 653683 219087 268582 524950 025537 > 3²⁰³ [i]