MinT - Best Known (139, 139+65, s)-Nets in Base 3

Best Known (139, 139+65, s)-Nets in Base 3

(139, 139+65, 164)-Net over F₃ — Constructive and digital

Digital (139, 204, 164)-net over F₃, using

3¹ times duplication [i] based on digital (138, 203, 164)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 39, 16)-net over F₃, using
    - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
        an explicitly constructive algebraic function field [i]
  2. digital (99, 164, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 82, 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+65, 374)-Net over F₃ — Digital

Digital (139, 204, 374)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+65, 6769)-Net in Base 3 — Upper bound on s

There is no (139, 204, 6770)-net in base 3, because

1 times m-reduction [i] would yield (139, 203, 6770)-net in base 3, but
- 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]