MinT - Best Known (249−83, 249, s)-Nets in Base 3

Best Known (249−83, 249, s)-Nets in Base 3

(249−83, 249, 164)-Net over F₃ — Constructive and digital

Digital (166, 249, 164)-net over F₃, using

3¹ times duplication [i] based on digital (165, 248, 164)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 48, 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 (117, 200, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 100, 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]

(249−83, 249, 385)-Net over F₃ — Digital

Digital (166, 249, 385)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−83, 249, 6166)-Net in Base 3 — Upper bound on s

There is no (166, 249, 6167)-net in base 3, because

1 times m-reduction [i] would yield (166, 248, 6167)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21235 935198 122340 167422 931829 150165 146457 977441 630241 825552 370413 794985 408384 465338 463039 893886 102989 203329 601332 023775 > 3²⁴⁸ [i]