MinT - Best Known (249−127, 249, s)-Nets in Base 3

Best Known (249−127, 249, s)-Nets in Base 3

(249−127, 249, 78)-Net over F₃ — Constructive and digital

Digital (122, 249, 78)-net over F₃, using

t-expansion [i] based on digital (121, 249, 78)-net over F₃, using
- net from sequence [i] based on digital (121, 77)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(249−127, 249, 120)-Net over F₃ — Digital

Digital (122, 249, 120)-net over F₃, using

t-expansion [i] based on digital (113, 249, 120)-net over F₃, using
- net from sequence [i] based on digital (113, 119)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 113 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(249−127, 249, 857)-Net in Base 3 — Upper bound on s

There is no (122, 249, 858)-net in base 3, because

1 times m-reduction [i] would yield (122, 248, 858)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22404 506162 039152 864391 112275 017535 948858 882233 451954 274433 218364 032074 723876 969307 792791 926475 093706 032884 282817 107161 > 3²⁴⁸ [i]