MinT - Best Known (248−137, 248, s)-Nets in Base 3

Best Known (248−137, 248, s)-Nets in Base 3

(248−137, 248, 74)-Net over F₃ — Constructive and digital

Digital (111, 248, 74)-net over F₃, using

t-expansion [i] based on digital (107, 248, 74)-net over F₃, using
- net from sequence [i] based on digital (107, 73)-sequence over F₃, using
  - base reduction for sequences [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]

(248−137, 248, 104)-Net over F₃ — Digital

Digital (111, 248, 104)-net over F₃, using

t-expansion [i] based on digital (102, 248, 104)-net over F₃, using
- net from sequence [i] based on digital (102, 103)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 102 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(248−137, 248, 642)-Net in Base 3 — Upper bound on s

There is no (111, 248, 643)-net in base 3, because

1 times m-reduction [i] would yield (111, 247, 643)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7513 276290 910609 418421 181175 377220 345211 301669 239255 642659 679498 180163 093069 146378 871801 302211 373054 707363 792375 200345 > 3²⁴⁷ [i]