MinT - Best Known (231−140, 231, s)-Nets in Base 3

Best Known (231−140, 231, s)-Nets in Base 3

(231−140, 231, 64)-Net over F₃ — Constructive and digital

Digital (91, 231, 64)-net over F₃, using

t-expansion [i] based on digital (89, 231, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(231−140, 231, 96)-Net over F₃ — Digital

Digital (91, 231, 96)-net over F₃, using

t-expansion [i] based on digital (89, 231, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(231−140, 231, 439)-Net in Base 3 — Upper bound on s

There is no (91, 231, 440)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 187 117785 020088 083704 902202 821946 690652 663818 809817 610479 002941 438434 167359 114685 341503 213874 918134 667801 843793 > 3²³¹ [i]