MinT - Best Known (222−57, 222, s)-Nets in Base 3

Best Known (222−57, 222, s)-Nets in Base 3

(222−57, 222, 288)-Net over F₃ — Constructive and digital

Digital (165, 222, 288)-net over F₃, using

9 times m-reduction [i] based on digital (165, 231, 288)-net over F₃, using
- trace code for nets [i] based on digital (11, 77, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]

(222−57, 222, 835)-Net over F₃ — Digital

Digital (165, 222, 835)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−57, 222, 32920)-Net in Base 3 — Upper bound on s

There is no (165, 222, 32921)-net in base 3, because

1 times m-reduction [i] would yield (165, 221, 32921)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2780 312787 615971 647068 707141 475762 073302 618959 413548 292716 940527 981985 654172 268814 120575 227779 630571 562865 > 3²²¹ [i]