MinT - Best Known (165, 165+65, s)-Nets in Base 3

Best Known (165, 165+65, s)-Nets in Base 3

(165, 165+65, 288)-Net over F₃ — Constructive and digital

Digital (165, 230, 288)-net over F₃, using

1 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]

(165, 165+65, 609)-Net over F₃ — Digital

Digital (165, 230, 609)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+65, 16573)-Net in Base 3 — Upper bound on s

There is no (165, 230, 16574)-net in base 3, because

1 times m-reduction [i] would yield (165, 229, 16574)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18 231813 188459 496013 961756 160336 594319 174043 584190 633850 690806 222690 343760 463576 197304 217140 137675 239638 865729 > 3²²⁹ [i]