MinT - Best Known (222−87, 222, s)-Nets in Base 3

Best Known (222−87, 222, s)-Nets in Base 3

(222−87, 222, 156)-Net over F₃ — Constructive and digital

Digital (135, 222, 156)-net over F₃, using

4 times m-reduction [i] based on digital (135, 226, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 113, 78)-net over F₉, using
  - net from sequence [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]

(222−87, 222, 217)-Net over F₃ — Digital

Digital (135, 222, 217)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−87, 222, 2349)-Net in Base 3 — Upper bound on s

There is no (135, 222, 2350)-net in base 3, because

1 times m-reduction [i] would yield (135, 221, 2350)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2827 095266 395330 207395 493100 230729 029339 349505 391025 804301 924785 034220 347761 965579 052043 978112 259031 826609 > 3²²¹ [i]