MinT - Best Known (162, 162+81, s)-Nets in Base 3

Best Known (162, 162+81, s)-Nets in Base 3

(162, 162+81, 164)-Net over F₃ — Constructive and digital

Digital (162, 243, 164)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 47, 16)-net over F₃, using
  - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (115, 196, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 98, 74)-net over F₉, using
    - net from sequence [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]

(162, 162+81, 377)-Net over F₃ — Digital

Digital (162, 243, 377)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(162, 162+81, 6032)-Net in Base 3 — Upper bound on s

There is no (162, 243, 6033)-net in base 3, because

1 times m-reduction [i] would yield (162, 242, 6033)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 103810 503185 562185 546257 345309 670947 517769 002825 179757 990574 859064 703747 784406 236907 158684 543242 769317 478576 494625 > 3²⁴² [i]