MinT - Best Known (168, 168+81, s)-Nets in Base 3

Best Known (168, 168+81, s)-Nets in Base 3

(168, 168+81, 172)-Net over F₃ — Constructive and digital

Digital (168, 249, 172)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 53, 24)-net over F₃, using
  - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [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]

(168, 168+81, 415)-Net over F₃ — Digital

Digital (168, 249, 415)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(168, 168+81, 7120)-Net in Base 3 — Upper bound on s

There is no (168, 249, 7121)-net in base 3, because

1 times m-reduction [i] would yield (168, 248, 7121)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21235 800511 491919 075205 628740 588230 549235 938014 322661 404002 853343 867874 169854 621661 794234 899018 077809 021260 895782 996001 > 3²⁴⁸ [i]