MinT - Best Known (166, 166+78, s)-Nets in Base 3

Best Known (166, 166+78, s)-Nets in Base 3

(166, 166+78, 176)-Net over F₃ — Constructive and digital

Digital (166, 244, 176)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 54, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (112, 190, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 95, 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]

(166, 166+78, 431)-Net over F₃ — Digital

Digital (166, 244, 431)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(166, 166+78, 7399)-Net in Base 3 — Upper bound on s

There is no (166, 244, 7400)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 262 014382 225026 646981 939922 380393 787561 614102 410872 998947 453882 178352 117666 428669 499290 924671 354761 591416 426726 418913 > 3²⁴⁴ [i]