MinT - Best Known (226−74, 226, s)-Nets in Base 3

Best Known (226−74, 226, s)-Nets in Base 3

(226−74, 226, 164)-Net over F₃ — Constructive and digital

Digital (152, 226, 164)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 44, 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 (108, 182, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 91, 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]

(226−74, 226, 374)-Net over F₃ — Digital

Digital (152, 226, 374)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(226−74, 226, 5977)-Net in Base 3 — Upper bound on s

There is no (152, 226, 5978)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 675319 352675 932953 653380 067081 057874 339651 320340 633376 307738 242795 385087 448956 233637 284134 193128 253806 073933 > 3²²⁶ [i]