MinT - Best Known (241−80, 241, s)-Nets in Base 3

Best Known (241−80, 241, s)-Nets in Base 3

(241−80, 241, 164)-Net over F₃ — Constructive and digital

Digital (161, 241, 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 (114, 194, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 97, 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]

(241−80, 241, 379)-Net over F₃ — Digital

Digital (161, 241, 379)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(241−80, 241, 5868)-Net in Base 3 — Upper bound on s

There is no (161, 241, 5869)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 734672 119529 329993 023359 111027 875547 229772 794377 656682 969227 515350 280757 720285 622338 460040 253456 144876 170145 428705 > 3²⁴¹ [i]