MinT - Best Known (241−111, 241, s)-Nets in Base 3

Best Known (241−111, 241, s)-Nets in Base 3

(241−111, 241, 85)-Net over F₃ — Constructive and digital

Digital (130, 241, 85)-net over F₃, using

5 times m-reduction [i] based on digital (130, 246, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 85, 37)-net over F₃, using
    - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]
  2. digital (45, 161, 48)-net over F₃, using
    - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
        an explicitly constructive algebraic function field [i]

(241−111, 241, 148)-Net over F₃ — Digital

Digital (130, 241, 148)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(241−111, 241, 1235)-Net in Base 3 — Upper bound on s

There is no (130, 241, 1236)-net in base 3, because

1 times m-reduction [i] would yield (130, 240, 1236)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 367280 687358 696248 175450 875876 717250 510258 622226 060938 655181 414643 038311 149626 640398 553934 389819 285440 141224 496177 > 3²⁴⁰ [i]