MinT - Best Known (248−112, 248, s)-Nets in Base 3

Best Known (248−112, 248, s)-Nets in Base 3

(248−112, 248, 86)-Net over F₃ — Constructive and digital

Digital (136, 248, 86)-net over F₃, using

t-expansion [i] based on digital (134, 248, 86)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (32, 89, 38)-net over F₃, using
    - net from sequence [i] based on digital (32, 37)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
        an explicitly constructive algebraic function field [i]
  2. digital (45, 159, 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]

(248−112, 248, 161)-Net over F₃ — Digital

Digital (136, 248, 161)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−112, 248, 1353)-Net in Base 3 — Upper bound on s

There is no (136, 248, 1354)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21779 625218 385435 049621 838710 110983 667480 414355 538384 013571 699511 606137 738748 031291 304136 645513 282619 959129 709899 278257 > 3²⁴⁸ [i]