MinT - Best Known (248−116, 248, s)-Nets in Base 3

Best Known (248−116, 248, s)-Nets in Base 3

(248−116, 248, 85)-Net over F₃ — Constructive and digital

Digital (132, 248, 85)-net over F₃, using

t-expansion [i] based on digital (131, 248, 85)-net over F₃, using
- 1 times m-reduction [i] based on digital (131, 249, 85)-net over F₃, using
  - (u,Â u+v)-construction [i] based on
    1. digital (27, 86, 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, 163, 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]

(248−116, 248, 146)-Net over F₃ — Digital

Digital (132, 248, 146)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−116, 248, 1174)-Net in Base 3 — Upper bound on s

There is no (132, 248, 1175)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21473 106325 175478 648730 502541 246842 317353 498643 768311 833239 809481 497185 794642 177643 335282 015076 158810 966473 165599 045181 > 3²⁴⁸ [i]