MinT - Best Known (131, 131+111, s)-Nets in Base 3

Best Known (131, 131+111, s)-Nets in Base 3

(131, 131+111, 85)-Net over F₃ — Constructive and digital

Digital (131, 242, 85)-net over F₃, using

7 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]

(131, 131+111, 151)-Net over F₃ — Digital

Digital (131, 242, 151)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+111, 1260)-Net in Base 3 — Upper bound on s

There is no (131, 242, 1261)-net in base 3, because

1 times m-reduction [i] would yield (131, 241, 1261)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 687981 706342 700127 365688 552404 204647 769489 062456 149074 973349 983640 076278 896785 540395 121489 052064 950556 823606 485563 > 3²⁴¹ [i]