MinT - Best Known (132, 132+109, s)-Nets in Base 3

Best Known (132, 132+109, s)-Nets in Base 3

(132, 132+109, 86)-Net over F₃ — Constructive and digital

Digital (132, 241, 86)-net over F₃, using

1 times m-reduction [i] based on digital (132, 242, 86)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (32, 87, 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, 155, 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]

(132, 132+109, 156)-Net over F₃ — Digital

Digital (132, 241, 156)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(132, 132+109, 1331)-Net in Base 3 — Upper bound on s

There is no (132, 241, 1332)-net in base 3, because

1 times m-reduction [i] would yield (132, 240, 1332)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 347817 074660 270984 911679 165604 226885 406023 226978 051448 987967 604252 586329 310601 453720 711926 293028 797910 435563 120889 > 3²⁴⁰ [i]