MinT - Best Known (131, 131+119, s)-Nets in Base 3

Best Known (131, 131+119, s)-Nets in Base 3

(131, 131+119, 85)-Net over F₃ — Constructive and digital

Digital (131, 250, 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, 164, 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+119, 140)-Net over F₃ — Digital

Digital (131, 250, 140)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+119, 1120)-Net in Base 3 — Upper bound on s

There is no (131, 250, 1121)-net in base 3, because

1 times m-reduction [i] would yield (131, 249, 1121)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 66574 453440 176093 078220 778969 158160 176111 445129 572161 810630 663247 364101 893982 599385 319122 137411 734884 598868 469101 775291 > 3²⁴⁹ [i]