MinT - Best Known (123, 123+99, s)-Nets in Base 3

Best Known (123, 123+99, s)-Nets in Base 3

(123, 123+99, 85)-Net over F₃ — Constructive and digital

Digital (123, 222, 85)-net over F₃, using

3 times m-reduction [i] based on digital (123, 225, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 78, 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, 147, 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]

(123, 123+99, 151)-Net over F₃ — Digital

Digital (123, 222, 151)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+99, 1308)-Net in Base 3 — Upper bound on s

There is no (123, 222, 1309)-net in base 3, because

1 times m-reduction [i] would yield (123, 221, 1309)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2880 683172 711635 057240 873555 906048 424373 697897 800783 926984 237508 912045 173143 885755 306391 766774 405143 399547 > 3²²¹ [i]