MinT - Best Known (124, 124+99, s)-Nets in Base 3

Best Known (124, 124+99, s)-Nets in Base 3

(124, 124+99, 85)-Net over F₃ — Constructive and digital

Digital (124, 223, 85)-net over F₃, using

5 times m-reduction [i] based on digital (124, 228, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 79, 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, 149, 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]

(124, 124+99, 153)-Net over F₃ — Digital

Digital (124, 223, 153)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(124, 124+99, 1338)-Net in Base 3 — Upper bound on s

There is no (124, 223, 1339)-net in base 3, because

1 times m-reduction [i] would yield (124, 222, 1339)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8416 500406 772071 795181 821459 467129 916081 822661 393958 928553 577860 915234 663025 393902 741261 256161 387212 144023 > 3²²² [i]