MinT - Best Known (125, 125+107, s)-Nets in Base 3

Best Known (125, 125+107, s)-Nets in Base 3

(125, 125+107, 85)-Net over F₃ — Constructive and digital

Digital (125, 232, 85)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 80, 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, 152, 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]

(125, 125+107, 143)-Net over F₃ — Digital

Digital (125, 232, 143)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(125, 125+107, 1185)-Net in Base 3 — Upper bound on s

There is no (125, 232, 1186)-net in base 3, because

1 times m-reduction [i] would yield (125, 231, 1186)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 169 933861 599139 732415 003782 117815 676946 138218 528975 099021 806092 271676 104531 257737 651032 986849 529292 108768 908669 > 3²³¹ [i]