MinT - Best Known (113, 113+101, s)-Nets in Base 3

Best Known (113, 113+101, s)-Nets in Base 3

(113, 113+101, 76)-Net over F₃ — Constructive and digital

Digital (113, 214, 76)-net over F₃, using

5 times m-reduction [i] based on digital (113, 219, 76)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 68, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (45, 151, 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]

(113, 113+101, 125)-Net over F₃ — Digital

Digital (113, 214, 125)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(113, 113+101, 1001)-Net in Base 3 — Upper bound on s

There is no (113, 214, 1002)-net in base 3, because

1 times m-reduction [i] would yield (113, 213, 1002)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 435811 309782 890504 775302 430071 491938 148003 682756 259249 352857 366917 280275 277907 805431 389081 569966 381349 > 3²¹³ [i]