MinT - Best Known (116, 116+107, s)-Nets in Base 3

Best Known (116, 116+107, s)-Nets in Base 3

(116, 116+107, 76)-Net over F₃ — Constructive and digital

Digital (116, 223, 76)-net over F₃, using

5 times m-reduction [i] based on digital (116, 228, 76)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 71, 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, 157, 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]

(116, 116+107, 124)-Net over F₃ — Digital

Digital (116, 223, 124)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(116, 116+107, 974)-Net in Base 3 — Upper bound on s

There is no (116, 223, 975)-net in base 3, because

1 times m-reduction [i] would yield (116, 222, 975)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8388 629817 541807 135743 949624 462116 218608 107309 593556 458599 005695 532505 600917 201776 509686 515479 669335 043767 > 3²²² [i]