MinT - Best Known (109, 210, s)-Nets in Base 3

Best Known (109, 210, s)-Nets in Base 3

(109, 210, 75)-Net over F₃ — Constructive and digital

Digital (109, 210, 75)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 77, 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 (32, 133, 38)-net over F₃, using
  - net from sequence [i] based on digital (32, 37)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
      - an explicitly constructive algebraic function field [i]

(109, 210, 116)-Net over F₃ — Digital

Digital (109, 210, 116)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 210, 913)-Net in Base 3 — Upper bound on s

There is no (109, 210, 914)-net in base 3, because

1 times m-reduction [i] would yield (109, 209, 914)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5466 726239 815747 030070 231723 751821 170213 796804 274361 359224 575323 975196 260982 301866 693850 536912 320981 > 3²⁰⁹ [i]