MinT - Best Known (209−97, 209, s)-Nets in Base 3

Best Known (209−97, 209, s)-Nets in Base 3

(209−97, 209, 76)-Net over F₃ — Constructive and digital

Digital (112, 209, 76)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (32, 80, 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]
2. digital (32, 129, 38)-net over F₃, using
  - net from sequence [i] based on digital (32, 37)-sequence over F₃ (see above)

(209−97, 209, 128)-Net over F₃ — Digital

Digital (112, 209, 128)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−97, 209, 1048)-Net in Base 3 — Upper bound on s

There is no (112, 209, 1049)-net in base 3, because

1 times m-reduction [i] would yield (112, 208, 1049)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1819 654506 707218 835687 839727 082414 024972 452517 669652 419882 847969 872341 803467 636600 807685 261837 234625 > 3²⁰⁸ [i]