MinT - Best Known (209−81, 209, s)-Nets in Base 3

Best Known (209−81, 209, s)-Nets in Base 3

(209−81, 209, 156)-Net over F₃ — Constructive and digital

Digital (128, 209, 156)-net over F₃, using

3 times m-reduction [i] based on digital (128, 212, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 106, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(209−81, 209, 212)-Net over F₃ — Digital

Digital (128, 209, 212)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−81, 209, 2347)-Net in Base 3 — Upper bound on s

There is no (128, 209, 2348)-net in base 3, because

1 times m-reduction [i] would yield (128, 208, 2348)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1759 228939 937409 880149 171149 708619 814802 114856 857140 437616 522782 693949 621047 952315 528733 563188 230849 > 3²⁰⁸ [i]