MinT - Best Known (209−75, 209, s)-Nets in Base 3

Best Known (209−75, 209, s)-Nets in Base 3

(209−75, 209, 156)-Net over F₃ — Constructive and digital

Digital (134, 209, 156)-net over F₃, using

15 times m-reduction [i] based on digital (134, 224, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 112, 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]

(209−75, 209, 265)-Net over F₃ — Digital

Digital (134, 209, 265)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−75, 209, 3487)-Net in Base 3 — Upper bound on s

There is no (134, 209, 3488)-net in base 3, because

1 times m-reduction [i] would yield (134, 208, 3488)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1742 871352 070250 346259 087851 306117 674205 311505 246574 964041 707195 781584 089802 924770 220140 682475 832129 > 3²⁰⁸ [i]