MinT - Best Known (209−77, 209, s)-Nets in Base 3

Best Known (209−77, 209, s)-Nets in Base 3

(209−77, 209, 156)-Net over F₃ — Constructive and digital

Digital (132, 209, 156)-net over F₃, using

11 times m-reduction [i] based on digital (132, 220, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 110, 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]

(209−77, 209, 246)-Net over F₃ — Digital

Digital (132, 209, 246)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−77, 209, 3034)-Net in Base 3 — Upper bound on s

There is no (132, 209, 3035)-net in base 3, because

1 times m-reduction [i] would yield (132, 208, 3035)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1755 057835 782077 461679 416099 685959 599355 310063 250090 144107 527765 078148 819193 212102 364582 453019 733325 > 3²⁰⁸ [i]