MinT - Best Known (170−61, 170, s)-Nets in Base 3

Best Known (170−61, 170, s)-Nets in Base 3

(170−61, 170, 156)-Net over F₃ — Constructive and digital

Digital (109, 170, 156)-net over F₃, using

4 times m-reduction [i] based on digital (109, 174, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 87, 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]

(170−61, 170, 222)-Net over F₃ — Digital

Digital (109, 170, 222)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−61, 170, 2905)-Net in Base 3 — Upper bound on s

There is no (109, 170, 2906)-net in base 3, because

1 times m-reduction [i] would yield (109, 169, 2906)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 433 666120 372674 704181 934557 351384 373899 383988 249649 261674 958366 097908 362700 186045 > 3¹⁶⁹ [i]