MinT - Best Known (161−53, 161, s)-Nets in Base 3

Best Known (161−53, 161, s)-Nets in Base 3

(161−53, 161, 156)-Net over F₃ — Constructive and digital

Digital (108, 161, 156)-net over F₃, using

11 times m-reduction [i] based on digital (108, 172, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 86, 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]

(161−53, 161, 276)-Net over F₃ — Digital

Digital (108, 161, 276)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(161−53, 161, 4528)-Net in Base 3 — Upper bound on s

There is no (108, 161, 4529)-net in base 3, because

1 times m-reduction [i] would yield (108, 160, 4529)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21901 390738 141836 338796 877383 964117 168545 039639 951082 663174 371984 437836 433865 > 3¹⁶⁰ [i]