MinT - Best Known (161, 161+85, s)-Nets in Base 3

Best Known (161, 161+85, s)-Nets in Base 3

(161, 161+85, 162)-Net over F₃ — Constructive and digital

Digital (161, 246, 162)-net over F₃, using

t-expansion [i] based on digital (157, 246, 162)-net over F₃, using
- 4 times m-reduction [i] based on digital (157, 250, 162)-net over F₃, using
  - trace code for nets [i] based on digital (32, 125, 81)-net over F₉, using
    - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
        F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(161, 161+85, 342)-Net over F₃ — Digital

Digital (161, 246, 342)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(161, 161+85, 4970)-Net in Base 3 — Upper bound on s

There is no (161, 246, 4971)-net in base 3, because

1 times m-reduction [i] would yield (161, 245, 4971)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 789 489065 942849 176581 834491 605033 768096 240959 575103 079025 751593 476212 561786 166206 378993 062325 052023 699529 752606 186981 > 3²⁴⁵ [i]