MinT - Best Known (156−51, 156, s)-Nets in Base 3

Best Known (156−51, 156, s)-Nets in Base 3

(156−51, 156, 156)-Net over F₃ — Constructive and digital

Digital (105, 156, 156)-net over F₃, using

10 times m-reduction [i] based on digital (105, 166, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 83, 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]

(156−51, 156, 275)-Net over F₃ — Digital

Digital (105, 156, 275)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(156−51, 156, 4596)-Net in Base 3 — Upper bound on s

There is no (105, 156, 4597)-net in base 3, because

1 times m-reduction [i] would yield (105, 155, 4597)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 90 063937 070931 449970 746010 882583 089777 170656 977479 891176 108649 774781 564939 > 3¹⁵⁵ [i]