MinT - Best Known (172−61, 172, s)-Nets in Base 3

Best Known (172−61, 172, s)-Nets in Base 3

(172−61, 172, 156)-Net over F₃ — Constructive and digital

Digital (111, 172, 156)-net over F₃, using

6 times m-reduction [i] based on digital (111, 178, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 89, 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]

(172−61, 172, 232)-Net over F₃ — Digital

Digital (111, 172, 232)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−61, 172, 3128)-Net in Base 3 — Upper bound on s

There is no (111, 172, 3129)-net in base 3, because

1 times m-reduction [i] would yield (111, 171, 3129)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3901 159420 554617 901394 711101 198435 226527 029706 283666 436478 890065 110331 746928 209769 > 3¹⁷¹ [i]