MinT - Best Known (89, 89+45, s)-Nets in Base 3

Best Known (89, 89+45, s)-Nets in Base 3

(89, 89+45, 156)-Net over F₃ — Constructive and digital

Digital (89, 134, 156)-net over F₃, using

trace code for nets [i] based on digital (22, 67, 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]

(89, 89+45, 223)-Net over F₃ — Digital

Digital (89, 134, 223)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+45, 3447)-Net in Base 3 — Upper bound on s

There is no (89, 134, 3448)-net in base 3, because

1 times m-reduction [i] would yield (89, 133, 3448)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2868 216741 373116 299687 779684 918639 941114 479311 943030 700307 303377 > 3¹³³ [i]