MinT - Best Known (109, 109+54, s)-Nets in Base 3

Best Known (109, 109+54, s)-Nets in Base 3

(109, 109+54, 156)-Net over F₃ — Constructive and digital

Digital (109, 163, 156)-net over F₃, using

11 times m-reduction [i] based on digital (109, 174, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 87, 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]

(109, 109+54, 274)-Net over F₃ — Digital

Digital (109, 163, 274)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+54, 4120)-Net in Base 3 — Upper bound on s

There is no (109, 163, 4121)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 590011 304803 792612 192355 352254 411314 185112 880063 410926 777558 319454 178766 313179 > 3¹⁶³ [i]