MinT - Best Known (111, 111+58, s)-Nets in Base 3

Best Known (111, 111+58, s)-Nets in Base 3

(111, 111+58, 156)-Net over F₃ — Constructive and digital

Digital (111, 169, 156)-net over F₃, using

9 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]

(111, 111+58, 252)-Net over F₃ — Digital

Digital (111, 169, 252)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 111+58, 3491)-Net in Base 3 — Upper bound on s

There is no (111, 169, 3492)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 431 036180 161151 815900 013117 771935 403826 532992 620872 935968 783161 778363 816961 655977 > 3¹⁶⁹ [i]