MinT - Best Known (126, 126+72, s)-Nets in Base 3

Best Known (126, 126+72, s)-Nets in Base 3

(126, 126+72, 156)-Net over F₃ — Constructive and digital

Digital (126, 198, 156)-net over F₃, using

10 times m-reduction [i] based on digital (126, 208, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 104, 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]

(126, 126+72, 243)-Net over F₃ — Digital

Digital (126, 198, 243)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+72, 2969)-Net in Base 3 — Upper bound on s

There is no (126, 198, 2970)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29555 742154 448299 809216 863869 800674 879146 287046 541053 927508 522558 017762 361877 701552 840265 991913 > 3¹⁹⁸ [i]