MinT - Best Known (126, 126+81, s)-Nets in Base 3

Best Known (126, 126+81, s)-Nets in Base 3

(126, 126+81, 156)-Net over F₃ — Constructive and digital

Digital (126, 207, 156)-net over F₃, using

1 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+81, 205)-Net over F₃ — Digital

Digital (126, 207, 205)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+81, 2219)-Net in Base 3 — Upper bound on s

There is no (126, 207, 2220)-net in base 3, because

1 times m-reduction [i] would yield (126, 206, 2220)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 194 022836 244526 955075 038804 155941 797055 439165 202930 542196 454249 193115 203207 507544 247424 468758 170305 > 3²⁰⁶ [i]