MinT - Best Known (137, 137+72, s)-Nets in Base 3

Best Known (137, 137+72, s)-Nets in Base 3

(137, 137+72, 162)-Net over F₃ — Constructive and digital

Digital (137, 209, 162)-net over F₃, using

1 times m-reduction [i] based on digital (137, 210, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 105, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(137, 137+72, 299)-Net over F₃ — Digital

Digital (137, 209, 299)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+72, 4168)-Net in Base 3 — Upper bound on s

There is no (137, 209, 4169)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5245 635915 983371 581956 115793 858538 668056 391105 329086 658425 331232 667197 095185 831314 780336 313029 094545 > 3²⁰⁹ [i]