MinT - Best Known (135, 135+67, s)-Nets in Base 3

Best Known (135, 135+67, s)-Nets in Base 3

(135, 135+67, 162)-Net over F₃ — Constructive and digital

Digital (135, 202, 162)-net over F₃, using

4 times m-reduction [i] based on digital (135, 206, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 103, 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]

(135, 135+67, 326)-Net over F₃ — Digital

Digital (135, 202, 326)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+67, 5269)-Net in Base 3 — Upper bound on s

There is no (135, 202, 5270)-net in base 3, because

1 times m-reduction [i] would yield (135, 201, 5270)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 799728 060925 675139 363286 983069 479312 505569 774081 104053 849604 095999 928264 861021 191904 965754 900717 > 3²⁰¹ [i]