MinT - Best Known (139, 139+109, s)-Nets in Base 3

Best Known (139, 139+109, s)-Nets in Base 3

(139, 139+109, 128)-Net over F₃ — Constructive and digital

Digital (139, 248, 128)-net over F₃, using

t-expansion [i] based on digital (138, 248, 128)-net over F₃, using
- 2 times m-reduction [i] based on digital (138, 250, 128)-net over F₃, using
  - trace code for nets [i] based on digital (13, 125, 64)-net over F₉, using
    - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
        a function field by GarcÃa and Quoos [i]

(139, 139+109, 173)-Net over F₃ — Digital

Digital (139, 248, 173)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+109, 1542)-Net in Base 3 — Upper bound on s

There is no (139, 248, 1543)-net in base 3, because

1 times m-reduction [i] would yield (139, 247, 1543)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7122 477993 380378 487197 545661 149429 336052 966269 665793 959203 760359 951377 971283 663512 460054 737202 511866 397962 107709 650789 > 3²⁴⁷ [i]