MinT - Best Known (137, 137+91, s)-Nets in Base 3

Best Known (137, 137+91, s)-Nets in Base 3

(137, 137+91, 156)-Net over F₃ — Constructive and digital

Digital (137, 228, 156)-net over F₃, using

2 times m-reduction [i] based on digital (137, 230, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 115, 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]

(137, 137+91, 211)-Net over F₃ — Digital

Digital (137, 228, 211)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+91, 2204)-Net in Base 3 — Upper bound on s

There is no (137, 228, 2205)-net in base 3, because

1 times m-reduction [i] would yield (137, 227, 2205)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 038213 117509 410079 361718 592735 710031 820264 971120 670096 256971 248011 499094 758376 769203 351580 231039 374196 447019 > 3²²⁷ [i]