MinT - Best Known (137, 137+89, s)-Nets in Base 3

Best Known (137, 137+89, s)-Nets in Base 3

(137, 137+89, 156)-Net over F₃ — Constructive and digital

Digital (137, 226, 156)-net over F₃, using

4 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+89, 217)-Net over F₃ — Digital

Digital (137, 226, 217)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+89, 2332)-Net in Base 3 — Upper bound on s

There is no (137, 226, 2333)-net in base 3, because

1 times m-reduction [i] would yield (137, 225, 2333)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 228738 020965 713702 162133 618679 886193 193152 233952 829516 572085 001253 612686 381425 429484 697499 625154 813534 118001 > 3²²⁵ [i]