MinT - Best Known (137, 137+77, s)-Nets in Base 3

Best Known (137, 137+77, s)-Nets in Base 3

(137, 137+77, 156)-Net over F₃ — Constructive and digital

Digital (137, 214, 156)-net over F₃, using

16 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+77, 268)-Net over F₃ — Digital

Digital (137, 214, 268)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+77, 3512)-Net in Base 3 — Upper bound on s

There is no (137, 214, 3513)-net in base 3, because

1 times m-reduction [i] would yield (137, 213, 3513)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 427421 746776 998052 869382 389088 103319 145342 513909 770249 004442 429258 749594 349690 243686 607595 284185 512841 > 3²¹³ [i]