MinT - Best Known (137, 137+79, s)-Nets in Base 3

Best Known (137, 137+79, s)-Nets in Base 3

(137, 137+79, 156)-Net over F₃ — Constructive and digital

Digital (137, 216, 156)-net over F₃, using

14 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+79, 258)-Net over F₃ — Digital

Digital (137, 216, 258)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+79, 3247)-Net in Base 3 — Upper bound on s

There is no (137, 216, 3248)-net in base 3, because

1 times m-reduction [i] would yield (137, 215, 3248)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 821485 044849 001982 649670 073527 787084 848338 225702 672024 955128 943901 371815 494933 837687 971959 958830 582849 > 3²¹⁵ [i]