MinT - Best Known (137, 137+69, s)-Nets in Base 3

Best Known (137, 137+69, s)-Nets in Base 3

(137, 137+69, 162)-Net over F₃ — Constructive and digital

Digital (137, 206, 162)-net over F₃, using

4 times m-reduction [i] based on digital (137, 210, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 105, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(137, 137+69, 322)-Net over F₃ — Digital

Digital (137, 206, 322)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+69, 5062)-Net in Base 3 — Upper bound on s

There is no (137, 206, 5063)-net in base 3, because

1 times m-reduction [i] would yield (137, 205, 5063)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 64 839350 880648 353952 649307 970280 880850 598812 270292 015921 130761 433133 133544 449903 221593 163891 412941 > 3²⁰⁵ [i]