MinT - Best Known (139, 139+71, s)-Nets in Base 3

Best Known (139, 139+71, s)-Nets in Base 3

(139, 139+71, 162)-Net over F₃ — Constructive and digital

Digital (139, 210, 162)-net over F₃, using

4 times m-reduction [i] based on digital (139, 214, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 107, 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]

(139, 139+71, 318)-Net over F₃ — Digital

Digital (139, 210, 318)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+71, 4878)-Net in Base 3 — Upper bound on s

There is no (139, 210, 4879)-net in base 3, because

1 times m-reduction [i] would yield (139, 209, 4879)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5250 358767 103815 392869 739826 869697 137415 550411 075954 653275 070140 943115 483094 122351 991148 453649 332987 > 3²⁰⁹ [i]