MinT - Best Known (139, 139+111, s)-Nets in Base 3

Best Known (139, 139+111, s)-Nets in Base 3

(139, 139+111, 128)-Net over F₃ — Constructive and digital

Digital (139, 250, 128)-net over F₃, using

t-expansion [i] based on digital (138, 250, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 125, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(139, 139+111, 170)-Net over F₃ — Digital

Digital (139, 250, 170)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+111, 1488)-Net in Base 3 — Upper bound on s

There is no (139, 250, 1489)-net in base 3, because

1 times m-reduction [i] would yield (139, 249, 1489)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 64363 813807 930898 206851 625441 401846 487682 405112 849111 179104 906114 994646 736625 969967 101051 555483 285635 486166 270505 469707 > 3²⁴⁹ [i]