MinT - Best Known (129, 129+63, s)-Nets in Base 3

Best Known (129, 129+63, s)-Nets in Base 3

(129, 129+63, 162)-Net over F₃ — Constructive and digital

Digital (129, 192, 162)-net over F₃, using

2 times m-reduction [i] based on digital (129, 194, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 97, 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]

(129, 129+63, 323)-Net over F₃ — Digital

Digital (129, 192, 323)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(129, 129+63, 5373)-Net in Base 3 — Upper bound on s

There is no (129, 192, 5374)-net in base 3, because

1 times m-reduction [i] would yield (129, 191, 5374)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13 556595 417508 175532 034504 954374 842106 487609 560964 126300 107547 749829 341842 507721 236128 670409 > 3¹⁹¹ [i]