MinT - Best Known (131, 131+61, s)-Nets in Base 3

Best Known (131, 131+61, s)-Nets in Base 3

(131, 131+61, 162)-Net over F₃ — Constructive and digital

Digital (131, 192, 162)-net over F₃, using

6 times m-reduction [i] based on digital (131, 198, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 99, 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]

(131, 131+61, 358)-Net over F₃ — Digital

Digital (131, 192, 358)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+61, 6538)-Net in Base 3 — Upper bound on s

There is no (131, 192, 6539)-net in base 3, because

1 times m-reduction [i] would yield (131, 191, 6539)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13 524348 971941 468815 271106 705442 678498 389866 868571 470769 008355 869462 195064 323057 516175 701885 > 3¹⁹¹ [i]