MinT - Best Known (126, 126+61, s)-Nets in Base 3

Best Known (126, 126+61, s)-Nets in Base 3

(126, 126+61, 162)-Net over F₃ — Constructive and digital

Digital (126, 187, 162)-net over F₃, using

1 times m-reduction [i] based on digital (126, 188, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 94, 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]

(126, 126+61, 322)-Net over F₃ — Digital

Digital (126, 187, 322)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+61, 5439)-Net in Base 3 — Upper bound on s

There is no (126, 187, 5440)-net in base 3, because

1 times m-reduction [i] would yield (126, 186, 5440)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55655 830991 126133 025099 608901 927249 408399 539847 582414 616859 682052 596941 122324 325132 184449 > 3¹⁸⁶ [i]