MinT - Best Known (194−63, 194, s)-Nets in Base 3

Best Known (194−63, 194, s)-Nets in Base 3

(194−63, 194, 162)-Net over F₃ — Constructive and digital

Digital (131, 194, 162)-net over F₃, using

4 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]

(194−63, 194, 337)-Net over F₃ — Digital

Digital (131, 194, 337)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(194−63, 194, 5770)-Net in Base 3 — Upper bound on s

There is no (131, 194, 5771)-net in base 3, because

1 times m-reduction [i] would yield (131, 193, 5771)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 122 040013 978304 173865 894690 756227 888117 615221 848832 311357 763446 747920 759386 959106 737180 522739 > 3¹⁹³ [i]