MinT - Best Known (169, 169+81, s)-Nets in Base 3

Best Known (169, 169+81, s)-Nets in Base 3

(169, 169+81, 172)-Net over F₃ — Constructive and digital

Digital (169, 250, 172)-net over F₃, using

3¹ times duplication [i] based on digital (168, 249, 172)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 53, 24)-net over F₃, using
    - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
  2. digital (115, 196, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 98, 74)-net over F₉, using
      - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(169, 169+81, 422)-Net over F₃ — Digital

Digital (169, 250, 422)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(169, 169+81, 7319)-Net in Base 3 — Upper bound on s

There is no (169, 250, 7320)-net in base 3, because

1 times m-reduction [i] would yield (169, 249, 7320)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 63573 889529 176711 745427 293195 097024 074257 228126 322560 685450 449784 075700 742439 183318 824352 379159 620559 329547 180121 090305 > 3²⁴⁹ [i]