MinT - Best Known (169, 169+80, s)-Nets in Base 3

Best Known (169, 169+80, s)-Nets in Base 3

(169, 169+80, 176)-Net over F₃ — Constructive and digital

Digital (169, 249, 176)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 55, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (114, 194, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 97, 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+80, 432)-Net over F₃ — Digital

Digital (169, 249, 432)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(169, 169+80, 7319)-Net in Base 3 — Upper bound on s

There is no (169, 249, 7320)-net in base 3, because

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]