MinT - Best Known (250−82, 250, s)-Nets in Base 3

Best Known (250−82, 250, s)-Nets in Base 3

(250−82, 250, 168)-Net over F₃ — Constructive and digital

Digital (168, 250, 168)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 52, 20)-net over F₃, using
  - net from sequence [i] based on digital (11, 19)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 9, N(F)Â =Â 19, and 1 place with degree 3 [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using an explicitly constructive algebraic function field [i]
2. digital (116, 198, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 99, 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]

(250−82, 250, 406)-Net over F₃ — Digital

Digital (168, 250, 406)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(250−82, 250, 6508)-Net in Base 3 — Upper bound on s

There is no (168, 250, 6509)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 191448 320343 918738 655977 881277 850537 329093 718176 875131 976145 778787 890361 598431 164802 852849 896820 805392 377550 287564 530747 > 3²⁵⁰ [i]