MinT - Best Known (245−80, 245, s)-Nets in Base 3

Best Known (245−80, 245, s)-Nets in Base 3

(245−80, 245, 168)-Net over F₃ — Constructive and digital

Digital (165, 245, 168)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 51, 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 (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]

(245−80, 245, 405)-Net over F₃ — Digital

Digital (165, 245, 405)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−80, 245, 6554)-Net in Base 3 — Upper bound on s

There is no (165, 245, 6555)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 788 079891 591946 558752 536833 535816 803578 555667 527413 809267 790719 225214 866761 129682 445509 706331 882157 574124 520294 634289 > 3²⁴⁵ [i]