MinT - Best Known (82, 82+78, s)-Nets in Base 3

Best Known (82, 82+78, s)-Nets in Base 3

(82, 82+78, 65)-Net over F₃ — Constructive and digital

Digital (82, 160, 65)-net over F₃, using

2 times m-reduction [i] based on digital (82, 162, 65)-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 (27, 107, 37)-net over F₃, using
    - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(82, 82+78, 89)-Net over F₃ — Digital

Digital (82, 160, 89)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+78, 660)-Net in Base 3 — Upper bound on s

There is no (82, 160, 661)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22723 855743 645699 102233 952639 409380 395355 880975 853158 070134 029975 843914 706075 > 3¹⁶⁰ [i]