MinT - Best Known (82, 82+77, s)-Nets in Base 3

Best Known (82, 82+77, s)-Nets in Base 3

(82, 82+77, 65)-Net over F₃ — Constructive and digital

Digital (82, 159, 65)-net over F₃, using

3 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+77, 90)-Net over F₃ — Digital

Digital (82, 159, 90)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+77, 687)-Net in Base 3 — Upper bound on s

There is no (82, 159, 688)-net in base 3, because

1 times m-reduction [i] would yield (82, 158, 688)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2552 211589 968328 023902 804587 156112 655989 299656 798547 197535 072141 664066 474529 > 3¹⁵⁸ [i]