MinT - Best Known (74, 137, s)-Nets in Base 3

Best Known (74, 137, s)-Nets in Base 3

(74, 137, 65)-Net over F₃ — Constructive and digital

Digital (74, 137, 65)-net over F₃, using

1 times m-reduction [i] based on digital (74, 138, 65)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 47, 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, 91, 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]

(74, 137, 92)-Net over F₃ — Digital

Digital (74, 137, 92)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(74, 137, 739)-Net in Base 3 — Upper bound on s

There is no (74, 137, 740)-net in base 3, because

1 times m-reduction [i] would yield (74, 136, 740)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 79109 202942 990554 296907 020439 728080 682698 578833 158674 755387 278897 > 3¹³⁶ [i]