The best possible tradeoff between the coding gain and trellis complex
ity for lattices is studied, Three trellis complexity functions are de
fined for lattices as a measure of minimum trellis decoding complexity
per dimension required for achieving a coding gain gamma. The propert
ies of these functions are studied from an analytic perspective, It is
also shown that the trellis decoding complexity per dimension is lowe
r-bounded by an explicit power of gamma.