Kraft inequality for decoding with respect to a fidelity criterion

Let
d_1(\alpha, \beta)
be a distortion measure that is row balanced if the source alphabet is a finite set and that has the form
d(\alpha - \beta)
if the source alphabet is the real line. For
N
-tuples, let the distortion measure
d_N
be the single-letter measure. Consider a variable-length code in which there are
D
code symbols and for which the length of the codeword for
\alpha
is
n(\alpha)
. Codewords
w(u)
for
N
-tuples
u
are formed by concatenating codewords for the individual letters. Let
E(N) = \sup d_N (u,v)
where the supremum is over all pairs
(u,v)
for which
w(u) = w(v)
. Call the code
\varepsilon
-decodable if
\lim E(N) = \varepsilon
. If
0 = d_1(\alpha,\beta) < d_1(\alpha,\beta)
for
\alpha \neq \beta
, and if the code is uniquely decipherable, then
\varepsilon = 0
. For a discrete source it is shown that
\sum D^{-n(i) -h(\varepsilon)} \leq 1
, where
h(0) = 0
and
h(\varepsilon) > 0
if
\varepsioln > 0
. For a continuous source for which values from the interval
[-c,c]
are encoded,
\int_{-c}^c D^{-n(z)-h_1 (varepsilon)} dz \leq 1
, where
h_1 (\varepsilon)
is a known function. These inequalities are used to obtain lower bounds on the mean length of any
\varepsilon
-decodable code. In many cases, these lower bounds coincide with the rate-distortion function
R(\varepsilon)
associated with the same distortion measure.