Combinatorial properties of good codes for binary autoregressive sources (Corresp.)

Let
M
be a binary autoregressive source to be encoded within a specified Hamming distortion
\delta
. A binary
n
-tuple is called
\sigma
-central if it is at distance
\leq n(\delta + \sigma)
from at least
2^{nH(\delta - \sigma)}
typical sequences produced by the source
M
. It is first shown that, in the region where the Shannon rate-distortion bound is achieved, there exist "good codes" consisting only of
\sigma
-central words. Next, the characterization problem is studied; the basic conjecture is that a central sequence is well-characterized by its level, which is the Hamming weight of an image sequence. The problem is solved for the memoryless source. In general, if
N(k,r)
is defined to be the mean number of typical
n
-tuples at distance
\leq r = n \delta
from the
n
-tuples of level
k=n \xi
, then it is shown that
n^{-l} \log N(k,r)
becomes arbitrarily close to
H(\delta)
for an explicitly determined unique value of
\xi
.