Problems of Information Transmission, Vol. 38, No. 3, 2002, pp. 182–193. Translated from Problemy Peredachi Informatsii, No. 3, 2002, pp. 20–33.
Original Russian Text Copyright
2002 by Griesser, Sidorenko.
A Posteriory Probability Decoding
of Nonsystematically Encoded Block Codes
Received June 13, 2001
Abstract—We consider the problem of symbol-by-symbol a posteriori probability (APP) de-
coding for information symbols of nonsystematically encoded block codes. This problem arises
at soft concatenated decoding of generalized concatenated block codes. The well-known BCJR
algorithm for eﬃcient APP decoding is not able to solve the problem if it runs on the minimal
code trellis of a block code. We introduce an extended trellis representation for block codes,
which includes encoding information and thus makes it possible to apply the BCJR algorithm
as well as trellis-based decoding in the dual code space. Complexity properties of the extended
trellis are investigated.
1. INTRODUCTION AND PROBLEM DESCRIPTION
AblockcodeC is a set of diﬀerent codewords of length n over an alphabet Q. We assume that
the cardinality of C is given by |C| = q
, q |Q|. For linear codes, k is called the dimension of the
code. An encoding scheme of a code C is deﬁned by any one-to-one mapping between the codewords
c ∈C and all possible information words u ∈ Q
. An information symbol u is called systematic if
it is contained in the codeword as a component. If all k information symbols are systematic, then
the encoding scheme is systematic, otherwise it is nonsystematic. Usually, the code is assumed to
be ﬁxed and the encoder is chosen arbitrarily. However, in some cases, an encoding scheme is also
ﬁxed. Thus, for example, properties of a concatenated code depend on the encoding scheme of the
inner code, which is often nonsystematic .
In what follows, we investigate the isolated problem of APP decoding of a component code with
respect to a ﬁxed encoding scheme. Changing this scheme is not allowed.
We consider a memoryless source that delivers information symbols u with known apriori
probability P (u). A block of k information symbols, u, is encoded into a vector c ∈Cof length n.
The discrete memoryless channel model is given by a transition probability function P (y
∈ Q, y
,whereQ is the input and Q
the output alphabet of the channel. (Generalization
to channel models with continuous output alphabets is straightforward.)
Let y be a received sequence. The goal of the APP decoding is calculating the probability of
symbol a to be the τth information symbol:
= a | y)=
P (c | y)=
P (y | c)P (c)=
It is assumed that the decoder knows the code, encoding scheme, received vector y, and probabilities
P (u)andP (y
Supported by the DFG, Germany, project BO 867/6-2. Research done in part during the author’s visit to
Supported in part by the Russian Foundation for Basic Research, project no. 99-01-00840, and DAAD,
2002 MAIK “Nauka/Interperiodica”