ISSN 0032-9460, Problems of Information Transmission, 2014, Vol. 50, No. 4, pp. 303–312.
Pleiades Publishing, Inc., 2014.
Original Russian Text
P.V. Trifonov, 2014, published in Problemy Peredachi Informatsii, 2014, Vol. 50, No. 4, pp. 3–14.
Successive Cancellation Decoding
of Reed–Solomon Codes
P. V. Trifonov
St. Petersburg State Polytechnical University, St. Petersburg, Russia
Received December 30, 2013; in ﬁnal form, August 12, 2014
Abstract—A novel soft-decision decoding algorithm for Reed–Solomon codes over GF (2
proposed, which is based on representing them as polar codes with dynamic frozen symbols and
applying the successive cancellation method. A further performance improvement is obtained
by exploiting multiple permutations of codewords which are taken from the automorphism
group of Reed–Muller codes. It is also shown that the proposed algorithm can be simpliﬁed in
the case of decoding a binary image of the Reed–Solomon code.
Despite of several years of research, the problem of soft-decision decoding of Reed–Solomon
codes still has no satisfactory solution. The decoding error probability of existing methods [1–4]
signiﬁcantly exceeds that of the maximum likelihood decoding. Reducing it requires a considerable
increase in the decoding complexity.
Polar codes introduced in  asymptotically achieve the capacity of a wide class of communica-
tion channels, but at moderate lengths provide inferior performance as compared to other known
classes of codes. The successive cancellation algorithm which was suggested for the decoding of
polar codes fails to provide the maximum likelihood performance. However, its list extension in-
troduced in , as well as analogs of sequential decoding presented in [7, 8], enable one to perform
near maximum likelihood decoding with complexity O(Ln log n), where n isthecodelengthandL
is the maximum number of branches at each level of the code tree considered by the decoder.
In this paper we consider application of the successive cancellation method and its analogs to
the decoding problem for Reed–Solomon codes over GF (2
). The proposed approach is based on
representing them as polar codes with dynamic frozen symbols . Furthermore, multiple decoding
attempts are performed for various permutations of the received sequence which are drawn from
the automorphism group of Reed–Muller codes.
The paper is organized as follows. Section 2 provides a survey of polar codes and their decoding
techniques. A new decoding method for Reed–Solomon codes is presented in Section 3. Numeric
results illustrating the performance of the proposed method are provided in Section 4.
2. POLAR CODES AND SUCCESSIVE CANCELLATION DECODING
An (n =2
,k) polar code over GF (2) is generated by k rows of a matrix A = B
where F =
, ⊗m denotes the mth Kronecker power of a matrix, and B
is a bit-reversal
Supported in part by the Russian Foundation for Basic Research, project no. 12-01-00365.