ISSN 0032-9460, Problems of Information Transmission, 2015, Vol. 51, No. 3, pp. 217–230.
Pleiades Publishing, Inc., 2015.
Original Russian Text
A. Zeh, M. Ulmschneider, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 3, pp. 15–30.
Decoding of Repeated-Root Cyclic Codes
up to New Bounds on Their Minimum Distance
and M. Ulmschneider
Computer Science Department, Technion, Haifa, Israel
Institute of Communications and Navigation, German Aerospace Center (DLR), Germany
Received November 22, 2013; in ﬁnal form, March 24, 2015
Abstract—The well-known approach of Bose, Ray-Chaudhuri, and Hocquenghem and its gen-
eralization by Hartmann and Tzeng are lower bounds on the minimum Hamming distance of
simple-root cyclic codes. We generalize these two bounds to the case of repeated-root cyclic
codes and present a syndrome-based burst error decoding algorithm with guaranteed decoding
radius based on an associated folded cyclic code. Furthermore, we present a third technique for
bounding the minimum Hamming distance based on the embedding of a given repeated-root
cyclic code into a repeated-root cyclic product code. A second quadratic-time probabilistic
burst error decoding procedure based on the third bound is outlined.
The length of a conventional linear cyclic block code C over a ﬁnite ﬁeld F
has to be coprime to
the ﬁeld characteristic p. This guarantees that the generator polynomial of C has roots of multiplic-
ity at most one, and therefore we refer to these codes as simple-root cyclic codes. The approaches
of Bose, Ray-Chaudhuri, and Hocquenghem (BCH) [1, 2] and of Hartmann and Tzeng (HT) [3, 4]
give lower bounds on the minimum Hamming distance of simple-root cyclic codes. Both approaches
are based on consecutive sequences of roots of the generator polynomial. We give—similarly to the
BCH and HT bound—two lower bounds on the minimum Hamming distance of a repeated-root
cyclic code, i.e., a cyclic code whose length is not relatively coprime to the characteristic p of the
and therefore its generator polynomial can have roots with multiplicities greater than one.
Repeated-root cyclic codes were ﬁrst investigated in . A special class of maximum distance
separable (MDS) repeated-root constacyclic codes was treated in [6, 7], and the advantages of
a syndrome-based decoding were outlined. An alternative derivation of the minimum Hamming
distance of these repeated-single-root MDS codes and their application to secret-key cryptosystems
was given in . In [9–11] an elaborated description of repeated-root cyclic codes including an
explicit construction of a parity-check matrix was given, which was investigated for the case q =2
slightly earlier in . Although asymptotic badness of repeated-root cyclic codes was shown
in [9–11], several good binary repeated-root cyclic codes were constructed in  with distances
close to the Griesmer bound. In  some of results of [9–11] were re-proved by cyclic group
algebra, and a squaring construction of all binary repeated-root cyclic codes was given in .
Recent publications  and [17, 18] consider repeated-root quasi-cyclic codes.
Supported by the German Research Council (DFG) under grants Bo867/22-1 and Ze1016/1-1; the work was
initiated when both authors were aﬃliated with the Institute of Communications Engineering, University
of Ulm, Ulm, Germany.