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Collaborative Decoding of Interleaved Reed–Solomon Codes And 1 Collaborative Decoding of Interleaved Reed–Solomon Codes and Concatenated Code Designs Georg Schmidt, Student Member, IEEE, Vladimir R. Sidorenko, Member, IEEE, and Martin Bossert Senior Member, IEEE Department of Telecommunications and Applied Information Theory, University of Ulm, Germany georg.schmidt,vladimir.sidorenko,martin.bossert @uni-ulm.de { } Abstract— Interleaved Reed–Solomon codes are applied in Yung [4] propose an algorithm based on the Welch–Berlekamp numerous data processing, data transmission, and data storage approach. Parvaresh, and Vardy [8] consider IRS decoding systems. They are generated by interleaving several codewords in the context of multivariate polynomial interpolation, which of ordinary Reed–Solomon codes. Usually, these codewords are decoded independently by classical algebraic decoding methods. yields a quite powerful list decoding algorithm. However, by collaborative algebraic decoding approaches, such In this paper, we consider IRS codes in a rather general interleaved schemes allow the correction of error patterns beyond way. More precisely we consider IRS codes consisting of l half the minimum distance, provided that the errors in the Reed–Solomon codes (1), (2),..., (l) of length N and received signal occur in bursts. In this work, collaborative dimensions K(1),K(2),...,KA A(l). WeA take a codeword from decoding of interleaved Reed–Solomon codes by multi-sequence each code (ℓ), ℓ =1,...,l, and arrange them row-wise into shift-register synthesis is considered and analyzed. Based on A the framework of interleaved Reed–Solomon codes, concatenated a matrix like depicted in Fig. 1. We call the set of matrices code designs are investigated, which are obtained by interleaving obtainable in this way an IRS code. If (1) = (2) = = several Reed–Solomon codes, and concatenating them with an (l), we call the code homogeneous IRSA code,A if the codes··· inner block code. areA different, we call the resulting IRS code heterogeneous. Index Terms— Interleaved Reed–Solomon codes, homogeneous Most previous publications consider only homogeneous IRS IRS codes, heterogeneous IRS codes, concatenated codes, collab- codes. Heterogeneous constructions have first been considered orative decoding, shift-register synthesis, multiple sequences in [3] (without calling them heterogeneous IRS codes), and some of their properties have been investigated in [12]. Het- I. INTRODUCTION erogeneous IRS codes may be interesting, e.g. when used as outer codes of generalized concatenated codes introduced and In recent years, code designs with interleaved Reed– described by Blokh and Zyablov [13] and by Zinoviev [14]. Solomon (IRS) codes were the topic of several scientific Another interesting application of heterogeneous IRS codes is publications. They are investigated by different authors like the decoding of a single Reed–Solomon code beyond half the Krachkovsky, Lee, and Garg [1], [2], [3], Bleichenbacher, minimum distance like described in [15], where the problem of Kiayias, and Yung [4], Brown, Minder and Shokrollahi [5], [6], decoding a single low-rate Reed–Solomon code is transformed Justesen, Thommesen, and Høholdt [7], as well as Parvaresh into the problem of decoding a heterogeneous IRS code. and Vardy [8]. Moreover, IRS codes are considered in [9], arXiv:cs/0610074v2 [cs.IT] 12 Oct 2006 We propose a method for collaborative decoding of both [10], [11], and other publications. Interleaved Reed–Solomon homogeneous and heterogeneous IRS codes, which is based codes are mainly considered in applications where error bursts on multi-sequence shift-register synthesis. To analyze the occur, since IRS codes are most effective if correlated errors behavior of this decoder, we derive bounds on the decoding affect all words of the interleaved scheme simultaneously. error and decoding failure probability. These bounds allow for In [3], [7], and [10], it is suggested to consider IRS codes estimating of the gain, which can be achieved by collaborative also as outer codes in concatenated code designs, since even decoding in comparison to decoding the l Reed–Solomon for channel models inducing statistically independent random codes independently up to half the minimum distance by a errors, the decoder for the inner code will usually create standard Bounded Minimum Distance (BMD) decoder. correlated burst errors at the input of the decoder for the outer If one column of the arrangement shown by Fig. 1 is cor- codes. For decoding IRS codes, Bleichenbacher, Kiayias, and rupted by a burst error, i.e., an error which affects a complete The work of Vladimir R. Sidorenko and Georg Schmidt is supported column of the arrangement, the l Reed–Solomon codewords by Deutsche Forschungsgemeinschaft (DFG), Germany, under project BO may have an erroneous symbol at the same position. Hence, a 867/14, and BO 867/15 respectively. V. R. Sidorenko is on leave from IITP, RAS RU. Some of the results presented in this paper have also collaborative decoding strategy can be applied, which locates been presented on the 2005 IEEE ITSOC Information Theory Workshop in the errors jointly in all Reed–Solomon codewords instead of Rotorua, New Zealand, and the 10th International Workshop on Algebraic and locating them independently in the several words. This allows Combinatorial Coding Theory (ACCT-10) 2006 in Zvenigorod, Russia. This work has been submitted to the IEEE for possible publication. Copyright may for uniquely locating up to t errors, in many cases even if t is be transferred without notice, after which this version will be superseded. larger than half the minimum distance of the Reed–Solomon PREPRINT -- PREPRINT -- PREPRINT -- PREPRINT -- 2 N Definition 1 (Discrete Fourier Transform (DFT) over Fq) n 1 information symbols redundancy symbols Let p(x) = p0 + p1x + + pn 1x − be a polynomial of PSfrag replacements ··· − K(1) degree deg (p(x)) n 1 with coefficients from Fq. Further, ≤ − (2) let α Fq be some element of order n. Then, the polynomial K ∈ K(3) n 1 l P (x)= F (p(x)) = P0 + P1x + + Pn 1x − ··· − i K(l) whose coefficients are calculated by Pi = p(α ) is called the Discrete Fourier Transform of p(x) over the field Fq. burst error The inverse of the Discrete Fourier Transform pursuant to Fig. 1. Interleaved Reed–Solomon code. Definition 1 can be calculated in the well-known way as follows: Let P (x) be a polynomial of deg (P (x)) n 1, and denote by ≤ − code with the largest dimension. 1 n 1 Algebraic decoding of a single Reed–Solomon codeword p(x)= F − (P (x)) = p0 + p1x + + pn 1x − ··· − can efficiently be performed by the Berlekamp–Massey al- the Inverse Discrete Fourier Transform. Then, the coefficients gorithm [16], [17], which is based on a single-sequence of are calculated by 1 i . shift-register synthesis approach. It is mentioned in [2] and pi p(x) pi = n− P (α− ) In the terminology of the Fourier Transform, we formally [3], that decoding of interleaved Reed–Solomon codes can be performed on the basis of a multi-sequence shift-register call p(x) time domain polynomial and P (x) the frequency synthesis algorithm. Such an algorithm is described by Feng domain polynomial or the spectrum of p(x). We now describe Reed–Solomon codes based on the Discrete Fourier Trans- and Tzeng in [18] and [19]. For homogeneous IRS codes, this algorithm provides an effective method for collaborative form. The following definition does not describe the class of decoding. However, it is shown in [20] and [21] that the Reed–Solomon codes in its most general way. However, with Feng–Tzeng algorithm does not always work correctly for regard to a concise notation, it is adequate for our purposes: sequences of varying length. Thus, it cannot be applied for Definition 2 (Reed–Solomon (RS) code) the heterogeneous case, since the different dimensions of the Let codes (1), (2),..., (l) result in syndrome sequences of A A A K 1 different lengths. Hence, we use the multi-sequence shift- − i A(x) = Aix , Ai Fq register synthesis algorithm proposed in [20] to decode both, { } ∈ ( i=0 ) homogeneous and heterogeneous IRS codes. The complexity X of this shift-register based approach is similar to the complex- be the set of all polynomials of deg (A(x)) < K with coeffi- F F ity of applying the Berlekamp–Massey algorithm to all words cients Ai from q, and let α q be some element of order N < q. Then, a Reed–Solomon∈ code = (q; N,K,D) of the interleaved Reed–Solomon code independently. A RS In the following sections, we describe the basic principles of length N, dimension K, and minimum Hamming distance D = N K +1 can be defined as the set of polynomials behind collaborative IRS decoding. Based on this, we derive − the maximum correction radius. Furthermore, we discuss how 1 , a(x)= F − (A(x)) A(x) A(x) . the code properties and the decoding performance is influenced A | ∈{ } by the IRS code design, i.e., by the choice of the parameters The codewords of are represented by the polynomials a(x)= (1) (2) (l) A n 1 of , ,..., . Moreover, we derive bounds on the a0 + a1x + + an 1x − , or alternatively by the n-tuples A A A ··· − failure and error probability, which allow us to estimate the a = (a0,a1,...,an 1). performance gain, which is achievable by collaborative decod- − ing. On the basis of these results, we investigate concatenated Reed–Solomon codes fulfill the Singleton Bound with equality, code designs with outer IRS codes and inner block codes, and are therefore Maximum Distance Separable (MDS). This and derive bounds on the overall decoding performance. These means that from any set of K correct symbols, a Reed– considerations are supplemented by Monte Carlo simulations, Solomon codeword can uniquely be reconstructed. to assess the quality of our bounds. An interleaved Reed–Solomon code is now obtained by taking l Reed–Solomon codes according to Definition 2 and grouping them row-wise into a matrix: II.
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