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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.INTERLEAVED REED–SOLOMON CODES As already mentioned, the codewords of an IRS code are Definition 3 (Interleaved Reed–Solomon (IRS) code) matrices whose rows are the codewords of the Reed–Solomon Let (ℓ) = q; N,K(ℓ),D(ℓ) , ℓ = 1,...,l, be l Reed– codes (1), (2),..., (l). Before we formally define IRS SolomonA codesRS of length N according to Definition 2. Then, an codes,A we brieflyA introduceA classical Reed–Solomon codes, interleaved Reed–Solomon code is the set of matrices basically to establish the notation we use in the following. a(1) Reed–Solomon codes can conveniently be defined in the a(2) frequency domain, i.e., by the Fourier Transform of their code- A , . , a(ℓ) (ℓ), ℓ =1,...,l . words. For this purpose, we consider the following definition . ∈ A of the Discrete Fourier Transform over finite fields: a(l) PREPRINT -- PREPRINT -- PREPRINT -- PREPRINT -- 3
(1) (2) If all l Reed–Solomon codes are equivalent, i.e., = = of the erroneous symbols. Therefore, Λ(x)=Λ0 +Λ1x + (l) A A t = = , the IRS code is called homogeneous. +Λtx is called error locator polynomial. An error locator ···Otherwise,A we sayA that the IRS code is heterogeneous. ···polynomial was first applied by Peterson [22] for decoding BCH codes. Instead of considering an IRS codeword as matrix l N Since the roots of the error locator polynomial are not A with elements from the extension field F , we× can also q modified by multiplying Λ(x) by a constant factor, and since consider it as a row vector a with elements from the extension Λ0 = 0, Λ(x) can always be normalized w.l.o.g. in such field F l . In this extension field representation, an IRS code q a way6 that Λ = 1. Equation (1) forms a linear system of l k(ℓ) 0 is a code of length N, cardinality ℓ=1 q , and minimum N equations. In this system, t equations only depend on the (ℓ) F distance min1 ℓ l N K +1 over ql . Hence, for the ≤ ≤ − M = N K known syndrome coefficients S0,...,SM 1 and homogeneous case, i.e., if (ℓ) =Q (q; N,K,D), ℓ = − − A RS the unknown coefficients Λ1,..., Λt. With these t equations, 1,...,l, we obtain a code of length N, dimension K, and we write the matrix equation minimum distance D = N K +1 over F l . This means − q that homogeneous IRS codes fulfill the Singleton Bound with S0 S1 ... St 1 Λt St − − equality and are therefore MDS. However, since the minimum S1 S2 ... St Λt 1 St+1 distance of a heterogeneous IRS code is determined by the . . . .− = − . , . . . . . weakest Reed–Solomon code, i.e., the Reed–Solomon code SM t 1 SM t ... SM 2 Λ1 SM 1 with the smallest minimum distance, heterogeneous IRS codes − − − − − − are generally not MDS. S Λ T (2) |which is a linear{z system of N } | K{z t} equations| {z and} t III. COLLABORATIVE DECODING OF INTERLEAVED − − EED OLOMON ODES unknowns. Hence, (2) cannot have a unique solution, if t > R –S C N K N K −2 and we are never able to correct more than −2 Before we explain the concept of collaborative interleaved N K ⌊ ⌋ errors. If t −2 , solving (2) yields a unique error locator Reed–Solomon decoding, we consider the conventional case of polynomial,≤⌊ and hence⌋ also the locations of the erroneous a single Reed–Solomon code = (q; N,K,D). Decoding A RS symbols. Basically, (2) can be solved by standard methods is usually performed by an algebraic Bounded Minimum Dis- from linear algebra. This approach, also known as Peterson tance (BMD) decoder, which corrects all errors with weights Algorithm, is described in [23]. However, today the Peterson up to half the minimum code distance. Algorithm is of minor practical relevance since there exist algorithms which utilize the structure of (2) to solve the system A. BMD Decoding up to Half the Minimum Distance of equations in a more efficient way. Assume that a codeword a(x) is transmitted over To understand how the structure of (2) can be exploited, we ∈ A a noisy channel, which adds an error polynomial e(x) = observe that we are able to state (2) in the form of the linear N 1 e0 + e1x + + eN 1x − over Fq, so that we observe recursion the word ··· − at the output of the channel. If r(x)= a(x)+ e(x) t e(x) 0, the Discrete Fourier Transform R(x)= F (r(x)) = (ℓ) ≡ Si = Λj Si j , i = t,...,N K 1 (3) R + R x + + R xN 1 will be a polynomial of degree − − − − 0 1 N 1 − j=1 smaller than···K due− to Definition 2, i.e., the coefficients X RK ,...,RN 1 will be zero. However, if e(x) / , we obtain of length t. In this way, the problem of calculating Λ(x) is − ∈ A deg (R(x)) K, which means that some of the coefficients transformed to the problem of finding the smallest integer t and ≥ RK ,...,RN 1 are non-zero. These coefficients only depend the connection weights Λ1,..., Λt for recursively generating − F N K 1 on the transformed error E(x) = (e(x)) = E0 + E1x + the syndrome sequence = Si i=0− − . This problem is N 1 S { } + EN 1x − . Hence, we can consider the coefficients equivalent to the problem of determining the shortest possible ··· − RK ,...,RN 1 as syndrome coefficients and denote them by linear feedback shift-register, which is capable of generating − Sj = RK j = EK j , j =0,...,N K 1. the syndrome sequence . This single-sequence shift-register + + − − To correct t errors, the standard approach for algebraic synthesis problem is solvedS very efficiently by the Berlekamp– Reed–Solomon decoding is to define the polynomial λ(x) = Massey algorithm [16], [17]. N 1 λ0 +λ1x+ +λN 1x − , such that it has a zero coefficient When Λ(x) is determined, and hence the erroneous posi- ··· − λj = 0, whenever the corresponding coefficient ej of the tions are known, the most difficult part of decoding is accom- error polynomial e(x) is non-zero. For all error free positions, plished. If the error locations are known, the error values can i.e., for all positions for which ej = 0, the corresponding always be uniquely determined, as long as t N K. Error ≤ − coefficient λj is defined to be non-zero. Consequently, λj ej = evaluation can be performed using several standard techniques · 0 holds for all positions j =0,...,N 1. Due to the properties like Recursive Extension [24] or the Forney algorithm [25]. of the Discrete Fourier Transform, this− relation is transformed into Λ(x) E(x) 0 mod xN 1 . (1) B. Collaborative Error Location · ≡ − The spectrum Λ(x) is a polynomial of degree t, whose roots Now, we consider an interleaved Reed–Solomon code pur- j1 j2 jt α− , α− ,...,α− correspond to the locations j1, j2,...,jt suant to Definition 3. We assume that we observe a received PREPRINT -- PREPRINT -- PREPRINT -- PREPRINT -- 4
word of length t. All l linear recursions use the same connection (1) (1) (1) r a e weights Λ1,..., Λt to combine the syndrome coefficients of r(2) a(2) e(2) the l received words. Hence, the error locator polynomial R = . = . + . , . . . Λ(x) can be calculated by finding the smallest integer t and (l) (l) (l) the connection weights Λ1,..., Λt for recursively generating r a e (ℓ) (ℓ) N K 1 all l different syndrome sequences = Si i=0− − . A E This is equivalent to synthesizing theS shortest linear{ } feedback where the ℓth row of R consists| {z of} the| coefficients{z } of the re- shift-register capable of generating the l syndrome sequences ceived word r(ℓ)(x)= a(ℓ)(x)+e(ℓ)(x). Each error polynomial (1),..., (l). For a homogeneous Reed–Solomon code, i.e., S S e(ℓ)(x) has t(ℓ) t non-zero coefficients located at the indices if all l sequences have the same length, this multi-sequence ≤ i1,i2,...,it(ℓ) j1, j2,...,jt . Hence, R = A + E shift-register problem can be solved by the Feng-Tzeng al- can{ be written} as ⊆ sum { of a codeword} A A and an error gorithm [18], [19]. However, for a heterogeneous IRS code, matrix E with exactly t non-zero columns.∈ For each row in the sequences (1),..., (l) may be of different length. It is S S R, we are able to calculate a syndrome sequence (ℓ) = demonstrated in [20] and [21] that the Feng–Tzeng algorithm (ℓ) N K(ℓ) 1 S (ℓ) does not always yield the shortest shift-register for varying Si i=0− − . However, since we allow the codes to{ have} different dimensions K(ℓ), the calculated syndromeA length sequences. Hence, we apply the varying length shift- sequences may be of different lengths. Nevertheless, as long register synthesis algorithm proposed in [20] to calculate as t
Once a t-valid error locator polynomial Λ(x) is successfully D. Joint Error and Erasure Correction calculated for an IRS code, error evaluation can be performed In [11], an algorithm for joint error and erasure decoding independently for all l Reed–Solomon codewords by the of IRS codes is proposed. Like the algorithm presented in standard techniques also used for classical BMD decoding. [4] for decoding errors only, the algorithm from [11] uses a Using Algorithm 1 for calculating Λ(x) yields the same Welch-Berlekamp approach, which is based on solving a linear computational complexity as locating the errors in the l code- system of equations. We briefly explain how Algorithm 2 can words independently by the Berlekamp–Massey algorithm. be modified to allow joint error and erasure correction based Consequently, we are able to increase the error correction on shift-register synthesis. radius above half the minimum distance of the Reed–Solomon Technically, erasures occur if the receiver is not able to code with the smallest distance, without increasing the decod- detect any meaningful symbol at the output of the channel. ing complexity. This means that the decoder does not know the value, but the Altogether, our collaborative decoding strategy for IRS position of an erased symbol. Formally, we define an erasure codes consists of four steps. First, we calculate l syndromes as follows: Let ξ denote an erased symbol, and define it to be (1),..., (l), next, we synthesize an error locator polynomial F F S S a special symbol ξ / q. Moreover, let the addition of α q Λ(x), then we check whether this polynomial is t-valid, and and ξ be defined by∈α + ξ = ξ + α = ξ. Now, assume that∈ we whether t is smaller than some maximum error correcting observe a received word radius tmax, which will be specified later. If we have a t- valid Λ-polynomial, we calculate values of the errors e(ℓ)(x) R = A + E + E , (ℓ) independently for all received words r (x), and obtain esti- where the rows of the matrix mates ǫ(1) aˆ(ℓ)(x)= r(ℓ)(x) e(ℓ)(x) − ǫ(2) E for ℓ = 1,...,l. If Λ(x) is not t-valid, we get a decoding = . . failure. The complete decoding algorithm is summarized by ǫ(l) Algorithm 2. (ℓ) (ℓ) are the coefficients of the erasure polynomials ǫ (x)= ǫ0 + Algorithm 2: Collaborative IRS Decoder (ℓ) (ℓ) n 1 ǫ1 x + + ǫn 1x − over 0, ξ , ℓ = 1,...,l. Each non- r(1) zero element··· in −E indicates an{ erased} symbol. In contrast to R . input: received word = . errors, the erased symbols are detectable in the received word r(l) ! R, which means that they do not have to be located, only use DFT to calculate syndromes (1),..., (l) their values have to be determined. For this reason, we do not synthesize t, Λ(x) by AlgorithmS 1 S require the erasures to occur in column bursts. if t tmax and Λ(x) is t-valid then To perform joint error and erasure decoding for IRS codes, for≤ each ℓ from 1 to l do we proceed like described in [26] and define an erasure locator evaluate errors, and calculate e(ℓ) polynomial (ℓ) (ℓ) (ℓ) calculate aˆ = r e (ℓ) (ℓ) i(ℓ) i(ℓ) i − Ψ (x) , 1 α 1 x 1 α 2 x 1 α ϑℓ x = else − − ··· − (ℓ) (ℓ) (ℓ) decoding failure =1+Ψ x +Ψ x2 + +Ψ xϑℓ 1 2 ··· ϑℓ aˆ(1) for each of the rows ℓ = 1,...,l in our IRS code. The . A (ℓ) (ℓ) (ℓ) output: A = . or decoding failure i i iϑ . roots α− 1 , α− 2 ,...,α− ℓ indicate the positions of the (l) ! ∈ aˆ erasures in the erasure polynomial ǫ(ℓ)(x), i.e., the coefficients b (ℓ) (ℓ) (ℓ) of ǫ(ℓ)(x) at the positions i ,i ,...,i are equal to ξ, and 1 2 ϑℓ Generally, depending on the errors added by the channel, all other coefficients are zero. Now we consider the received Algorithm 2 may yield three different results: polynomial r(ℓ)(x) whose coefficients are the ℓth row of R. (ℓ) (ℓ) 1) The algorithm may obtain a correct result, i.e., a(ℓ)(x) From r (x) we create the polynomial r¯ (x) by replacing aˆ(ℓ)(x) ℓ =1,...,l. ≡ all erasure elements by an arbitrary field element. Then, we ¯(ℓ) F (ℓ) ˘(ℓ) ¯(ℓ) 2) The algorithm∀ may obtain an erroneous result, i.e., ℓ calculate R (x)= r¯ (x) and R (x)= R (x)Ψ(x). (ℓ) (ℓ) (ℓ) (ℓ) ∃ ∈ It is explained in [26] that R˘(ℓ)(x) = R˘ + R˘ x + + [1,...,l]: a (x) aˆ (x). 0 1 6≡ ˘(ℓ) n 1 (···ℓ) 3) The algorithm may not yield a result at all, i.e., it may Rn 1x − is the spectrum of a modified received word r˘ (x) yield a decoding failure. with− errors at the same positions as r(ℓ)(x). Moreover, the last (ℓ) ˘(ℓ) All three events occur with a certain probability. In the N K ϑℓ coefficients of R (x) are only influenced by the errors,− so− that we are able to obtain the modified syndromes following, we denote the probability for a correct decision by (ℓ) ˘(ℓ) ˘(ℓ) N K 1 ϑℓ ˘(ℓ) ˘(ℓ) = S − − − , where S = R (ℓ) , i = Pc, the probability for an erroneous decision by Pe, and the i i=0 i K +ϑℓ+i S { }(ℓ) probability for a decoding failure by Pf . Hence, the probability 0,...,N K ϑℓ 1, ℓ = 1,...,l. This means that − − − Pw for obtaining a wrong decoding result is calculated by each erasure symbol in the row ℓ reduces the length of the (ℓ) Pw =1 Pc = Pe + Pf . usable syndrome ˘ by one. In this way, we obtain l (varying PREPRINT− -- PREPRINT -- PREPRINTS -- PREPRINT -- 6
length) syndrome sequences, which we feed into Algorithm 1 Proof: Assume that Algorithm 2 is applied to the error to calculate a pair t, Λ(x). Once, a t-valid polynomial Λ(x) matrix e(1) is calculated and t tmax, we proceed like described in (2) [26] to obtain the error≤ and erasure values independently for e E = . . each row, using the error and erasure locator polynomial . (ℓ) (ℓ) Ω (x)=Ψ (x)Λ(x). e(l) In some sense, error and erasure decoding of an IRS code is equivalent to error only decoding of a heterogeneous IRS In its fist step, Algorithm 2 computes the Discrete Fourier code, since in both cases we process varying length syndrome Transform for every row of E: sequences. For this reason, we do not further consider the error (1) (1) (1) (1) F e E0 E1 ... EN 1 and erasure case explicitly, but keep in mind that it is closely F e(2) (2) (2) (2)− E0 E1 ... EN 1 related to the case of heterogeneous IRS codes. F (E)= = − . . . . . . . . (l) (l) (l) (l) F e E E ... E IV. ERROR CORRECTING RADIUS 0 1 N 1 − The last N K(ℓ ) elements of the ℓth row coincide with the If the number of errors t is smaller than half the mini- elements of− the syndrome sequence (ℓ). Hence, we formally mum distance of the Reed–Solomon code with the smallest define the syndrome matrix S redundancy, we are always able to correct all errors by solving (1) (1) (1) (3) separately for all l Reed–Solomon words using standard ξ ξ EK(1) EK(1) +1 ... EN 1 techniques like the Berlekamp–Massey algorithm. Certainly, it (2) (2) (2) (2)− ξ EK(2) EK(2) +1 EK(2) +2 ... EN 1 is important that Algorithm 2 is also capable of decoding all S(E)= − . . . errors in this case. To prove that Algorithm 2 actually has this . . . property, we first consider the following lemma: (l) (l) ξξ ξ EK(l) ... EN 1 − (1) (1) (1) ξ ξ S S ... S Fn 0 1 N K(1) 1 Lemma 1 Let v, and w be two vectors in q , such that the (2) (2) (2) (2)− − ξ S0 S1 S2 ... SN K(2) 1 Hamming weights of v and w satisfy wt (v) < q 1 or = − − − F . . . wt (w) < q 1. Then, there always exists an element β q, . . . − ∈ β =0, such that the support of v + βw satisfies ξξ ξS(l) ... S(l) 6 0 N 1 − supp (v + βw) = supp (v) supp (w) . (6) by replacing the first K(ℓ) elements of the ℓth row by erasure ∪ symbols, i.e., by symbols ξ / Fq, which fulfill α+ξ = ξ+α = ∈ ξ, and α ξ = ξ α = ξ for an arbitrary element α Fq. In other words,· S·(E) is an l N matrix containing∈ the l Proof: Since wt (v) < q 1, or wt (w) < q 1 the cardi- syndrome sequences aligned to× the right, and prepended by nality of the intersection =− supp (v) supp (w−) of the sup- D ∩ erasure symbols. From the linearity of the Discrete Fourier ports of v = (v0, v1,...,vN 1) and w = (w0, w1,...,wN 1) Transform we know that is at most = min wt (v−) , wt (w) < q 1. To obtain− (6), |D| { } − F F F vi + βwi =0 has to be satisfied for all i . In other words, E = (LE)= L (E) , 6 ∈ D vi β = i . (7) and consequently alsoe S(E)= LS(E). 6 −wi ∀ ∈ D Since (5) is a linear recursion, we know (see e.g. [27]) that Equation (7) can always be satisfied choosing a suitable β, if the rows of S(E) satisfye (5) with some pair t, Λ(x), then since < q 1, and therefore there exists at least one the rows of LS(E) also satisfy (5) with the same pair t, 1 β =0,|D| which fulfills− (7). Λ(x). Since L is a non-singular matrix, E = L− E, and we 6 Now, we consider a received word R = A + E, where A conclude that S(E) satisfies (5) with the pair t, Λ(x), if and is a codeword of an IRS code and E is an error matrix. Since only if S(E) satisfies (5) with the same pair t, Λ(ex). Thus, the syndromes (1),..., (l) calculated by Algorithm 2 only if Algorithm 2 has a unique solution t, Λ(x) for S(E), then depend on E andS not onSA, we can assume w.l.o.g. the case Algorithme 2 has the same unique solution for S(E), and vice A = 0 to analyze Algorithm 2. In other words, we are able versa. to apply Algorithm 2 directly to the error matrix E instead of e Theorem 1 (Guaranteed Correcting Radius) Consider an the received word R. interleaved Reed–Solomon code pursuant to Definition 3. Assume that this code is corrupted by an error matrix E with Lemma 2 Assume that E is an l N error matrix and L is × t non-zero columns. Then, Algorithm 2 always yields a unique a non-singular l l matrix over Fq. Moreover, let E = LE × and correct solution, i.e., all t column errors can be corrected, be another error matrix, whose rows are linear combinations as long as t satisfies of the rows of E. Then, Algorithm 2 applied to Eeyields a unique pair t, Λ(x), if and only if Algorithm 2 applied to E N K t tg = − , (8) yields the same unique pair t, Λ(x). ≤ $ 2 % PREPRINT -- PREPRINT --e PREPRINT --b PREPRINT -- 7
where dimension. This gives rise to the following theorem on the K = max K(ℓ) maximum error correcting radius: 1 ℓ l ≤ ≤ n o is the maximum dimensionb among the l Reed–Solomon codes Theorem 2 (Maximum Correcting Radius) Consider an (1), (2),..., (ℓ). interleaved Reed–Solomon code pursuant to Definition 3. A A A Assume that a word of this code is corrupted by t column Proof: Assume that a codeword of the interleaved Reed– errors. Then, Algorithm 2 may only find a unique and correct Solomon code is corrupted by an error matrix E with t non- solution, if t satisfies zero columns. We distinguish two different cases: (1) (1) l 1) If wt e = supp e = t, then the Berlekamp– t tmax = min N K¯ ,N K , (10) (1) ≤ l +1 − − Massey algorithm applied to the sequence yields a pair S t, Λ(x) which is the shortest length solution of (3), provided where b l that t satisfies (8). Since we consider column errors, the pair 1 K¯ = K(ℓ) t, Λ(x) is also a solution of (3) for the remaining sequences l (2),..., (l). In other words, t, Λ(x) is the shortest length ℓ=1 S S X solution of (5), which is also obtained by applying Algorithm1 is the average dimension of the l Reed–Solomon codes. to the l sequences (1), (2),..., (l). Consequently, Algo- rithm 2 yields the correctS S solution.S Proof: To find a unique and correct solution, the linear system of equations (4) with t unknowns and the set of linear e(1) e(1) 2) Now, we consider the case wt = supp < recursions (5) of length t must have a unique solution, which t, i.e., the case that the column errors only affect t(1) < t means that rank (Sl) = t. This is only possible, if (9) is symbols in e(1). Since satisfied, since otherwise the matrix Sl has less than t rows. N K Hence, in order to be able to locate the errors, t must not be t − < q 1 larger than the first argument in (10). Even if the locations of ≤ 2 − $ % the errors are known, the error values can only be calculated b is always satisfied due to Definition 2, we are able to recur- if t is not larger than the number of redundancy symbols in sively apply Lemma 1 to obtain coefficients , every Reed–Solomon code. Thus, if t>N K, it is not l β1,β2,...,βl − β1 =1, such that the linear combination possible to evaluate the errors in at least one Reed–Solomon codeword. Consequently, t must also not be largerb than the l second argument in (10). e˜(1) = β e(ℓ) ℓ The Theorems 1 and 2 characterize the collaborative de- ℓ X=1 coding approach described by the linear system of equations is an error vector with t non-zero coefficients, i.e., we have (4) or equivalently by the set of linear recursions (5). If the (1) supp e˜ = t. Hence, we define the non-singular matrix number of column errors satisfy (8), i.e., if the number of errors is smaller than half the minimum distance of the Reed– 1 β ... β 2 l Solomon code with the largest dimension, (4) and hence also 0 1 ... 0 (5) have a unique solution, and we are always able to correctly L = . . . . , . . .. . reconstruct the transmitted codeword A. If the number of 0 0 ... 1 errors lies between the guaranteed error correcting radius specified by (8) and the maximum correcting radius specified and use it to calculate the error matrix E = LE, which has t by (10), (4) and (5) may still have a unique solution, which non-zero error symbols in its first row. Thus, we know from enables us to correct errors beyond half the minimum distance case 1) that Algorithm 1 synthesizes thee correct pair t, Λ(x) of the component codes. In this sense, our collaborative for E, and that Algorithm 2 consequently yields the correct decoding approach yields a Bounded Distance (BD) decoder, solution for the matrix E. Due to Lemma 2, we know that whose maximum error correction radius lies beyond half the Algorithme 2 also yields the correct solution for the error matrix minimum distance. However, as soon as the number of column E, which proves Theoreme 1 for the case 2). errors exceeds the limit specified by (8), we are not able any more to guarantee that (4) and hence also (5) has a unique Beyond the guaranteed error correcting radius we may be solution. In fact, the solutions of (4) and (5) will be ambiguous able to locate the errors, as long as the number of unknowns with some probability Pf > 0, since rank (Sl) < t for some in (4) is not larger than the number of equations, i.e., as long error patterns. as The possibility to obtain an ambiguous solution is inherent l l 1 to Bounded Distance decoders which correct errors within a t N K(ℓ) (9) radius beyond half the minimum distance, since some correct- ≤ l +1 − l ℓ ! X=1 ing spheres are inevitably overlapping in this case. Basically, is satisfied. Moreover, we are only able to evaluate the errors, three strategies are conceivable to cope with this problem: if the number of errors does not exceed the number of 1) The decoder selects on single codeword out of all valid redundancy symbols in the component code with the largest decoding results PREPRINT -- PREPRINT -- PREPRINT -- PREPRINT -- 8
2) The decoder returns a list of all valid decoding results over all component codes. Though, if l is small enough, the that is, a list of all codewords which lie inside a sphere redundancy may be distributed unequally, as long as we only with radius tmax around the received word y. care about the maximum error correcting radius, and not about 3) If the decoding result is not unique, i.e., if there exists the minimum distance of the IRS code. However, the minimum more than one valid decoding result, the decoder yields distance is only optimal in the homogeneous case. This is a decoding failure explained by the fact that homogeneous IRS codes are MDS, Since finding all valid solutions is algebraically difficult with while the minimum distance of heterogeneous IRS codes our approach and usually requires a high computational effort, is determined by the Reed–Solomon code with the largest the decoders we consider here apply the third strategy: when- dimension. Nevertheless, heterogeneous IRS codes can be ever (4) does not have a unique solution, Algorithm 2 may interesting e.g. in the context of generalized concatenated code yield a decoding failure (we formalize this statement later in constructions introduced and described by Blokh and Zyablov Lemma 5). [13] and Zinoviev [14], for decoding single Reed–Solomon codes beyond half the minimum distance like described in V. DISTRIBUTIONOFTHE REDUNDANCY [15], and for several other applications. Definition 3 does not restrict the choice of the dimensions of VI. COLLABORATIVE DECODING PERFORMANCE the l Reed–Solomon codes used to construct a heterogeneous IRS code. However, to ensure a good performance under To evaluate the performance of our decoding strategy, we collaborative decoding, we should apply some restrictions on would like to estimate the error probability Pe, and the failure the dimensions K(1),...,K(l). probability Pf , which together result in the probability Pw From (10) we observe that the maximum error correction for obtaining a wrong decoding result. We are particularly radius depends on the average redundancy N K¯ and the interested in the failure probability Pf (t) in the range tg is a solution of the system of equations (4) with t unknowns, Proof: For proving Lemma 4, we assume w.l.o.g. that and of the set of linear recursions (5) of length t. the spheres s1 and s2 are centered around the origin. Then, we consider a vector v on s1. Since s1 is centered around Lemma 3 is helpful for proving the following theorem: the origin, wt (v) = r1, and hence v has exactly r1 non- zero components. Now, we consider another vector w on the Theorem 4 (ML Certificate of Algorithm 2) The decoder surface of s2, i.e., wt (w)= r2, and assume that v and w are specified by Algorithm 2 exhibits the ML certificate property. overlapping in exactly i non-zero coordinates. The Hamming distance d (v, w) is calculated by Proof: If Algorithm 2 does not fail while decoding R, it yields a pair , , where is a -valid polynomial d (v, w) = (r1 i) + (r2 i)+ δ(i) , t Λ(x) Λ(x) t − − satisfying the system of equations (4) with t unknowns, and where δ(i) is number of differing symbols in the overlapping the set of linear recursions (5) of length t. From Λ(x), the coordinates. Then, we count how many vectors w exist, for decoder computes an error word E with t non-zero columns, which d (v, w)= ρ, or equivalently and a codeword A = R E which differs in t columns from − a r δ(i)= ρ (r1 + r2)+2i . (12) R. Hence, the extension field representations , and of the − matrices A and R fulfill d (a, r)= t. Hence, we have Now, we assume that there exists a codeword A′ whose i ρ (r +r )+2i extension field representation a has a smaller Hamming (q 2) − 1 2 ′ ρ (r + r )+2i − distance to r than a, i.e., − 1 2 possibilities to choose non-zero symbols in the overlapping d (a′, r)= t′ < t . coordinates which are different in the two vectors, and N r1 r i Applying Lemma 3 to R and A′ yields that there exists a − (q 1) 2− r i − t′-valid solution Λ′(x) of the system of equations (4) with t′ 2 − unknowns and the set of linear recursions (5) of length t′, non-zero symbols in the non-overlapping coordinates of w, where t′ < t. However, since Algorithm 1 always finds a in order to obtain a vector with Hamming distance ρ to v. solution of (5) with the smallest t, this yields a contradiction Consequently, to calculate U(q, r2, r1,ρ), we have to sum up to the assumption that dist (a′, r) < dist (a, r). Hence, if the product Algorithm 2 finds a solution, this solution corresponds to the i N r codeword with the smallest Hamming distance to R. − 1 ρ (r + r )+2i r i · 1 2 2 − − ρ (r +r )+2i r i (q 2) − 1 2 (q 1) 2− B. Error Probability · − − for all possible numbers of overlapping coordinates. To obtain Now, we derive an upper bound on P for the general e the summation limits, we observe that δ(i) is limited by 0 case that a linear block code is decoded by a BD decoder δ(i) i. By inserting (12), we obtain ≤ which exhibits the ML certificate property. For this purpose, ≤ we assume that the transmitted word is corrupted by t errors, (r + r ρ)/2 i r + r ρ . 1 2 − ≤ ≤ 1 2 − and that the BD decoder is able to correct errors up to the r +r ρ Hence we have to sum up from 1 2− to r + r ρ. radius . We consider a pair of codewords, and count 2 1 2 tmax Consequently, U(q, r , r ,ρ) is calculated⌈ by⌉ the sum − the number of received vectors which may be decoded into 2 1 wrong codewords. Then, we use a union bounding technique r1+r2 ρ − r1 i N r1 to overbound the error probability Pe(t). The bound obtained − r r −ρ i ρ (r1 + r2)+2i r2 i · in this way is directly applicable to IRS codes by interpreting i= 1+ 2 − − ⌈ X2 ⌉ the IRS codewords as vectors over the extension field F l . ρ (r1+r2)+2i r2 i q (q 2) − (q 1) − . · − − This proves Lemma 4. Lemma 4 Consider two concentric spheres s1 and s2 in the FN Theorem 4 and Lemma 4 enable us to prove the following Hamming space q . Assume that s1 has radius r1, and s2 theorem, which overbounds the error probability Pe(t): has radius r2. Fix an arbitrary point v on the surface of s1, and let U(q, r , r ,ρ) be the number of points on the surface 2 1 Theorem 5 (Error Probability) Let (Q; N,K,D) be a lin- of s with distance ρ to v. Then, U(q, r , r ,ρ) is calculated 2 2 1 ear block code of length N, dimensionC K, and minimum by distance D over the field FQ, decoded by a BD decoder which exhibits the ML certificate property. Assume that the decoding U(q, r2, r1,ρ)= radius of this decoder is tmax, and that it decodes a received r1+r2 ρ − r1 i N r1 word, which is corrupted by t errors. Then, the probability for − a decoding error is overbounded by − i ρ (r1 + r2)+2i r2 i · i= r1+r2 ρ − − ⌈ X2 ⌉ t+tmax min t,tmax A { } U(Q,t,w,ρ) ρ (r1+r2)+2i r2 i w=D w ρ=0 (q 2) − (q 1) − . Pe(t) P e(t)= · , · − − ≤ N t P Pt (Q 1) − (13) PREPRINT -- PREPRINT -- PREPRINT -- PREPRINT -- 10 where Aw describes the weight distribution of the code, i.e., To calculate Pe(t), we divide Nt by the number Sn(t) of Aw is the number of codewords of weight w. points on the sphere s with radius t, which is calculated by Proof: Consider two codewords c and c′ with N t Sn(t)= (Q 1) . Hamming distance dist (c, c )= w to each∈C other. Assume∈C that t − ′ the codeword c is corrupted by t errors, and that the word y Since Nt is overbounded by (14), we obtain the statement of is observed at the output of the channel. For this arrangement, Theorem 5. we overbound the pairwise error probability P (c c′), i.e, → Note that if we set tmax = tg in Theorem 5, (13) coincides the probability that c′ is decoded under the condition that c with the error probability for classical BMD decoding as has been transmitted. To do this, we consider a sphere s of described in [24]. The expression is actually exact in this case, radius t centered at c. We count all points on the surface of since the spheres around the surrounding codewords do not s, which lie inside a sphere s′ of radius tmax around c′, since overlap in the case tmax = tg. they may be decoded into c′ (see Fig. 2). However, we only Theorem 5 basically holds for arbitrary linear block codes count the points on s′, which have a maximum distance of t over the field Fq, which are decoded up to the maximum to c′, since otherwise they are closer to c than to c′, and due error correcting radius tmax by a decoder exhibiting the ML to the ML certificate property we know that they are correctly certificate property. To apply this theorem to IRS codes, we decoded into the codeword c. F interpret them as codes over the extension field ql , and set In other words, we count all points on s, whose distance ρ Q = ql. Due to Theorem 4, Algorithm 2 exhibits the ML to c′ is in the range 0 ρ min t,t . For this purpose, ≤ ≤ { max} certificate property, and hence (13) provides us with an upper we consider the two concentric spheres s1 and s2 around c. bound on the error probability Pe(t). However, to calculate We select the sphere s1 to have radius w, and the sphere s2 (13), we require knowledge about the weight distribution of the to have radius t. This means that c′ is a point on s1, and code. A homogeneous IRS code can be interpreted as an MDS s2 coincides with the sphere s. Hence we are able to apply F code over the extension field ql . Since the weight distribution Lemma 4 to count the number of points Np on the surface of of MDS codes is well–known (see e.g. [28]), the weights Aw s2 with distance ρ to c′ on s1, where 0 ρ min t,tmax . can be calculated for this case by This is illustrated by Fig. 2. ≤ ≤ { } w D N l − i w 1 l(w D i) s Aw = (q 1) ( 1) − q − − , ′ w − − i = i=0 s s2 X where D = N K +1. Unfortunately, heterogeneous IRS − codes are usually not MDS. Hence Aw can generally not be w obtained in such a simple way. However, even if the weight tmax distribution is not known for a heterogeneous IRS code, we are able to overbound Pe(t) on the basis of (13), by replacing cw c′ Aw by Aw, where Aw is such that Aw Aw w =0,...,N. t 2 ≥ ∀ PSfrag replacements t C. Failureb Probabilityb b Collaborative decoding of IRS codes provides us with a method of decoding errors beyond half the minimum distance, if the errors affect the columns of the interleaved scheme like depicted in Fig. 1. If the number of errors is smaller than the s1 guaranteed correcting radius (8), we know from Theorem 1 that we are always able to correct these errors. Contrariwise, Fig. 2. Number of points which may contribute to P (c → c′) if the number of errors exceeds the maximum correcting radius (10), the decoder will always fail or take an erroneous decision. In this way, we obtain In order to analyze the probability Pf (t) in the range tg < min t,tmax t t , we assume that for each column of the interleaved { } ≤ max Np = U(Q,t,w,ρ) . Reed–Solomon code, each error pattern occurs equiprobable. ρ=0 More precisely, we assume, that the burst errors X To overbound the total number Nt of points which are decoded (1) ej into wrong codewords, we use a union bounding technique and . e = . consider all codewords within distance D w t + tmax to j . c, since the sphere s does not overlap with≤ the≤ correcting (l) ej spheres of all other codewords. As there are Aw codewords Fl 0 c are random vectors, uniformly distributed over q . Under with distance w to , we obtain \{ } t+tmax this assumption, bounds on Pf have been derived in [4], and [5] for homogeneous IRS codes. However, since these bounds Nt AwNp . (14) ≤ w D are rather weak or do not depend on the error weight t, they PREPRINT -- PREPRINTX= -- PREPRINT -- PREPRINT -- 11 generally do not yield a good estimation of the decoding Further, let performance. w(1) To obtain a bound for heterogeneous IRS codes, we gener- . W = . = (w0,..., wω 1) alize a bound from [10] for homogeneous IRS codes, which . − allows for estimating the failure probability P (t) in depen- w(l) f dence of t. For proving this generalized bound, we use similar be a l ω matrix, such that w = 0, j = 0,...,ω 1, but simpler techniques, which have been applied in [2] to j i.e., that×W does not have any all-zero6 column.∀ Furthermore,− bound the failure probability for folded Reed–Solomon codes. assume that all columns of W are uniformly distributed over These techniques are based upon analyzing the linear system all non-zero vectors of length l. Then, the probability P that of equations (4), and overbounding the number of cases, in ω which the system of equations (4) with t unknowns yields w(ℓ) (ℓ), ℓ =1,...,l (15) multiple solutions. To obtain an upper bound on the failure ∈ K ∀ is overbounded by probability of Algorithm 2 from this analysis, we have to lω show, that whenever Algorithm 2 yields a decoding failure, q l ̺(ℓ) Pω q− ℓ=1 . (16) then there exist multiple solutions for (4). ≤ (ql 1)ω · − P Lemma 5 Consider a codeword A A of an IRS code Proof: Let be the set of all l ω matrices whose rows pursuant to Definition 3. Assume that∈ this word is corrupted fulfill Equation (15).L Further, let ×be the set of all l ω by an error matrix E with t non-zero columns, and that ω matrices with elements from F , andS let the subset v × Algorithm 2 yields a decoding failure. Then, the linear system q ω ω be the set of matrices without any non-zero column. Then,S ⊂ theS of equations (4) with t unknowns has multiple solutions. probability Pω that a matrix W without any all-zero column fulfills (15) can be calculated by Proof: Assume that a codeword A A has been ∈ v transmitted and Y = A + E is received, where E has t non- ω Pω = |L ∩v S | |L|v . zero columns. Algorithm 2 only yields a decoding failure, if ω ≤ ω Algorithm 1 computes a pair t , Λ (x), such that Λ (x) is not |S | |S | ′ ′ ′ The cardinality is obtained by t′-valid. |L| l First we show that t′ t. Applying Lemma 3 to Y and A ℓ lω l ̺(ℓ) ≤ ( ) ℓ=1 yields that there exists a t-valid polynomial Λ(x), which is a = = q − , |L| K P solution of (5). Since Algorithm 1 always yields the smallest ℓ=1 Y length shift-register which is a solution of (5), we know that and the cardinality v is calculated by |Sω| t′ t. v l ω ≤ ω = q 1 . Second, Λ′(x) is a solution of the set of linear recursions |S | − (5) of length t′, and hence also for the system of equations (4) Consequently, Pω is overbounded by with t′ unknowns. However, if Λ′(x) is a solution for the set of lω q l ̺(ℓ) linear recursions (5) of length t′, it is also a solution for length ℓ=1 Pω l ω q− . t t′, since if Λ′(x) fulfills (5) for i t′, it obviously also ≤ (q 1) · P ≥ ≥ − fulfills it for i t t′. Hence, Λ′(x) is not only a solution ≥ ≥ of the system of equations (4) with t′ unknowns and the set Lemma 6 enables us, to state and prove the following of linear recursions (5) of length t′, but also for the system theorem: (4) with t unknowns, and the set of recursions (5) of length t. Consequently, the system of equations (4) with t unknowns Theorem 6 (Failure Probability) Consider an interleaved has at least the two solutions Λ(x) and Λ′(x). This proves the Reed–Solomon code pursuant to Definition 3, which is decoded statement of Lemma 5. by Algorithm 2. Assume that the codes (1),..., (l) are Note that we only show that if Algorithm 2 yields a decoding chosen such that (11) is satisfied. FurthermoreA assumeA that failure, then (4) has multiple solutions. Contrariwise this does A is corrupted by t column errors, where each column vector not necessarily mean that Algorithm 2 always fails if (4) has is an independent random vector uniformly distributed over multiple solutions. However, since we are only interested in Fl 0 . Then, the probability for a decoding failure is q \{ } overbounding the failure probability Pf , Lemma 5 is sufficient overbounded by for our purposes, since the number of events for which (4) has t l 1 (l+1)(t t) multiple solutions is an upper bound for the number of events q q q− max− Pf (t) P f (t)= − , (17) in which Algorithm 2 fails. ≤ ql 1 · q 1 − ! − Hence, to overbound the probability P that Algorithm 2 f l ¯ fails, we consider the following lemma: where tmax = l+1 (N K) is the maximum error correcting radius. − Lemma 6 Let (1), (2),..., (l) be l q-ary linear codes of Proof: According to Lemma 5, the failure probability length ω, and letK theK dimensionK of the code (ℓ) be ω ̺(ℓ). of Algorithm 2 can be overbounded by considering the cases, PREPRINT -- PREPRINTK − -- PREPRINT -- PREPRINT -- 12 in which the system of equations (4) with t unknowns has matrix of a (shortened) Reed–Solomon code of length t and (ℓ) (ℓ) multiple solutions. We have such a case whenever rank (Sl) < dimension t ̺ , which we denote by . We observe t, i.e., whenever there exists a column vector u = 0, such that that the matrices− V and D both have full rank.K Therefore, the 6 Sl u = 0. Equivalently we can say that (4) cannot have a product v = D V u defines a one-to-one mapping u v, unique· solution, if such that 0 0·. Consequently,· the statement 7→ 7→ u = 0 : S(ℓ) u = 0 ℓ =1, . . . , l . (18) v = 0 : H(ℓ) F (ℓ) v = 0 ℓ =1,...,l (19) ∃ 6 · · ∀ ∃ 6 · ∀ T (1) (2) (l) Since the syndrome matrices S , S ,..., S directly de- is equivalent to Equation (18). With w(ℓ) = F (ℓ) v , and pend on the error matrix E, we are able to express the failure · the fact that H(ℓ) is a parity-check matrix of the code (ℓ), we probability P (t) in a general way by f can state another equivalent condition for a decodingK failure: number of matrices E satisfying (18) t (ℓ) (ℓ) Pf (t)= , v = 0 : w ℓ =1, . . . , l . total number of matrices Et ∃ 6 ∈ K ∀ where Et denotes an error matrix with exactly t non-zero Assume that we have a vector v with Hamming weight columns. Now, we consider matrices with non-zero columns wt(v) = ω. Then, the vectors w(ℓ), ℓ = 1,...,l, have at at fixed indices j1, j2,...,jt. More precisely, for a fixed most ω non-zero components. Now consider the matrix set j , j ,...,jt of t indices, we consider the ensemble (1) { 1 2 } w t (j1,...,jt) of matrices, in which every column with in- . E W = . . dex j j1, j2,...,jt is an independent random vector . uniformly∈ { distributed over} Fl 0 , and all other columns w(l) q \{ } are zero vectors. Then, the probability that (18) is satisfied for T Since we know that all vectors e (1) (l) , matrices Et from the ensemble t (j1,...,jt) is calculated by ji = eji ,...,eji i = E 1,...,t, are non-zero, and that all non-zero error patterns are E t (j1,...,jt): E satisfies (18) Pf (j1,...,jt)= |{ ∈E }| . distributed uniformly, we also know that W contains exactly t (j ,...,jt) |E 1 | ω non-zero columns uniformly distributed over all non-zero Fl We will now derive an upper bound on Pf (j1,...,jt), which vectors in q. Assume that the non-zero columns in W are does not depend on the selection of the indices j1, j2,...,jt, located at the indices i1,i2,...,iω, let W ω be a l ω matrix × (ℓ) but only on the number of erroneous columns t. Hence, this consisting of the non-zero columns of W , and let Hω be (ℓ) bound will directly provide us with the upper bound on Pf (t), obtained from H by removing all columns whose indices (ℓ) in which we are actually interested in. are not in the set i1,i2,...,iω . Furthermore, denote by ω { }(ℓ) K(ℓ) For calculating Pf (j1,...,jt), let the number of rows in the code defined by the ω ̺ parity-check matrix Hω . (ℓ) (ℓ) (ℓ) (ℓ) (ℓ) × T 0 (ℓ) T 0 S be denoted by ̺ , i.e., ̺ = N K t. It is known Then, the statements H W = , and Hω W ω = −(ℓ) − · · (ℓ) (cf. e.g. [24]) that a syndrome matrix S can be decomposed are equivalent. Consequently, we can apply Lemma 6 on ω , K into and W ω to overbound the probability Pω that a fixed vector S(ℓ) = H(ℓ) F (ℓ) D V . v of weight ω satisfies · · · At this, the matrix, H(ℓ) F (ℓ) v =0 ℓ =1, . . . , l . (20) · · ∀ j 2j (t 1)j 1 α 1 α 1 ... α − 1 We observe that the probability P (v) for a vector v to fulfill j 2j (t 1)j 1 α 2 α 2 ... α − 2 (20) is independent of the positions of the non-zero symbols V = . . . in v, but only depends on the weight wt (v)= ω, i.e., . . . 1 αjt α2jt ... α(t 1)jt P (v : wt(v)= ω)= P . − ω is a t t Vandermonde matrix, the matrices Hence, the probability P (j ,...,j ) that (18) or equivalently × f 1 t (19) is satisfied can be overbounded using a union bounding D = diag αj1 , αj2 ,...,αjt technique, by summing up over all non-zero vectors v: and t (ℓ) (ℓ) (ℓ) (ℓ) F = diag e ,e ,...,e Pf (j ,...,jt) P (v)= Pω . (21) j1 j2 jt 1 ≤ v Fl 0 ω=1 v:wt(v)=ω are t t diagonal matrices, and the matrix ∈Xq\{ } X { X } × Since the right side of (21) is independent of the indices 1 1 ... 1 j1, j2,...,jt but only depends on t, we see that (21) is also αj1 αj2 ... αjt an upper bound on the failure probability Pf (t). 2j1 2j2 2jt H(ℓ) = α α ... α To improve (21), we should take care about the following . . . . . . fact: if a vector v fulfills (20), a vector v′ = αv also (ℓ) (ℓ) (ℓ) F (̺ 1)j1 (̺ 1)j2 (̺ 1)jt fulfills (20) for all α q 0 . Therefore, we call v and α − α − ... α − ∈ \{ } αv equivalent vectors. Since there are q 1 different non- (ℓ) (ℓ) − is a ̺ t matrix consisting of ̺ rows of a transposed zero elements in Fq, there exist q 1 equivalent vectors × (ℓ) − Vandermonde matrix. Hence, H represents a parity-check for each non-zero vector over Fq. Thus, the number Mω of PREPRINT -- PREPRINT -- PREPRINT -- PREPRINT -- 13 0 non-equivalent vectors of length t with a certain weight ω is 10 calculated by 10-10 ω t (q 1) t ω 1 -20 Mω = − = (q 1) − . (22) 10 ω q 1 ω − − 10-30 Hence, to obtain a better upper bound on Pf (t), we can e -40 multiply the probabilities Pω bounded by (16) by the number 10 , P f of non-equivalent words of weight ω calculated by (22), and P 10-50 sum up over all weights 1 ω t. In this way we obtain ≤ ≤ -60 t 10 Pf (t) Pω Mω -70 ≤ · ≤ PSfrag replacements 10 ω=1 Upper bound (17) on Pf (t) X l (ℓ) -80 ̺ t l ω 10 Upper bound (13) on Pe(t) q− ℓ=1 t (q 1)q − = 17 18 19 20 21 22 23 24 ≤ Pq 1 ω ql 1 ω=0 t − X − l+1 t l ̺(ℓ) q 1 q− ℓ=1 = l − P = Fig. 3. Upper bounds on the failure and error probability for an IRS code q 1 · q 1 RS 8 − − composed of three codewords from the code 2 ; 255, 223, 33 . l 1 t ¯ q q l(N K)+(l+1)t = − q − − = ql 1 · q 1 − ! − for obtaining a decoding error or a decoding failure. For errors l 1 t q q (l+1)(tmax t) of weight t tg, we know from Theorem 1 that we never − q − − ≤ = l . fail or get an erroneous decoding result, and consequently q 1 ! · q 1 − − Pw(t)= 0. For a received word R with t >tg column This proves Theorem 6. errors, the probability Pw(t) is simply the sum of the error For the case of homogeneous IRS codes, the bound (17) probability Pe(t) and the failure probability Pf (t). Hence, was mentioned the first time in [9] without proof. A similar Pw(t) is overbounded by the sum P e(t)+ P f (t), or more but somewhat weaker bound is presented in [6] for decoding precisely by homogeneous interleaved Algebraic Geometry (AG) codes. Applied to homogeneous IRS codes, this bound basically 0 t tg 1 Pw(t) P w(t) , ≤ . (23) differs from (17) by the factor q 1 . ≤ min P e(t)+ P f (t), 1 t>tg − ( A D. Comparison of Pf , and Pe Now, we consider a codeword A with elements from F , which is corrupted by an error matrix∈ E. We assume that The decoding performance of an IRS scheme is mainly q a non-zero column in E occurs with probability p, and that influenced by the probabilities P (t) and P (t) in the range f e the non-zero column vectors are distributed uniformly over t t t . To get an impression of these probabilities, g max Fl 0 . With this error model, the probability for column we≤ consider≤ a homogeneous IRS code composed of three q t errors\{ is} codewords of the code = 28;255, 223, 33 . This A RS N t N t Reed–Solomon code is used in several practical applications, p (1 p) − , t · − and guarantees to decode all error patterns with a weight up to 16. The resulting IRS code A can be considered as MDS code and we can overbound the overall word error probability by of length 255, dimension 223, and minimum distance 33 over F N the field (28)3 . According to Theorem 2, our collaborative N t N t Pw P w(t) p (1 p) − . (24) decoding strategy is able to correct up to 24 errors. Hence, ≤ t · · · − t=tg+1 we calculate the upper bound on Pf (t) pursuant to (17), and X the upper bound on Pe(t) pursuant to (13) in the interval 17 t 24 to figure out, in how many cases the decoder VII. IRS CODES IN CONCATENATED CODE DESIGNS ≤ ≤ fails or yields an erroneous result in the worst case. The results Reed–Solomon codes are commonly used as outer codes are depicted by Figure 3. We observe that both probabilities in concatenated code designs. In many applications, we even decrease exponentially with decreasing t, and that the the error find schemes with interleaved Reed–Solomon codes of some probability Pe(t) is smaller than the failure probability Pf (t) kind. However, usually the Reed–Solomon codewords of such by several orders of magnitude in the complete range. Hence, schemes are decoded independently, i.e., the special inter- the decoder performance is usually dominated by the failure leaved structure is typically not taken into account by the probability Pf , and not by the error probability Pe. decoder. To investigate the applicability of collaborative decoding E. Overall Word Error Probability strategies for IRS codes in concatenated code designs, we The probability Pw = Pe + Pf is the probability for Algo- consider simple concatenated schemes with outer IRS codes rithm 2 to yield a wrong decoding result, i.e., the probability and inner block codes. PREPRINT -- PREPRINT -- PREPRINT -- PREPRINT -- 14 A. Concatenated Code Construction for all columns of Y . Then, we use the inverse mapping 1 T T l g− (b ): b a F to obtain the matrix To construct a concatenated code, we consider a codeword ∈ B 7→ ∈ q A F m A of an IRS code over the field q, where q = p . We r(1) group∈ the elements of the codeword matrix T T 1 ˆ 1 ˆ . R = g− (b0 ),...,g− (bN 1) = . . (27) a(1) − r(l) . A = . = (a0,..., aN 1) , ℓ − Each row r( ), ℓ =1,...,l, of R corresponds to an input word (l) a of the Reed–Solomon code (ℓ). Hence, basically any row r(ℓ) A into the column vectors can be decoded independently from all others with respect to the corresponding Reed–Solomon code (ℓ). However, a(1) j since an erroneous decision of the inner MLA decoder may . aj = . , j =0,...,N 1 . affect a complete column of the matrix R, it occurs to be . − (l) more expedient to apply the collaborative decoding strategy a j described in Section III to decode all rows of the IRS code Then, we select a linear code of length n, dimension k = lm simultaneously. In this case, the word error probability p B F w and minimum distance d over the field p and define a one-to- at the output of the inner decoder will be the column burst one mapping g(a): a Fl b . We apply this mapping ∈ q 7→ ∈B error probability at the input of the collaborative decoder for to the column vectors aj , j =0,...,N 1, of the matrix A. − the outer IRS code. In other words, the probability pw is the In this way, we obtain a matrix probability that a column of R is erroneous. T T Unfortunately, the exact analytical calculation of pw is C = g(a0),...,g(aN 1) = b0 ,..., bN 1 , (25) − − generally not feasible. However, there are several known in which all columns are codewords of the inner code . We techniques to bound pw from both sides. One of the easiest and define the set C of all matrices obtainable in this wayB to be most widely known techniques to overbound the ML decoding the concatenated{ } code . For practical applications, usually error probability is the Union Bound. it is preferable to haveC codes with a binary representation. Therefore, we consider binary inner codes, i.e., p = 2, and C. Error Probability after Inner ML Decoding m outer IRS codes over an extension field of F2, i.e., q =2 . We Unfortunately, the Union Bound is not very tight for chan- assume that the concatenated code is used for transmission nels with a bad to moderate signal-to-noise ratio, so that it is C over an Additive White Gaussian Noise (AWGN) channel with not suitable for analyzing our concatenated scheme. Therefore, Binary Phase Shift Keying (BPSK) modulation. In this case, a we consider Poltyrev’s Tangential Sphere Bound (TSB) [29]. codeword Using this bound, pw can be overbounded by T T t 0 n 1 C = b0 ,..., bN 1 = ci,j , ⌈ t0+1 − ⌉ − , 2 2 pw pTSB Awγ(w, σ ,t0)+ ζ(σ ,t0) , (28) ci,j F2, is mapped to the signal matrix ≤ w=1 ∈ X T T where X = x0 ,..., xN 1 = xi,j − c 2 2 n by x = ( 1) i,j , and transmitted over the AWGN channel. γ(w, σ ,t0) , exp i,j π −2σ2 · At the output,− we observe the matrix √t0 ∞ T T x 2 n Y = y0 ,..., yN 1 = yi,j , exp x cosh 2x − · 1+ y2 − 2σ2(1 + y2) · Z0 Zw r with yi,j = xi,j + ni,j . Since we use energy 1 to transmit √ n−w 2 a binary code symbol, the variables ni,j are statistically n 2 t y 1 Γ − , 0 − x2 dydx , independent Gaussian random variables with zero mean and · − 2 1+ y2 2 variance σ = N0/2, where N0 is the single sided noise power spectral density. Moreover, for a concatenated code ∞ 2 Kk C 2 1 n of rate R = Nn , the effective energy for transmitting a single ζ(σ ,t ) , exp x + 0 √π − 2σ2 · information bit is specified by the code rate R. Hence, we are Z r ! −∞ able to characterize the AWGN channel by its signal-to-noise 2 Eb 1 Γ (n 1)/2,t0x dx , ratio, or more precisely by the ratio = 2 . N0 2Rσ · − and 1 B. Decoding of the Concatenated Code Γ(a,z) , Γ(a,z) Γ(a) To decode Y with respect to the inner code, we use a is the normalized incomplete Gamma function. The parameter Maximum Likelihood (ML) decoder for the code to find t is calculated from the cone half-angle the ML estimates B 0 T w ˆ T T θ = arccos bj = argmax P (b yj ) (26) w b | t0(n w) ∈B PREPRINT -- PREPRINT -- PREPRINTr -- − PREPRINT -- 15 by solving the equation To obtain a lower bound on pw, we take the maximum of the Sphere Packing bound (29), and the L2 bound (30), i.e. n 1 θw − t0 n 3 , Aw 1 u w n sin − (ϑ)dϑ = pw p max p ,p . (31) − − t +1 ≥ w SPB L2 w=1 0 X Z0 Since the TSB is quite tight forn all channelo conditions, it Γ ((n 2)/2) = − , directly serves us as upper bound on pw, i.e., √πΓ ((n 1)/2) − pw p , p . (32) where ≤ w TSB 0 x< 0 The bounds (31) and (32) turn out to be beneficial to derive u(x)= (1 x 0 bounds on the decoding performance after outer decoding. For ≥ decoding the outer code, we basically have two options. Either is the Heaviside step function. we apply Algorithm 2 for collaboratively decoding the outer To bound pw from below for channels with bad and moder- IRS code, or we use a classical BMD decoder to decode the ate signal-to-noise ratios, Shannon’s Sphere Packing Bound l outer Reed–Solomon words independently row by row. [30] is well suited for our purposes. The Sphere Packing Bound is calculated by D. Word Error Probability with Collaborative Outer Decoding , 1 We first analyze the collaborative decoding strategy, which pw p − SPB n 1 ≥ 2 2 Γ((n 1)/2)· applies Algorithm 2 for decoding the outer code. To overbound − c ∞ 2 n the word error probability Pw in this case, it would be helpful n 2 x σ2 cot(θ)x x − exp − erfc − dx , (29) to apply the Theorems 5 and 6 derived in Section III. However, · 2 √2 ! Theorem 6 requires that all column errors occur equiproba- Z0 p ble in the IRS scheme. Unfortunately, this is not true after where the half angle θ is obtained by solving the equation decoding the inner code, since due to the characteristics of θ ML decoding, low-weight error patterns occur more frequently n 2 √πΓ((n 1)/2) sin − (ϑ)dϑ = − , than high-weight patterns. To be able to apply Theorem 6 MΓ(n/2) anyway, we slightly modify the our concatenated coding Z0 scheme to randomize the error patterns after inner decoding. with M = . For this purpose, let l be the set of all non-singular l l |B| M × For good channels, i.e., high signal-to-noise ratios, the matrices with elements from the field Fq. Now we modify the Sphere Packing bound is known to be rather weak. Therefore, encoding rule given by Equation (25) into we use S´eguin’s L2–Bound [31] to bound pw from below for C′ = (g(M 0 a0),...,g(M N 1 aN 1)) good channels. The L2 bound is calculated by · − · − T T (33) n 2 w = β0 ,..., βN 1 , 1 Aw erfc 2σ2 − pw p , . (30) ≥ L2 2 Ω where the matrices M are statistically independent random w=1 p i X matrices, uniformly distributed in . The reverse mapping At this, the denominator is calculated by l Ω after inner decoding described byM Equation (27) is modified to , w Ω erfc + T T 2σ2 1 1 ˆ 1 1 ˆ r R = M 0− g− (β0 ),..., M N− 1 g− (βN 1) . (34) w w · − · − + 2(Aw 1)ψ α(w, w), , + − σ2 σ2 This randomization procedure does not influence the un- n r r corrupted columns in R, but only ensures the erroneous w w +2 A ψ α(w, v), , , columns after inner decoding to be transformed into uniformly v σ2 σ2 v=1 r r distributed error patterns. Since the number of erroneous vX=w 6 columns is not changed by randomization, it should have no where negative impact on the decoding performance. However, this randomization procedure allows us to apply Theorem 6 for i j i + j d α(i, j) , min , , − , estimating the failure probability after outer decoding. Later, (sj r i 2√ij ) we will observe by means of experimental results that from a and practical point of view randomization is not necessary, since the decoding results will be virtually the same, regardless ψ (ρ,x,y) , whether we use randomization or not. ∞ ∞ 1 u2 2ρuv + v2 Theorem 7 (Upper Bound on P c ) Consider a concate- exp − dudv w 2π 1 ρ2 − 2(1 ρ2) nated code design with randomization as described above, and Z Z − x y − assume that the outer component codes are chosen such that is the bivariatep Gaussian function. (11) is satisfied. Furthermore, assume that the words of the PREPRINT -- PREPRINT -- PREPRINT -- PREPRINT -- 16 inner code are decoded by a Maximum-Likelihood Decoder E. Word Error Probability with Independent Outer Decoding and the words of the outer interleaved Reed–Solomon code Now, we consider the case that we decode each row of R are decoded jointly by Algorithm 2. Then, the word error independently by a classical BMD decoder. We overbound the probability after decoding is overbounded by i word error probability Pw for independent decoding, i.e., the N probability that BMD decoding fails or is erroneous for at least c N one of the l rows, by the following theorem: P P w(t) p t, p ,p , (35) w ≤ t · · N w w t=tg+1 i X Theorem 8 (Upper Bound on Pw) Consider a concate- where nated code design as described above, and assume that the , outer component codes are chosen such that (11) is satis- P w(t) min P f (t)+ P e(t), 1 , fied. Furthermore, assume that the words of the inner code are decoded by a Maximum-Likelihood Decoder and that and the words of the outer interleaved Reed–Solomon code are t (N t) decoded independently by a BMD decoder. Then, the word p (1 p ) − , i p N w · − w ≤ w error probability after decoding is overbounded by p t, p ,p = t (N t) . N w w pw (1 p ) − , p N where pN t, pw,p is calculated by (36). Proof: For proving Theorem 7, we consider the fact that w the word error probability p after inner decoding coincides Proof: If the number of erroneous columns in R is t w ≤ with the column error probability p at the input of the tg, we always obtain a correct decoding result. To overbound i outer IRS decoder. Hence, according to (24) the word error Pw, we assume that for t >tg the independent decoding c probability Pw after outer decoding is overbounded by strategy based on standard BMD decoders never takes a correct decision. In other words, we simply replace P w(t) N in Theorem 7 by one, which directly yields the statement of c N t N t P P w(t) p (1 pw) − , (37) w ≤ t · · w − Theorem 8. t=t +1 Xg Theorem 8 allows us to estimate the worst case decoding performance for independent decoding of the outer Reed– where Solomon codes. However, to be able to assess the potential of P w(t) , min P f (t)+ P e(t), 1 . our collaborative decoding strategy, we also require a lower bound on P i . For this purpose, we first consider the following w To further overbound (37), we define the function lemma: , t N t f(pw) pw (1 pw) − , Lemma 7 Consider an IRS code according to Definition 3, · − and assume that we have t column errors, uniformly dis- and calculate the derivative Fl 0 tributed over q . Further, assume that the Reed–Solomon codewords are\{ decoded} independently by a BMD decoder. d t 1 N t t N t 1 f(pw)= tpw− (1 pw) − pw(N i)(1 pw) − − . dpw − − − − Then, the probability that a corrupted IRS word is not decoded (38) correctly, i.e., the probability that the BMD decoder fails to From this derivative we observe that f(pw) is monotonically decode at least one Reed–Solomon word, is lower bounded by non-decreasing as long as t , and monotonically non- pw N (ℓ) t ≤ l tg t i increasing, if pw . Consequently, to overbound f(pw), (q 1) ≥ N i i , ℓ=1 i=0 i we replace p by p if t p N, and by p if t p N. In Pe (t) P e (t) 1 t − , (40) w w w w w ≥ − (ql 1) the range ≤, we do not know, whether≥ Q P − pwN t different error patterns with t erroneous columns. The total monotonically non-increasing for pw N . This allows us to l t ≥ number of matrices without any all-zero columns is q 1 . lower bound f(pw) by Hence, the probability for an error pattern with t column− errors f(pw) p t, p ,p = to be decoded is ≥ N w w t (N t) N N p (1 p ) − , t p N P i (t)= c c∗ . w · − w ≤ w c l t ≤ l t = t (N t) . (q 1) (q 1) pw (1 pw) − , pwn 100 100 10-1 10-1 10-2 10-2 w w 10-3 10-3 P P 10-4 10-4 PSfrag replacements PSfrag replacements 10-5 10-5 Upper bound (39), independent decoding Upper bound (39), independent decoding Simulation, independent decoding Simulation, independent decoding Upper bound (35), collaborative decoding Upper bound (35), collaborative decoding Sim., collaborative dec., randomized errors Sim., collaborative dec., randomized errors Sim., collaborative dec., no randomization Sim., collaborative dec., no randomization 10-6 10-6 2 2.5 3 3.5 4 4.5 2.5 3 3.5 4 4.5 5 Eb/N0 [dB] Eb/N0 [dB] Fig. 4. Simulated decoding performance of the concatenated code C1 Fig. 5. Simulated decoding performance of the concatenated code C2 com- composed of two codewords from the outer code RS 26; 63, 54, 10 , and posed of three codewords from the outer code RS 28; 255, 223, 33 , and an binary inner Golay code G (23), AWGN channel, BPSK modulation. an binary linear inner (30, 24, 4) code, AWGN channel, BPSK modulation. range. However, the horizontal offset of the two bounds is also performance. The bounds derived in Section VII provide us in the range of 0.5 to 0.6 dB, so that the bounds can be used with such tools, since they become tight for high signal-to- for a rough estimation of the expected gain of collaborative noise ratios. To see this, we calculate the bounds (41), (39), decoding. and (35) for the codes 1 and 2. The results for 1 are For the code , we observe a collaborative decoding gain of C C C 2 depicted by Fig. 6, and the results for 2 are depicted by C 6 about 0.3 dB at Pw = 10− , or in other words, the word error Fig. 7. C rate is about 1000 times smaller with collaborative decoding We observe for both codes that the lower bound (41) for compared to independent decoding. The upper bound (39) independent decoding is not very tight for channels with small is about 0.4 dB worse than the actual decoding performance signal-to-noise ratios, but converges to the upper bound (39) if for independent decoding, while the gap between the bound the signal-to-noise ratio increases. For the code 1, the lower (35) and the collaborative decoding performance is about bound (41) and the upper bound (39) coincide beginningC from 0.3 dB. Hence, an estimation of the collaborative decoding an Eb –ratio of about 8 dB, for the two bounds coincide N0 2 gain obtained by comparing (39) and (35) is a bit optimistic a little later at about 9 dB. ThisC means that asymptotically, in this case. i.e., for Eb , we exactly know the performance of N0 We observe for both codes that we obtain virtually the independent decoding.→ ∞ same error probabilities Pw for collaborative decoding with Since we have the upper bound (35) on the collaborative randomized errors, and for decoding without randomization. error probability, and the lower bound (41) on the error This means that from a practical point of view, randomization probability of independent decoding, we are able to calculate has no significant influence on the collaborative decoding the guaranteed collaborative decoding gain, i.e., the horizontal performance. Hence, randomization is only necessary to fulfill gap between (41) and (35). In Fig. 6 we observe that for 1 the statistical requirements to derive a bound on the failure the collaborative decoding gain first increases, if the signal-to-C probability Pf . noise ratio increases, and then vanishes again asymptotically, i.e., for Eb . This behavior is explained by the fact N0 B. Asymptotic Considerations that the asymptotic→ ∞ gain is specified by the guaranteed error Clearly, Monte Carlo simulations are only feasible, if correction radius, which in turn is specified by the minimum the target block error probabilities are not too small. For distance. In this sense, the asymptotic gain is rather a code applications which require very low block error rates like property than a decoder property. data storage systems or optical data transmission systems, However, for all target block error rates which could be we need analytical tools to estimate the obtainable decoding relevant for practical applications, we obtain a collaborative PREPRINT -- PREPRINT -- PREPRINT -- PREPRINT -- 19 100 100 10-20 10-20 10-40 10-40 10-60 10-60 w -80 w -80 P 10 P 10 10-100 10-100 10-120 10-120 PSfrag replacements 10-140 PSfrag replacements 10-140 Upper bound (39), independent decoding Upper bound (39), independent decoding Lower bound (41), independent decoding Lower bound (41), independent decoding Upper bound (35), collaborative decoding Upper bound (35), collaborative decoding 10-160 10-160 4 6 8 10 12 14 4 5 6 7 8 9 10 Eb/N0 [dB] Eb/N0 [dB] Fig. 6. Bounds on the decoding performance of the concatenated code C1 Fig. 7. Bounds on the decoding performance of the concatenated code C2 composed of two codewords from the outer code RS 26; 63, 54, 10 , and composed of three codewords from the outer code RS 28; 255, 223, 33 , an binary inner Golay code G (23), AWGN channel, BPSK modulation. and an binary linear inner (30, 24, 4) code, AWGN channel, BPSK modula- tion. decoding gain by applying Algorithm 2. In Fig. 7 we see that radius the collaborative decoding gain, which can be guaranteed by l t = (N K¯ ) . our bounds, grows to more than 1 dB, when the signal-to- max l +1 − noise ratio improves to 9 or 10dB. Since the computational If the decoding radius is increased beyond half the minimum complexity of Algorithm 2 is of the same order as the compu- distance, it is in principle not possible to decode all error tational complexity of the Berlekamp–Massey algorithm, this patterns uniquely, since the received vector may lie in a region, gain can be obtained virtually for free. where several correction spheres are overlapping. In this case, three basic strategies are conceivable: the decoder selects one IX. CONCLUSIONS solution from the list, if several solutions exist; the decoder yields all possible solutions and leaves it to the succeeding In this paper, we presented a decoding algorithm for IRS data processing unit to cope with this list; or the decoder codes, which is based on varying length multi-sequence shift- does not take a decision at all and yields a decoding failure. register synthesis. This algorithm allows for efficiently de- Our algorithm applies the third strategy. Whenever it is not coding both homogeneous and heterogeneous IRS codes. We able to find a unique solution for a received word with t explained that homogeneous IRS constructions yield MDS errors, it declares a decoding failure. Clearly, this strategy codes with an optimum minimum distance, while heteroge- makes only sense, if the failure probability is small enough, neous IRS code designs provide a high degree of freedom for and hence most of the error patterns can be corrected in the constructing generalized concatenated codes and other special range tg t tmax. Therefore, we overbounded the failure applications like the decoding of low rate Reed–Solomon ≤ ≤ 1 probability Pf (t) and showed that it is in the order of q , codes as explained in [15]. for t = tmax, and decreases exponentially with decreasing t. The decoding of heterogeneous IRS codes is closely related Besides of the failure probability Pf , we also overbounded to joint error and erasure correction. Hence, it is also possible the error probability Pe to be able to analyze the overall to adopt our algorithm for performing joint error and erasure performance of collaborative decoding, or more precisely the decoding. probability Pw = Pe + Pf for a wrong decoding result. The decoder described by Algorithm 2 is able to correct all Moreover, we considered concatenated code designs with error patterns within a sphere of radius tg, where tg is half the outer IRS codes and inner block codes, since inner decoding minimum distance of the IRS code. Furthermore, it is able to causes column burst errors, which are necessary for effective correct errors beyond half the minimum distance, as long as collaborative decoding. We applied the bound on Pe(t) and the number of errors is below the maximum error correcting Pf (t) to analyze the decoding performance of such concate- PREPRINT -- PREPRINT -- PREPRINT -- PREPRINT -- 20 nated schemes. This analysis has been performed by deriving [10] G. Schmidt, V. R. Sidorenko, and M. Bossert, “Interleaved Reed– bounds on the word error probability after independent and Solomon codes in concatenated code designs,” in Proc. IEEE ITSOC Inform. Theory Workshop, (Rotorua, New Zealand), pp. 187–191, Au- collaborative outer decoding, and by using them to investi- gust 2005. gate, which decoding gains can be achieved by collaborative [11] G. Schmidt, V. R. Sidorenko, and M. Bossert, “Error and erasure correc- decoding in comparison to an independent decoding strategy. tion of interleaved Reed–Solomon codes,” in Coding and Cryptography, vol. 3969 of Lecture Notes in Computer Science, pp. 22–35, Berlin: We complemented these considerations by Monte Carlo simu- Springer Verlag, 2006. lations for two specific code designs. We observed that for all [12] G. Schmidt, V. R. Sidorenko, and M. Bossert, “Heterogeneous inter- operating points which could be interesting for practical appli- leaved Reed–Solomon code designs,” in Proc. 10th Int. Workshop on Algebraic and Combinatorial Coding Theory (ACCT-10), (Zvenigorod, cations, we are able to achieve a collaborative decoding gain Russia), pp. 230–233, September 2006. without increasing the decoding complexity in comparison to [13] E. L. Blokh and V. V. Zyablov, “Coding of generalized concatenated codes,” Problems of Information Transmission, vol. 10, pp. 218–222, independent decoding. The probabilities Pw achievable by our July–September 1974. Translated from Russian, original in Problemy collaborative decoding strategy are 100–1000 times smaller Peredachi Informatsii, pp. 45–50. than the probabilities achievable by independent decoding. [14] V. A. Zinoviev, “Generalized cascade codes,” Problems of Information The concatenated code designs considered here are rather Transmission, vol. 12, pp. 2–9, January–March 1976. Translated from Russian, original in Problemy Peredachi Informatsii, pp. 5–15. simple, to be able to analyze them theoretically. However, [15] G. Schmidt, V. 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