Decoding Algorithm and Architecture for BCH Codes under the Lee Metric

Transcription

1 Decoding Algorithm and Architecture for BCH Codes under the Lee Metric Yingquan Wu and Christoforos N. Hadjicostis Coordinated Science Laboratory and Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Keywords BCH codes, Lee distance, algebraic decoder, decoder architecture Abstract The Lee metric measures the circular distance between two elements in a cyclic group and is particularly appropriate as a measure of distance for data transmission under phase-shift keying modulation over a white noise channel. In this paper, using newly derived properties on Newton s identities, we initially investigate the Lee distance properties of a class of BCH codes and we show that (for an appropriate range of parameters) their minimum Lee distance is at least twice their designed Hamming distance. We then make use of properties of these codes to devise an efficient algebraic decoding algorithm that successfully decodes within the above lower bound of the Lee error-correction capability. Finally, we propose an attractive design for the corresponding VLSI architecture that is only mildly more complex than popular decoder architectures under the Hamming metric; since the proposed architecture can be re-used for decoding under the Hamming metric without extra hardware, one can use the proposed architecture to decode under both distance metrics (Lee and Hamming). This material is based upon work supported in part by the National Science Foundation under NSF Career Award and NSF ITR Award No Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of NSF. Corresponding author: C. N. Hadjicostis, 357 Coordinated Science Laboratory, 1308 West Main Street, Urbana, IL , USA. Tel: Fax: chadjic@uiuc.edu. 1

2 I. Introduction The Lee metric was introduced in [1] as an alternative to the Hamming metric when non-binary signals (usually taken from GF(p)) are transmitted over certain noisy channels. For two arbitrary elements a, b GF(p), the Hamming distance between a and b is defined as 1 if a b or 0 if a = b, whereas the Lee distance between a and b is defined as the smaller value between a b and p a b. Clearly, the Lee metric is more appropriate in cases where large Lee distances are less likely (e.g., when data is transmitted over a white noise channel using phase-shift-keying modulation). In [], Chiang and Wolf derived all discrete, memoryless, symmetric channels matched to the Lee metric and investigated the general properties of Lee metric block codes. For instance, when the channel noise samples are independent and identically distributed according to a bilateral exponential density with zero mean, Chiang and Wolf showed that maximum-likelihood decoding is equivalent to finding a codeword which is at the smallest Lee distance from the received word []. Let α be a primitive element of the finite field GF(p m ), so that GF(p m ) is represented by the set {0, 1, α, α,..., α pm }. In [3], Berlekamp introduced negacyclic codes for which the generator polynomial contains the roots α, α 3, α 5,..., α t 3. The lower bound on the minimum Lee distance of this class of codes is t 1 when t p 1. The core of the decoding procedure in [3] is the application of the Berlekamp-Massey algorithm to a polynomial congruence that is very similar to the key equation for decoding BCH codes under the Hamming metric. It is worth noting that negacyclic codes are not cyclic. In [4], Roth and Siegel characterized the Lee distance properties of a class of BCH codes whose generator polynomial contains 1, α, α,..., α t 1 as its roots. When t p 1 or when the code lies in GF(p), the minimum Lee distance is shown to be at least t. The authors also developed a decoding algorithm utilizing Euclid s algorithm to correct up to t 1 Lee errors. In contrast to the performance of negacyclic codes, this class of codes has twice the length of the negacyclic construction. In [5], Byrne showed that, under certain restrictions on t, the t lower bound on the minimum Lee distance also holds for codes on a Galois ring and devised a decoding algorithm in light of a Gr.. obner basis. In this paper we first extend the Lee distance properties of the class of BCH codes studied in [4] to a more general class of BCH codes. We then devise a novel decoding algorithm which is efficient and amenable to an attractive decoding architecture that is only mildly more complex than popular decoder architectures for BCH codes under the Hamming metric. The paper is organized as follows. In Section II, we briefly state Newton s identities and review two properties, namely the uniqueness and binomial decomposition properties that will be used in subsequent analysis. In Section III, using the uniqueness property, we characterize the Lee distance properties of a class of BCH codes that properly includes the codes studied in [4]. We show that the bound on the minimum Lee distance for the class of BCH codes in [4] can be generalized to the family of extended BCH codes. More specifically, for the extended BCH code whose generator polynomial contains roots α, α,..., α t 1, the minimum Lee distance is at least t when t p 1 or when the code lies in the prime field GF(p). As we show in Section III, this bound can be improved for

3 special cases. In Section IV, by applying the decomposition property, we devise an efficient decoding algorithm to correct up to t 1 Lee errors; in particular, we develop an error evaluation algorithm that is significantly less complex than the decoding algorithm proposed in [4] because it avoids applying Chien search multiple times. Using similar techniques, we are also able to show that although the minimum Lee distance of a code can be as small as p when p 1 <t p, there are at most two codewords within Lee distance t 1 to an arbitrary received word. In Section V, we present an efficient decoder architecture that is only mildly more complex than of the popular decoder for BCH codes under the Hamming metric, rendering it very attractive for VLSI implementation. In Section VI, we conclude with pertinent remarks. II. Newton s Identities and Extensions Throughout this paper we assume that all algebraic operations are defined over a finite field GF(p m ), where p is a prime greater than. Power polynomials are defined, for k =1,,..., as S k (x 1,x,...,x n ) = n x k i. (1) i=1 Elementary symmetric polynomials are defined as Λ k (x 1,x,...,x n ) =( 1) k x i1 x i...x ik, k =1,,...,n () 1 i 1 <...<i k n (for consistency, we set Λ 0 = 1). Newton s identities reveal a fundamental relation between elementary symmetric polynomials and power polynomials. More specifically, if we use Λ k, S k to denote concisely Λ k (x 1,x,...,x n ), S k (x 1,x,...,x n ) respectively, Newton s identities take the form [6] S k +Λ 1 S k Λ k 1 S 1 + kλ k =0, 1 k n, S k +Λ 1 S k Λ n S k n =0, k > n. (3) If S 1, S,...,S n are known and n<p(so that 1 n! is well-defined in GF(p m )), we can use (3) to obtain Λ k (1 k n) as a function of S 1, S,...,S k : S S S Λ k F k (S 1, S,...,S k ) = ( 1)k S 3 S S k! (4).. S k 1 S k S k 3... S 1 k 1 S k S k 1 S k... S S 1 We remark that the solution for Λ k in terms of {S i } k i=1 is independent of the number of original variables n. Once {Λ i (x 1,x,...,x n )} n i=1 are available, the underlying x 1,x,...,x n are exactly the 3

4 roots of the characteristic polynomial n σ x1,x,...,x n (x) = (x x i )=x n +Λ 1 (x 1,...,x n )x n Λ n (x 1,...,x n ). (5) i=1 The following developments for Newton s identities were derived in [7] and are included here for completeness. Note that even though Λ k (x 1,x,...,x n ) is not defined when k > n, F k (S 1, S,...,S k ) is. For consistency, in our development we will extend the definition of Λ k (x 1,x,...,x n ) for k >n by setting it equal to F k (S 1, S,...,S k ). In fact, based on this extension, we have Lemma 1 Let S 1, S,...,S k be the power polynomials of x 1,x,...,x n. Then, Λ k (x 1,x,...,x n ) F k (S 1, S,...,S k )=0, k > n. (6) Proof: We can regard {S i } k i=1 as a power polynomial of x 1,...,x n, 0 n+1,...,0 k, which immediately leads to F k (S 1, S,..., S k )=Λ k (x 1,...,x n, 0 n+1,...,0 k )=0 for k > n. The above viewpoint also enables us to simplify Newton s identities as S k +Λ 1 S k Λ k 1 S 1 + kλ k =0, k, n 1. (7) We now use Lemma 1, together with Newton s identities, to analyze the following system of equations: x 1 + x x k = y 1 + y y l where k l. x 1 + x x k = y1 + y y l x x x3 k = y1 3 + y y3 l. x k 1 + xk xk k = y1 k + yk yk l, (8) Theorem 1 The system of equations in (8) has the unique solution {x i } = {y j } when k = l, and the unique solution {x i } k i=1 = {y i} l i=1 {0} when k > l (up to permutations). Proof: For 1 i k, Newton s identities indicate Λ i (x 1,x,...,x k )=F i (S 1 (x 1,...,x k ), S (x 1,...,x k ),...,S i (x 1,...,x k )) = F i (S 1 (y 1,...,y l ), S (y 1,...,y l ),...,S i (y 1,...,y l )) =Λ i (y 1,y,...,y l ). In particular, Lemma 1 indicates that Λ i (x 1,x,...,x k )=Λ i (y 1,y,...,y l )=0, l < i k. 4

5 Thus, the characteristic polynomial of x 1,x,...,x k is given by k (x x i )=x k + x k 1 Λ 1 (x 1,x,...,x k )+...+Λ k (x 1,x,...,x k ) i=1 which yields the desired conclusion. = x k + x k 1 Λ 1 (y 1,y,...,y l )+...+ x k l Λ l (y 1,y,...,y l ) l = x k l (x y i ), i=1 Note that in Theorem 1 values in the set x 1,x,..., x k (or the set y 1,y,..., y l ) are not required to be distinct. The following theorem indicates another significant property of function F. Theorem Let x 1,x,...,x n be n variables and s 1,s,...,s k be k given parameters. Then, F k (s 1 + S 1,s + S,..., s k + S k )= min{n, k} i=0 F k i (s 1,s,...,s k i ) Λ i. (9) We provide an intuitive proof which only applies in the field of complex numbers. A lengthy inductive proof which applies in an arbitrary field is given in [7]. Proof: Let y 1,y,...,y k be a (possibly complex) solution 1 to the following equation array: y 1 + y y k = s 1 y1 + y y k = s Then, we have. y k 1 + yk +...+yk k = s k. F k (s 1 + S 1,..., s k + S k )=Λ k (x 1,...,x n,y 1,...,y k ) = = min{n, k} i=0 min{n, k} i=0 Λ k i (y 1,...,y k ) Λ i (x 1,...,x n ) F k i (s 1,s,...,s k i ) Λ i. III. Lower Bounds on Minimum Lee Distance Let α be a primitive element of the finite field GF(p m ), so that GF(p m ) can be represented by the set {0, 1, α, α,..., α pm }. In this section, we are interested in investigating the Lee distance properties 1 It is clear that there is always a solution y 1,y,...,y k : from s 1,s,...,s k, we can find Λ 1, Λ,...,Λ k (through the system of equations (3)); then y 1,y,...,y k are the roots of polynomial (5). 5

6 of the class of extended BCH codes with generator polynomials containing roots, α, α,...,α t 1. We will show shortly that this class of BCH codes, which we will denote by C(N, t, p), properly contains the class of BCH codes with generator polynomial containing roots 1, α, α,..., α t 1, which are investigated in [4]. The code C(N, t, p) has the parity check matrix of the form α 1 α... α N 1 H(N, t, p) = 0 α1 α... αn 1, (10) α1 t 1 α t 1... α t 1 N 1 where α 1,α,...,α N 1 are distinct nonzero elements in GF(p m ) with m being minimal, i.e., p m 1 < N p m. A codeword c =[c N 1 c N c N 3... c 1 c 0 ] satisfies Hc T = 0. Note that c N 1 is the parity symbol. Remark: The class of BCH codes with generator polynomial containing roots 1, α, α,..., α t 1, as studied in [4], are a subclass of the codes of our interest (with the key difference being that N can equal p m for the latter but not for the former). To see this, observe that the corresponding parity check matrix is of the form α 1 α α 3... α N H = α1 α α3... αn, α1 t 1 α t 1 α3 t 1... α t 1 N where α 1,α,...,α N are distinct nonzero elements in GF(p m ). Through linear operations on the rows of H, we can obtain another parity-check matrix which is exactly in the form of (10) (α α 1 ) (α 3 α 1 )... (α N α 1 ) H = 0 (α α 1 ) (α 3 α 1 )... (α N α 1 ), (α α 1 ) t 1 (α 3 α 1 ) t 1... (α N α 1 ) t 1 As opposed to Hamming weight which measures the number of nonzero positions in an N-dimensional vector y, Lee weight is defined as the sum of the Lee values of all entries, i.e., w L (y) = N 1 i=0 ρ(y i ), (11) 6

7 where ρ(x) = x, if x p 1 x p, otherwise., Clearly, the Lee distance d L (y, x) is at least as large as the Hamming distance d H (y, x): d L (y, x) = N 1 i=0 ρ(y i x i ) N 1 i=0 I yi x i = d H (y, x), (1) where I y x is 1 if y x, or 0 if y = x. Intuitively speaking, the Lee metric measures the number of + and differences. The following examples shed some light on the Lee distance d L ([0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 4, 0, 0, 4, 0, 0, 0, 0]) = 3 (p = 5) d L ([0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 3, 0, 0, 0, 0, 0, 0, 0]) = 3 (p = 5) d L ([1,,, 3, 4, 0, 4,, 0, 1], [1, 4,,, 0, 0, 4,, 0, 1]) = 4 (p = 5) It is well-known that the designed minimum Hamming distance (a lower bound) of the class of BCH codes C(N, t, p) defined in (10) is t + 1 [3, 8]. We next establish a bound for the minimum Lee distance for this class of codes. Three operators will be used repeatedly in the sequel: y + and y are defined element-wise, for i =0, 1,,...,N 1, as = max{0, ρ(y i )}, y + i y i = max{0, ρ(y i )}. Also, we define y =[y N y N 3... y 1 y 0 ] to be an operation that eliminates the parity symbol. Theorem 3 The minimum Lee distance of the extended BCH code C(N, t, p) satisfies the following bound t, if t p 1 d L (N, t, p), p, if p 1 < t < p. Proof: Due to the linearity of the code, the minimum Lee distance between two arbitrary distinct codewords is equal to the minimum Lee weight of nonzero codewords. (13) (14) Let c be a codeword with minimum Lee weight. Let k = w L (c + ) and l = w L (c ). Let {x i } k i=1 and {y i} l i=1 be associated to the locators of the positive entries of c + and c respectively. Note that if a location is associated with a Lee weight w>1, then the corresponding locator x k (or y l ) will appear exactly w times in the set {x i } k i=1 (or {y i} l i=1 ). Using this convention, {x i} k i=1 and {y i} l i=1 must satisfy the following parity checks: x x k y 1... y l = 0 x x k y 1... y l = 0 x t x t 1 k y1 t 1... y t 1 l = 0,. (15) 7

8 where {x i } k i=1 {y i} l i=1 = by definition of + and in (13). Theorem 1 implies that {x i } k i=1 = {y i} l i=1 if k, l < t. Thus, either k or l must be at least t, and, without loss of generality, we assume k t. Note that the Lee weight of c is in the form of w L (c) =k + l + ρ(c N 1 ), where c N 1 denotes the parity symbol. The conclusion is trivial if l t. The following argument assumes l t. Note that the parity symbol satisfies c N 1 +(k l) = 0 (mod p). Thus, ρ(c N 1 ) is either (k l) or p (k l). Consequently, w L (c) k + l + min{k l, p (k l)} = min{k, p +l}, which immediately concludes the theorem. We next show that the above bound is tight for some codes. To see this, let t divide p m 1. Then the equation x t = α tf, where α is a primitive element in GF(p m ) and f is an arbitrary integer within [1, pm 1 t ], has t distinct roots in GF(p m ): x i = α i(pm 1)/t+f, i =1,,..., t. Note that Λ i (x 1,x,...,x t ) = 0 for i =1,,...,t 1. Thus, Newton s identities indicate S i (x 1,x,...,x t )= t j=1 xj i = 0 for i =1,,...,t 1. Therefore, if the locator set {α i } i=1 N 1 contains {x i } t i=1 with an arbitrary f, then it can be easily verified that c which has value 1 in the positions associated with x 1,x,...,x t, value p t in the parity position and value 0 in the remaining positions is a nonzero codeword that achieves exactly the bound on the minimum Lee weight t, if t p 1 w L ( c) =t +0+ ρ(p t) =, p, if p 1 < t < p. Remark: The bound established in Theorem 3 is the same as the bound derived for the subclass of BCH codes studied in [4]. We next establish a tighter bound on the minimum Lee distance for the special case when t = p. Theorem 4 The minimum Lee distance of the extended BCH code C(N, p, p) is bounded by d L (N, p, p) p +. (16) Proof: We will be focusing on the minimum Lee weight of a nonzero codeword c. Let k = w L (c + ) and l = w L (c ), and without loss of generality, let k l. Let {x i } k i=1 and {y i} l i=1 be associated to 8

9 the locators of the positive entries of c + and c respectively. Note that {x i } k i=1 and {y i} l i=1 must satisfy the following parity checks: k x j i = i=1 l i=1 y j i, j =1,,...,p 1 and that k p (otherwise {x i } k i=1 = {y i} l i=1 {0} as indicated by Theorem 1). If we can show that there does not exist a solution with k = p, l = 0, then the proof will be completed. This is due to the following fact: for the case when k = p and l = 1, w L (c) =k +l + ρ(c N 1 ) = p+1+ ρ(p 1) = p+; for the case when k = p + 1 and l = 0, w L (c) =k + l + ρ(c N 1 ) =(p + 1) = p + (our argument yields a lower bound since both cases may not exist at all). We proceed to exclude the case k = p and l = 0. If there exists a solution x 1,x,...,x p, then we have Λ i (x 1,x,...,x p ) = 0, i =1,,...,p 1. From (5), we conclude that x 1,x,...,x p must be the p roots of an equation x p = β for some β GF(p m ). Let β = α n (where α is a primitive element of GF(p m )); then, the above equation is equivalent to pi n (mod p m 1), 1 i p m 1. Since p does not divide p m 1, there exists at most one solution. This results in a contradiction. Note that for p = 3 we have d L (N,3, 3) 5. (17) This indicates that p = 3 suffices to correct two Lee errors; this should be contrasted with the requirement that p = 5 according to Theorem 3. In the special case when m = 1, i.e., when the code is in the prime field GF(p), the bound on the minimum Lee distance can be improved, as stated below. Theorem 5 For t N p, the minimum Lee distance of the code C(N, t, p) satisfies d L (N, t, p) t. (18) Sketch of proof: Let k = w L (c + ) and l = w L (c ), and, without loss of generality, let k l. It suffices to consider the case w L (c) =k + l +(p (k l)) = p +l. The remainder of the proof follows exactly the proof of Theorem 3 in [4]. IV. Decoding Algorithm In this section we present an efficient decoder that corrects up to t 1 errors for the case when t p 1. The extension to the alternative case when p 1 < t p is straightforward. We first establish some notation. Let r =[r N 1 r N r N 3... r 1 r 0 ] be the received word after transmitting a codeword c. The error pattern is given by e = r c. Let k = w L (e + ) with 9

10 x 1,x.,..., x k as the corresponding locators, and l = w L (e ) with y 1,y,...,y l as the corresponding locators (here means that e N 1 is not counted). We have the following syndromes: N 1 s 0 = j=0 N s i = Consider the decomposition of syndromes for i =1,,...,t 1: s i = = j=0 N j=0 N j=0 r j, (19) r j α i j, i =1,,...,t 1. (0) N c j αj i + j=0 ρ(e j )α i j = e j α i j = k x i j j=1 N j=0 e j α i j (1) l yj. i () When decoding the received word r under the Hamming metric, the Berlekamp-Massey algorithm is used to correct Hamming errors on r, and then the parity symbol is corrected according to the parity of the corrected BCH codeword [8]. However, under the Lee metric this procedure is not applicable. This is because the parity information of r, namely s 0 (r ), is crucial for the decoding process (as will become clear shortly) and is not available without exploring the parity symbol r N 1. We next give a unified system of equations which account for all possibilities and allow us to proceed as if the Lee errors on the parity symbol were known a priori. Theorem 6 Let r = c + e be the received word after transmission of a codeword c in C(N, t, p). If the Lee weight w L (e) of the error pattern e is less than t, then the error locators (apart from the parity position 0) correspond to the unique solution of the following system of equations x x κ y 1... y ι = s 1 x x κ y1... yι = s where κ = t 1+δ, ι = t 1 δ and δ = ρ(s 0 ). j=1 x t x t 1 κ y1 t 1... yι t 1 = s t 1 Proof: Note that if we let κ and ι be the largest possible values satisfying κ + ι<t and κ ι = δ, then, κ = t 1+δ and ι = t 1 δ. Assume that τ Lee errors occurred at r N 1 (τ is possibly 0), i.e., ρ(e N 1 ) = τ; then, the Lee weight w L (e ) is at most t 1 τ. If we let k = w L (e + ) and l = w L (e ), then, the error pattern (ignoring the parity symbol) corresponds to the unique solution of x x k ȳ 1... ȳ l = s 1. (3) x x k ȳ 1... ȳ l = s x t x t 1 k ȳ1 t 1... ȳ t 1 l = s t (4)

11 where again the sets { x 1, x,..., x k } and {ȳ 1, ȳ,..., ȳ l } do not share any elements but the values in x 1, x,..., x k (and ȳ 1,ȳ,..., ȳ l ) are not necessarily all distinct. Without loss of generality, we assume ρ(e N 1 ) to be positive. Then, we have k l = δ τ, which, in conjunction with k + l < t τ, indicates k t 1+δ τ and l t 1 δ. To show that the unique solution of system (4) is identical to the (unique) solution of system (3), simply re-arrange (3) and (4) in the following form: x x κ +ȳ ȳ l = x x k +y y ι x x κ +ȳ ȳl = x x k +y yι. x t x t 1 κ +ȳ1 t ȳ t 1 l = x t x t 1 k +y1 t yι t 1. Since κ + l < t and k + ι<t, Theorem 1 indicates that {x i } κ i=1 {ȳ i } l i=1 = { x i } k i=1 {y i } ι i=1, which, in conjunction with the fact that {x i } κ i=1 {y i} ι i=1 = and { x i} k i=1 {ȳ i} l i=1 =, immediately leads to the equivalence between system (3) and system (4). Theorem 6 indicates that system (3) is the key for decoding. Applying Theorem to system (3), we conclude that for all τ > κ 0=Λ + τ (x 1,...,x κ ) = F τ (s 1 + S 1 (y 1,...,y ι ),...,s l + S τ (y 1,...,y ι )) ι = F τ i (s 1,...,s τ i )Λ i (y 1,...,y ι ) i=0 = F τ (s 1,...,s τ )+ ι F τ i (s 1,...,s τ i )Λ i (y 1,...,y ι ). i=1 By choosing τ = κ +1,κ +,...,t 1, we obtain the following system of equations involving {Λ i (y 1,y,...,y ι )} l i=1 ι i=1 F κ+1 i(s 1,s,...,s κ+1 i )Λ i (y 1,y,...,y ι )= F κ+1 (s 1,s,...,s κ+1 ) ι i=1 F κ+ i(s 1,s,...,s κ+ i )Λ i (y 1,y,...,y ι )= F κ+ (s 1,s,...,s κ+ ) ι i=1 F t 1 i(s 1,s,...,s t 1 i )Λ i (y 1,y,...,y ι ) = F t 1 (s 1,s,...,s t 1 ).. (5) Since κ t 1 (due to the assumption that δ 0), the above system has order at most t 1. We note that system (5) is of the same form as the system for decoding a non-binary BCH code under the Hamming metric and thus can be efficiently solved using popular algorithms, such as the Berlekamp-Massey algorithm or the extended Euclidean algorithm (cf. [8]). Once we have obtained 11

12 {Λ i (y 1,y,...,y ι )} ι i=1, we can solve for y 1,y,...,y ι. On the other hand, by applying Theorem with τ =1,,...,κ, we can easily obtain Λ + i (x 1,x,...,x κ ), i =1,,...,κ, via Λ + 1 (x 1,x,...,x κ )=F 1 (s 1 )+Λ 1 (y 1,y,...,y ι ) Λ + (x 1,x,...,x κ )= i=0 F i(s 1,s,...,s i )Λ i (y 1,y,...,y ι ). Λ + κ (x 1,x,...,x κ )= ι i=0 F κ i(s 1,s,...,s κ i )Λ i (y 1,y,...,y ι ). (6) Thus, x 1,x,...,x κ can be readily computed. Note that solving for y 1,y,...,y ι (or x 1,x,...,x κ ) has to account for repeated roots; we describe how this can be done efficiently after a final observation on the distance properties of extended BCH codes. For codes with p 1 <t p, it is possible that both s 0 and p s 0 are less than t (note that k l = δ <tis a prerequisite, otherwise there must be at least t errors since k + l k l t). In such cases, we need to take into account both δ = s 0 and δ = s 0 p, so there exist at most two candidate codewords within Lee distance t 1. Note that the minimum Lee distance can only be bounded by p (this bound was shown earlier to be strict for some codes). We re-state this observation in the form of the following theorem. Theorem 7 For the extended BCH code C(N, t, p) with p t> p 1 and m, there exist no more than two codewords within Lee distance t 1 from an arbitrary pattern in {GF(p)} N. The approach in [4] utilizes derivatives to account for the repeated roots of Λ + (x) and Λ (x). In the worst case, Chien search is employed t 1 times. The following approach effectively eliminates such tremendous computational burden: we first compute the product Λ(x) =Λ + (x)λ (x) (7) and then apply Chien search on Λ(x) to determine all distinct roots {x 1 i } f i=1. These roots represent all erroneous locations and, since the Hamming distance of the code (ignoring the parity symbol) is t, we can apply erasure decoding under the Hamming metric to determine the magnitude (equivalent to the Lee errors) of the error locations. More specifically, we nullify all erroneous locations and then determine magnitudes of the erased locations by Forney s formula (cf. [8]). Remark: Clearly, the decoding of the class of BCH codes studied in [4] is equivalent to solving the system in (3) and can be performed following the proposed approach without involving any transformation (the most straightforward approach would be to transform the received word to the one corresponding to the space of the extended BCH codes as argued in the beginning of Section III). Example: To illustrate the proposed decoding process, consider the extended BCH code C(31, 6, 31) with generator polynomial g(x) =(x 3)(x 3 )(x 3 3 )(x 3 4 )(x 3 5 )=x 5 +9x 4 4x x 13x +1, 1

13 where 3 is used as the primitive element in GF(31). Let m(x) =x 7 +9x 6 4x x 4 13x 3 + 3x 13x 8 be the message to be sent. Systematic encoding produces the corresponding transmitted codeword as c(x) =4x 31 + x 1 +9x 11 4x x 9 13x 8 +3x 7 13x 6 8x 5 x 4 +5x 3 +x, where the first term 4x 31 represents the parity symbol and the remaining expression is x 5 m(x) R g (x 5 m(x)), where R g (x 5 m(x)) denotes the remainder polynomial after dividing x 5 m(x) by g(x). Case 1. Let the received sequence r(x) be r(x) =4x 31 x 5 + x 0 + x 1 +9x 11 4x x 9 13x 8 +4x 7 13x 6 8x 5 +5x 3 +x and let r (x) =r(x) (mod x 31 ); then, we have syndromes s 0 = r(1) = 1, s 1 = r (3) =, s = r (3 ) = 14, s 3 = r (3 3 ) = 6, s 4 = r (3 4 ) = 1, s 5 = r (3 5 ) = 4. So we choose only δ = ρ(s 0 ) = 1 and exclude the alternative 30. This enables us to determine the maximum possible number of positive Lee errors to be κ = t 1+δ = 3 and the maximum possible number of negative Lee errors to be ι = t 1 δ = (assuming we are operating within the decoding power). Therefore, the errors are determined by the system x 1 + x + x 3 y 1 y = x 1 + x + x 3 y 1 y = 14 x x3 + x3 3 y3 1 y3 = 6 x x4 + x4 3 y4 1 y4 = 1 x x5 + x5 3 y5 1 y5 =4. In light of Newton s identities, we compute F 1 =, F = 5, F 3 = 11, F 4 = 17, F 5 = 11, and from (5) we have { 5Λ (y 1,y ) 11Λ 1 (y 1,y ) = 17 11Λ (y 1,y ) + 17Λ 1 (y 1,y ) = 11. Applying the Berlekamp-Massey algorithm, we obtain Λ 1 (y 1,y )= 1, Λ (y 1,y )=5. Thus, y 1, y are the two roots of the characteristic equation y 1y + 5 = 0, which immediately results in y 1 = y = 6 = 3 5 by exhaustive search through GF(31) (note that in this case it is easy to determine the double root simply by knowing the order of the polynomial and by performing Chien search once). Substituting Λ 1 (y 1,y )= 1, Λ (y 1,y ) = 1 into the system (6), we obtain Λ + 1 (x 1,x,x 3 )= 10, Λ + (x 1,x,x 3 )=7, Λ + 3 (x 1,x,x 3 )= 3. 13

14 Thus, x 1,x,x 3 are the three roots of the characteristic equation x 3 10x +7x 3=0, which has roots x 1 = 19 = 3 4, x = 17 = 3 7, x 3 =5=3 0. Therefore, the error polynomial is and the corresponding codeword polynomial is e(x) = x 5 + x 0 + x 7 + x 4, c(x) =r(x) e(x) =4x 31 + x 1 +9x 11 4x x 9 13x 8 +3x 7 13x 6 8x 5 x 4 +5x 3 +x, which coincides with the transmitted codeword. Case. Let the received sequence r(x) be r(x) =3x 31 + x 7 x 5 + x 15 + x 1 +9x 11 4x x 9 13x 8 +3x 7 1x 6 8x 5 x 4 +5x 3 +x. Then, we have syndromes s 0 = r(1) = 1, s 1 = r (3) = 1, s = r (3 ) = 6, s 3 = r (3 3 ) = 19, s 4 = r (3 4 ) = 13, s 5 = r (3 5 ) = 4. We first determine the maximum possible number of positive Lee errors κ = 3 and the maximum possible number of negative Lee errors ι =. Therefore, the errors are determined by the system x 1 + x + x 3 y 1 y =1 x 1 + x + x 3 y 1 y =6 x x3 + x3 3 y3 1 y3 = 19 x x4 + x4 3 y4 1 y4 = 13 x x5 + x5 3 y5 1 y5 =4. We next compute F 1 = 1, F = 13, F 3 = 1, F 4 = 10, F 5 =, in light of Newton s identities. We have from (5) { 13Λ (y 1,y ) + 1Λ 1 (y 1,y ) = 10 1Λ (y 1,y ) + 10Λ 1 (y 1,y ) =. Applying the Berlekamp-Massey algorithm, we obtain { Λ 1 (y 1,y )= (y 1 + y )= 6 Λ (y 1,y )=y 1 y =0, which immediately indicates y 1 =6=3 5, y = 0. Substituting Λ 1 (y 1,y )= 6, Λ (y 1,y ) = 0 into the system (6), we obtain Λ + 1 (x 1,x,x 3 )= 7, Λ + (x 1,x,x 3 )= 1, Λ + 3 (x 1,x,x 3 )= 4. Thus, x 1,x,x 3 are the three roots of the characteristic equation x 3 7x 1x 4=0, 14

15 r 0,r 1,...,r N,r N 1 α 1 α α t 1 s 0 s 1 s s t 1 Figure 1: Architecture for the block that generates the syndromes. and are easily evaluated to be x 1 = 8 =3 7, x = 1 =3 15, x 3 = 16 = 3 6. Therefore, the error polynomial (without counting the parity symbol) is e(x) =x 7 x 5 + x 15 + x 6. Note that the error pattern does not satisfy the parity check s 0 = 1. Therefore, a negative Lee error has occurred on the parity symbol, and the entire error polynomial is e(x) = x 31 + x 7 x 5 + x 15 + x 6. Finally, by subtracting the error polynomial from the received polynomial, we successfully retrieve the transmitted codeword c(x). =0? Σ Λ 0 Λ 1 Λ Λ h 1 Λ h α 1 α α (h 1) α h Figure : Architecture for the block that performs the Chien search. 15

16 s 1,s,s 3,...,s t Λ k Σ 1 k Control Figure 3: Architecture for the block that evaluates the elementary symmetric polynomials via Newton s identities. V. Decoder Architecture Notice that for the class of extended BCH codes the parity symbol needs to processed quite differently than the remaining symbols. As a result, the corresponding encoder and decoder architecture are significantly more complex than the subclass of BCH codes studied in [4]. In this section we focus on the class of BCH codes studied in [4] but, with appropriate modifications, our approach can also be applied to extended BCH codes (i.e., when the code length N satisfies N = p m ). In this section, we assume that systematic encoding is carried out and we investigate the architecture for the proposed decoder (under the Lee metric). Specifically, if we let σ(x) =σ K 1 x K 1 + σ K x K +...+σ 1 x+σ 0 denote the message polynomial, and ψ(x) =ψ N K x N K +ψ N K 1 x N K ψ 1 x + ψ 0 denote the remainder after dividing x N 1 K σ(x) by g(x) (the generator polynomial), then systematic encoding results in the following codeword polynomial c(x) =x N K σ(x) ψ(x). The alternative vector representation gives more insight c =[σ K 1,σ K,..., σ 0, ψ N K 1, ψ N K,..., ψ 0 ]. We first briefly review the prevalent decoder architecture under the Hamming metric (cf. [9, 10, 11]), since it forms the foundation of our proposed architecture for decoding under the Lee metric. This decoder architecture has three blocks: (i) syndrome computation, (ii) key-equation solver, and (iii) 16

17 F 1, F, F 3,...,F t 1 Λ t 1 Λ t Λ 3 Λ Λ 1 Σ Λ t 1, Λ t,..., Λ, Λ 1 Figure 4: Architecture for the block that is used to evaluate (6). Chien search and error correction. Note that we will use this notation when describing the decoder for the Lee metric even though this might be a bit awkward in our context. Let h = (t 1)/ denote the Hamming error correction capability of the code. Following Horner s rule, the syndrome is computed as s i = ((...((0 α i + r N 1 ) α i + r N ) α i +...+) α i + r 1 ) α i + r 0 for i = 1,,..., t 1; the corresponding architecture design is depicted in Figure 1. Note that this is done during the demodulation phase and causes no real delay. The block for the key-equation solver usually implements the inversionless Berlekamp-Massey algorithm or the extended Euclidean algorithm. Its goal is to determine the error locator polynomial Λ(x). This block is far more intricate than the other two and has been studied quite extensively (cf. [9, 10, 11]). Arguably, the two dominant design architectures exhibit similar implementation complexity and delay. Chien search determines all error locations (i.e., the roots of the error locator polynomial) by exhaustively searching through all locations. This search has been widely adopted due to its hardware simplicity (the corresponding architecture involves no full-multipliers but only half-multipliers, as depicted in Figure ). The error evaluation is based on Forney s original formula or one of its many variations (cf. [8]). We next devise a decoder architecture under the Lee metric. In addition to the syndrome, which can be computed as in the Hamming case, we also need to obtain the values of the Λ i s. Figure 3 depicts a systolic architecture to evaluate the Λ i s via Newton s identities in (3). Essentially, we use a shift register array to achieve the evaluation of Λ i in one clock cycle. Note that the control broadcasts an enabling signal that ensures that each time only one register is enabled to load the other input (from the product). The enabling signal is (left) shifted by one after each clock cycle. Note also 1 k, k = 1,,..., t 1, can be pre-computed and saved in the (t 1)-dimensional shift register array, to eliminate inversion operations. 17

18 Equation system (5) is precisely the key equation under the Hamming metric, and thus can be efficiently solved by either the Berlekamp-Massey algorithm or the extended Euclidean algorithm. Efficient architectures for these can be found in many publications, including [9, 10, 11]. Figure 4 illustrates the block which is used to evaluate (6) in t 1 clock cycles. double the hardware size of this block in order to be able to re-use it later on. Note that we artificially The architecture in Figure 4 can be re-used to compute the coefficients of the product (7) in t 1 clock cycles. We note that the degree of Λ(x) can be as high as t 1, almost twice h value. Therefore, the Chien search block for the Lee metric uses twice number of half-multipliers (i.e., one of the two input ends is fixed) as that of the Hamming metric. We proceed to account for the error magnitude of the erasure locations (that correspond to the roots of Λ(x)) using Forney s formula. Let x 1 1,x 1,...,x 1 f be the roots of Λ(x). We first compute the error locator polynomial Λ (x) = f (1 x 1 i x) = 1 + Λ 1x +Λ x +...+Λ f xf (8) i=1 which requires t 1 clock cycles. We then compute the derivative of the error locator polynomial [Λ ] (x) =Λ 1 + Λ x fλ f xf 1. (9) This may be completed in one cycle with parallel operation or in t 1 cycles with serial operation. We next compute the error evaluator polynomial in t 1 cycles by re-using the architecture in Figure 4 Γ(x) =Λ t 1 (x) s i x i 1 (mod x t 1 ). (30) i=1 Finally, the evaluation of the erased locations is performed utilizing Forney s formula as follows: Remark: e i = x 1 i Γ(x 1 i ) [Λ ] (x 1 i ). (31) It is worth noting that in [4] the extended Euclidean algorithm is employed to identify up to t 1 errors, whereas in our approach either the Berlekamp-Massey algorithm or the extended Euclidean algorithm is designed to correct up to h errors, which effectively cuts the hardware complexity by half. Moreover, the method in [4] requires multiple Chien searches (up to t 1 times), rendering it impractical for applications. Overall, the proposed decoding algorithm exhibits noticeably smaller hardware and computational complexity than the one proposed in [4]. Remark: The proposed decoder architecture for the Lee metric is only mildly more complex than the standard decoder architecture under the Hamming metric. In fact, the proposed architecture can A better strategy is to perform Chien search only for Λ + (x) and Λ (x), while evaluating the derivatives only at the roots of Λ + (x) and Λ (x) respectively. This requires us to perform Chien search twice and to compute derivatives of Λ + (x) and Λ (x) up to order of t 1. Nevertheless, it is still significantly more complex than the proposed erasure solution. 18

19 also be used to decode under the Hamming metric without any extra hardware. This could be useful in case where decoding results under both distance metrics (Hamming and Lee) can effectively be combined to achieve better decoding performance. VI. Conclusions In this paper we have investigated a class of non-binary BCH codes under the Lee metric. We first generalized the results about the Lee distance properties of the class of BCH codes studied in [4] to more general class of BCH codes. Specifically, we showed that for the extended BCH code whose generator polynomial has as roots α, α,...,α t 1 (where α is the primitive element of GF(p m )), the minimum Lee distance is at least t if t p 1 or if the code lies in the prime field GF(p). This lower bound on the minimum Lee distance is twice as large as the designed minimum Hamming distance. We next devised an efficient algebraic decoding algorithm that corrects up to t 1 Lee errors, and in particular, an error evaluation algorithm that avoids applying Chien search multiple times and is thus far more efficient than the approach in [4]. We showed that there may exist at most two codewords within Lee distance t 1 to an arbitrary received word, even though the minimum Lee distance can be as small as p for many codes when p 1 <t p. We also presented an efficient VLSI architecture for the proposed decoder which is only mildly more complex than the popular decoder architectures under the Hamming metric, rendering it very attractive for VLSI implementation. Moreover, the proposed architecture can be re-used for decoding under the Hamming metric without extra hardware; therefore, one can use the proposed architecture to decode under both distance metrics (Lee and Hamming) to achieve superior performance. References [1] C. Y. Lee, Some properties of non-binary error-correcting codes, IRE Trans. Inform. Theory, vol. 4, pp. 77-8, June [] J. C. Chiang and J. K. Wolf, On channels and codes for the Lee metric, Inform. and Control, vol. 19, pp , Sept [3] E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York [4] R. M. Roth and P. H. Siegel, Lee-metric BCH codes and their application to constrained and partialresponse channels, IEEE Trans. Inform. Theory, vol. 40, pp , July [5] E. Byrne, Decoding a class of Lee metric codes over a Galois ring, IEEE Trans. Inform. Theory, vol. 48, pp , Apr. 00. [6] D. Dobbs and R. Hanks, A Modern Course on the Theory of Equations, nd edition, Polygonal Publishing House, Washington, NJ, 199. [7] Y. Wu and C. N. Hadjicostis, On solving composite power polynomial equations, Mathematics of Computation, vol. 74, no. 50, pp , 005. [8] R. E. Blahut, Algebraic Codes for Data Transmission, Cambridge University Press, Cambridge, UK,

20 [9] H. M. Shao, T. K. Truong, L. J. Deutsch, J. H. Yuen, and I. S. Reed, A VLSI design of a pipeline Reed-Solomon decoder, IEEE Trans. Computers, vol. 34, pp , May [10] E. R. Berlekamp, G. Seroussi, and P. Tong, Reed-Solomon Codes and Their Applications, S. B. Wicker and V. K. Bhargava, Eds. Piscataway, NJ: IEEE Press, 1994, A hypersystolic Reed-Solomon decoder. [11] D. V. Sarwate and N. R. Shanbhag, High-speed architectures for Reed-Solomon decoders, IEEE Trans. VLSI Systems, vol. 9, pp , Oct