A Deviation-Based Conditional Upper Bound on the Error Floor Performance for Min-Sum Decoding of Short LDPC Codes

Transcription

1 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 12, DECEMBER A Deviation-Based Conditional Upper Bound on the Error Floor Performance for Min-Sum Decoding of Short LDPC Codes Eric T. Psota, Member, IEEE, Jędrzej Kowalczuk, Student Member, IEEE, and Lance C. Pérez, Senior Member, IEEE Abstract Conditional upper bounds are given for min-sum decoding of low-density parity-check codes in the error floor region. It is generally thought that absorbing sets, i.e., small collections of variable nodes connected to a relatively small number of unsatisfied check nodes, are the primary source of errors in the error floor region. The conditional upper bounds presented here are based on the assumption that all error floor errors are caused by absorbing sets. In order to bound the probability of error associated with each absorbing set, a directed-edge Tanner graph is used to link absorbing sets to low-weight deviations. These low-weight deviations result when a proportionally large number of nodes within a stopping set belong to an absorbing set contained inside the stopping set. A complete collection of the most problematic absorbing sets, and the minimum deviation weights that result from them, are used to derive a conditional upper bound on the probability of error for min-sum decoding of low-density parity-check codes. Simulation results are given to demonstrate the accuracy of the bound. Index Terms LDPC codes, iterative message-passing decoding, stopping sets, absorbing sets, computation trees, deviations. I. INTRODUCTION A well-known problem with low-density parity-check (LDPC) codes is the lack of reliable performance analyses of iterative decoders at high signal-to-noise ratios (SNR) [1]. As a result, the performance of LDPC codes with iterative decoding is largely a mystery at high SNR where computer simulations are computationally intractable. Consequently, the unpredictability of the iterative decoders makes LDPC codes risky candidates for many applications requiring near errorfree performance [2]. Since the re-emergence of low-density parity check codes [3], there have been many attempts to characterize the behavior of iterative decoders. Namely, stopping sets, trapping sets, and absorbing sets have been identified as subsets of variable nodes in the Tanner graph that are particularly problematic to decoding. While stopping sets have been shown to be the precise cause of errors for iterative decoding over the binary erasure channel [4] [5], the cause of errors over the additive Paper approved by H. Pishro-Nik, the Editor for LDPC Codes and Iterative Decoding of the IEEE Communications Society. Manuscript received December 6, 2011; revised May 11, 2012 and July 15, The authors are with the Department of Electrical Engineering, University of Nebraska-Lincoln, Lincoln, NE , USA ( {epsota2, jkowalczuk2, lperez}@unl.edu). Digital Object Identifier /TCOMM /12$31.00 c 2012 IEEE white Gaussian noise (AWGN) channel is not as precisely known. Experimental evidence obtained through computer simulations suggest that absorbing sets are the primary cause of errors in the error floor region for iterative decoding of LDPC codes over the AWGN channel [1] [2]. An error floor can be roughly defined as the decoding performance curve in a range of SNR where the minimum-weight error events become the dominant cause of errors. The precise meaning of minimum-weight error event is a topic of significant interest that is examined in this paper. In 1996, computation trees were introduced as a method for modeling the exact behavior of the two most popular iterative decoders used to decode low-density parity-check codes: the min-sum (MS) and sum-product (SP) decoders [6]. It was shown that by using the entire set of deviations on the computation tree, along with their respective weights, it is theoretically possible to bound the performance of MS decoding. Unfortunately, for most practical LDPC codes it becomes computationally intractable to enumerate the entire set of deviations, even after just a small number of iterations [7]. In this paper, a combination of absorbing set enumeration and deviation-based upper bounds is used to generate a conditional upper bound on the error rate performance of minsum decoding of short low-density parity-check codes in the error floor region. The conditional assumption made in this paper is that all errors in the error floor region are caused by a dominant absorbing set with size less than or equal to some threshold. Enumeration of all small absorbing sets is performed using a variant of the algorithm presented in [8]. Then, error rates associated with each absorbing set are upper bounded by finding the minimum-weight deviation that results from each absorbing set. In Section II, several existing methods are presented for characterizing the error-causing mechanisms of low-density parity-check codes. Then, the new conditional upper bounding method is described in Section III. Finally, simulation results are presented in Section IV to demonstrate the accuracy of the new deviation-based conditional upper bounds, and concluding remarks are given in Section V. II. BACKGROUND Analysis and results presented in this paper are restricted to iterative decoding of binary low-density parity-check codes

2 3568 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 12, DECEMBER 2012 over the binary-input additive white Gaussian noise channel. The length of LDPC codes is denoted by N, andk denotes their dimension. LDPC codes are often defined by their paritycheck matrix H, and a codeword c C is defined as any binary vector c F2 N, such that HcT = 0. Codeword transmission over the AWGN channel is modeled by y = x + n, where x = 2c 1, n is zero-mean white Gaussian noise, and y is the real-valued received vector. The log-likelihood ratio (LLR) vector used for iterative decoding is given by λ i = P Y X(y i 1) P Y X (y i 1) = 2 σ 2 y i for all i =1,...,N. Low-density parity-check codes can also be represented by their bipartite Tanner graph T = (V F, E) [9]. The set of variable nodes V corresponds to the columns of H, in that each column corresponds to a unique variable node v V. Similarly, each row of H corresponds to a unique check node f F. H serves as an adjacency matrix, where nonzero elements in H define edges e E between variable nodes and check nodes in the Tanner graph. Codes with d V edges connected to each variable node and d F edges connected to each check node are referred to as (d V,d F )-regular. In the following sections, existing methods for characterizing the errors of iterative decoding are examined. A. Stopping Sets The notion of stopping sets was first introduced by Forney et al. [10] in It has been shown that the error rates of iteratively decoded LDPC codes on the binary erasure channel (BEC) can be determined exactly from stopping sets. Definition 1 (Stopping Sets [10]): A set S V is a stopping set if all check nodes connected to S are connected to at least two variable nodes in S. During iterative message-passing decoding, information passed from a check node to a variable node is derived from all other variable nodes connected to that check node. If two or more of the variable nodes connected to a check node contain erasures, the check node will send erasure messages to each of its neighboring variable nodes. Therefore, if all variable nodes in S begin with erasures, the check nodes connected to S are incapable of resolving the erasures using message passing. B. Absorbing Sets In an effort to classify sets that cause errors when using min-sum and sum-product decoding, MacKay and Postol identified small collections of variable nodes that resemble codewords, known as trapping sets [11]. Much like codewords, the number of check nodes connected to the trapping set an odd number of times is relatively small. While trapping sets have been experimentally shown to cause errors during iterative decoding [12], their non-restrictive definition makes it difficult to determine which trapping sets will be most problematic to iterative message-passing decoding. To further clarify the ambiguity of trapping sets, Zhang et al. introduced the notion of absorbing sets [1]. Definition 2 (Absorbing Sets [1]): Let A V be a set containing A = a variable nodes. Also, let O(A) F be a set of check nodes such that O(A) = w a, where each check node in the set O(A) has an odd number of edges connected to A. If each variable node in A is connected to strictly more check nodes in F \O(A) than in O(A), theset A is an (a, w a ) absorbing set. A fully absorbing set satisfies the condition that each variable node in V is connected to strictly more check nodes in F \O(A) than in O(A). In simulations, the majority of errors encountered in the error floor region during quantized sum-product decoding of the IEEE 802.3an low-density parity-check code (N = 2048, K = 1723, (6, 32)-regular) can be attributed to absorbing sets [1] [2]. In particular, absorbing sets that are both dominant and minimal appear to be the primary cause of errors as SNR increases. Definition 3 ([2]): An (a, w a ) absorbing set is called minimal if no (a,w a ) absorbing set exists with a <aand w a /a w a /a. An(a, w a ) absorbing set is called dominant if no (a, w a) absorbing set exists with w a <w a. In order to use dominant absorbing sets to predict performance at high SNR, it is often necessary to make assumptions regarding the behavior of iterative decoders. Many approaches, including those based on variations of importance sampling, often assume that the probability of error associated with a particular absorbing set can be derived by partially isolating the variable nodes in the absorbing set from the rest of the variable nodes in the Tanner graph [12] [2]. However, this isolation of variable nodes prevents a complete understanding of the behavior of iterative decoders [8]. C. Computation Trees and Deviations Computation trees and deviations are tools that enable the direct analysis of iterative decoders without requiring the isolation of variable nodes [6]. Computation trees are locally identical to the Tanner graph at all nodes except the leaf nodes. They perfectly model the behavior of MS and SP decoding, in that MS decoding chooses the minimum-cost valid binary configuration on the computation tree, and SP chooses the highest-probability binary assignment to each variable node after considering all valid binary configurations on the computation tree. A valid binary configuration is any binary assignment to each of the variable nodes on the computation tree, such that every check node is connected to an even number of binary 1s. Fig. 1 shows a Tanner graph, and a corresponding computation tree after two iterations of decoding. Node v 1 is the root node and nodes at the bottom of the computation tree are referred to as leaf nodes. An example of a deviation on the computation tree is shown in Fig. 1(b) using thicker edges, and larger, darker variable nodes. Definition 4 (Deviation [6]): A deviation is any set of variable nodes on the computation tree satisfying the following conditions. 1) Each check in the computation tree is adjacent to either two or zero variable nodes in the deviation set. 2) A deviation contains the root node of the computation tree.

3 PSOTA et al.: A DEVIATION-BASED CONDITIONAL UPPER BOUND ON THE ERROR FLOOR PERFORMANCE FOR MIN-SUM DECODING v 1 v 2 v 3 f 1 f 2 f 3 (a) Tanner graph v 1 f 1 f 2 f 3 v 2 v 3 v 2 v 3 v 2 v 3 f 2 f 3 f 2 f 3 f 1 f 3 f 1 f 3 f 1 f 2 f 1 f 2 v 1 v 3 v 1 v 3 v 1v 2 v 1 v 2 v 1 v 3 v 1v 3 v 1 v 2 v 1 v 2 v 1 v 3 v 1 v 3 v 1v 2 v 1v 2 (b) Computation tree Figure 1. Computation tree of a simple repetition code after l =2iterations. An example of a deviation on the computation tree is illustrated using thicker edges, and larger, darker variable nodes. 3) No proper and non-empty subset of variable nodes in the deviation form a valid configuration on the computation tree. The set of deviations on the computation tree can be used to derive an upper bound on the performance of the min-sum decoder. Assuming the all-zeros codeword was transmitted it is necessary, but not sufficient, for at least one deviation δ in the set of all deviations Δ to have negative cost for an error to occur at the root node. It follows that the probability of error at the root node of a computation tree is upper bounded by P (v i =1) δ Δ P (G(δ) < 0) (1) where the cost of a deviation is G(δ) = v i δ λ i. Note that the cost λ i of a variable node v i is added to G(δ) as many times as v i appears in δ. In theory, the deviation bound given by (1) can be used to upper bound the performance of min-sum decoding of finitelength low-density parity-check codes. Unfortunately, the size of the computation trees and the number of configurations on them grows too large for practical analysis. While deviations provide a simplified approach to the analysis of computation trees, in that they are a relatively small subset of the total number of valid binary configurations on the computation tree, the number of deviations also grows exponentially with the number of iterations [7]. D. Related Work In [12], Richardson determines the impact of trapping sets on the probability of error using simulations that inject noise into the trapping set, effectively biasing the variable nodes in the trapping set towards errors. This importance-sampling based approach results in an estimate of the probability of error caused by trapping sets at high SNR. Schlegel and Zhang take a more analytical approach to estimating the probability of error caused by absorbing sets [2]. Their dynamic analysis operates under the assumption that messages passed between nodes within the absorbing set converge slower than messages passed between nodes outside of the absorbing set. This assumption is a key to isolating the nodes within the absorbing set from the rest of the graph. Once isolated, the graph structure of the absorbing set is used to determine the necessary channel conditions for erroneous messages to remain trapped within the absorbing set. An alternative approach to estimating the error rates of minsum decoding of small-to-moderate length LDPC codes was introduced by Xiao and Banihashemi [13]. Their method operates under the assumption that quantized MS decoding is used, and results are given for quantizations of 3, 4, and 5 bits per received channel value for codes of length up to N = 1008 bits. To find errors, short cycles within the Tanner graph are first enumerated. Then, noise is gradually increased within the cycle until MS decoding fails, and the probability of the resulting noise is used to generate codeword error probabilities. Error rate approximations are given in both the waterfall and error floor regions, demonstrating a high level of accuracy when compared to computer simulation results using 3, 4, and 5-bit decoding. The approximations are most accurate for 3-bit decoding, likely due to the fact that the 5-bit error approximation is generated using error patterns obtained from more coarse quantization. Coarse quantization is used for error pattern detection to reduce the size of the search space, which grows exponentially with the number of bits used for quantization. While quantized min-sum decoding can be used to accelerate decoding, it is known that the amount of quantization has a significant effect on the error floor behavior of iterative decoders [12]. Thus, the method presented in this paper assumes that the MS decoder uses real-valued channel information with no quantization. In [14], Stepanov et al. introduced instanton analysis as a method for predicting the error floor for min-sum decoding of low-density parity-check codes, and later generalized the analysis for a variety of other decoders [15] [16]. An instanton is an error-causing noise configuration that lies between a transmitted codeword and a pseudocodeword, such that any amount of shift toward the codeword results in correct decoding. When considering MS decoding, pseudocodewords correspond to valid configurations on the computation tree that include the root node. Instanton analysis proceeds by generating a large set of instantons in order to approximate an error surface surrounding each codeword, where noise configurations contained within the error surface are decoded correctly and noise configurations outside the surface cause errors. Using the resulting error surface, it is possible to approximate the error rate of MS decoding in the error floor region. In order to characterize the error surface, it is necessary to generate a sufficiently large number of instantons. When considering MS decoding for a large number of iterations (l > 20), it becomes increasingly difficult to generate a sufficient set of instantons due to the irregular behavior of the MS output [16]. To overcome this difficulty, the search space can either be limited to linear programming instantons or sets of variable nodes can be isolated from the rest of the Tanner graph based upon the decoding behavior for a small

4 3570 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 12, DECEMBER 2012 number of iterations. In contrast, the method presented in this paper makes no assumptions about the number of iterations, and does not require the isolation of variable nodes from the rest of the Tanner graph. III. THE WEIGHT OF DEVIATIONS INDUCED BY DOMINANT ABSORBING SETS In this section, a new method is used to generate conditional upper bounds on the probability of error for min-sum decoding of low-density parity-check codes in the error floor region. When an absorbing set causes a decoding error, it is often assumed that the noise within the absorbing set is strong enough to prevent the rest of the graph from correcting errors within the absorbing set [2] [12]. The approach taken in this paper is to find low-weight deviations corresponding to each absorbing set. When a decoding error occurs, it is known from Equation (1) that there exists a deviation on the computation tree with negative cost. If the error was caused by an absorbing set, it is very likely that the deviation with cost less than zero is composed of many of the same variable nodes as the absorbing set. By finding minimum-weight deviations in the computation tree that contain a majority of variable nodes in each absorbing set, it is no longer necessary to make assumptions about the effect of nodes outside of the absorbing set, because only the nodes included in the deviation are necessary to generate an upper bound on the decoding performance. The effective weight of the minimum-weight deviation induced by each absorbing set can be used to conditionally upper bound the probability of error for MS decoding. The following steps are required to generate the conditional upper bound. 1) Find all absorbing sets that are both dominant and minimal with size less than or equal to a predefined threshold, along with stopping sets that contains these absorbing sets. 2) For each absorbing set, create a directed-edge subset of the Tanner graph using the stopping set that contains the absorbing set. 3) Orient the directed edges in a such a way that nodes in the absorbing set are given the highest priority, and iteratively adjust the orientations to minimize the weight of the deviation created from this directed graph. 4) Use the collection of deviation weights to upper bound the probability of error for each absorbing set. Each of these steps is examined in detail in the following subsections. Put together, these steps form a complete algorithm, given in Section III-D, for generating the conditional upper bound on the performance of MS decoding of LDPC codes. A. Enumerating Dominant Absorbing Sets There exists many methods for enumerating small absorbing sets of low-density parity-check codes [8] [17] [18] [19]. A variant of the method given in [8] is used in this paper because, along with finding small absorbing sets, it also finds stopping sets that contain each absorbing set. It will be shown later that these stopping sets are very important when determining the weight of deviations induced by absorbing sets. Algorithm 1 Iterative Search Method for Finding Dominant Absorbing Sets and Corresponding Superset Stopping Sets Given T, enumerate a complete set of short cycles {X 1, X 2,...}, where each set X i contains all the variable node indices in cycle i. SetPairs for each cycle X i do m min while m min > 0 do j 1 for n =1,...,N do X i X{ i {n} 0.0 if k X i λ k for all k =1,...,N 1.0 if k V \X i Perform MS decoding for l iterations producing variable node output costs m 1,m 2,...,m N. if min m k <m min then k=1,...,n m min min m k k=1,...,n k min argmin k=1,...,n m k end if X i = X i \{n} end for X i X i {k min } A i,j X i if m min =0then S i X i else j j +1 end if end while SetPairs SetPairs {(A i,1, S i ),...,(A i,j, S i )} end for Remove elements (A i,j, S i ) from SetPairs if A i,j dominant absorbing set. The method used in this paper for enumerating trapping sets is given in Algorithm 1. The first step is enumerating all short cycles in the Tanner graph T, such that each cycle contains no repeated variable nodes. It has been shown that stopping sets must include a set of variable nodes that form a cycle [20], and it has also been shown that the majority of small absorbing sets also include variable nodes from short cycles [19]. In particular, it is shown in [19] that cycles with length G+4,whereG is the girth of the Tanner graph, are sufficient for enumerating low-weight trapping sets. Later, it is shown that, with this initialization, Algorithm 1 produces all minimal and dominant absorbing sets below a given size. For each cycle, Algorithm 1 initializes a set with variable nodes encountered in that cycle. Variable nodes are then added one-by-one that result in the largest reduction in the minimum deviation weight formed using nodes within the set. This method relies on the following two properties of the MS decoder: 1) when initialized with costs of 0.0 and 1.0, the MS output for a particular variable node is the sum of nodes with cost 1.0 in the minimum-weight deviation on a computation tree rooted at that variable node, and 2) when the MS output is

5 PSOTA et al.: A DEVIATION-BASED CONDITIONAL UPPER BOUND ON THE ERROR FLOOR PERFORMANCE FOR MIN-SUM DECODING for a particular variable node, a stopping set exists within the set and it contains that variable node. These two properties allow the MS decoding algorithm to be used as a tool for intelligently forming sets of variable nodes that produce lowweight deviations. B. Generating Directed-Edge Subgraphs from the Stopping Sets It was shown in [8] that absorbing sets alone can not be used to construct deviations. Instead, stopping sets are required to construct deviations. This is because, like deviations, each of the check nodes connected to stopping sets is connected to at least two different variable nodes. Consider an absorbing set A and a stopping set S such that A S. Without loss of generality, let {v 1,v 2,...,v A } Abe the variable nodes in the absorbing set, let {f 1,f 2,...,f N(A) } N(A) be the check nodes connected to A, let{v 1,v 2,...,v S } S be the variable nodes in the stopping set, and let {f 1,f 2,...,f N(S) } N(S) be the check nodes connected to S. Now, consider a directed-edge graph consisting of all the variable nodes in S, denoted by T S. The directed-edge graph T S also contains d fi copies of each check node f i N(S), where d fi is the number of times that f i is connected to S. Thus, T S =(V S,F S,E S ) where V S = {v 1,v 2,...,v S },and F S = {f 1,1,...,f 1,df1,...,f S,1,...,f S,df S }. The following set of constraints apply to the edge set E S. 1) There is exactly one outgoing edge and one incoming edge connected to each check node in N(S). 2) A directed edge (v i f j,k ) E S or (f j,k v i ) E S can exist only if the undirected edge (v i,f j ) E exists in the original Tanner graph. 3) If there is a directed edge (v i f j,k ) E S, then (f j,k v i ) / E S 4) If there exists an edge (v i,f j ) E, wherev i S and f j N(S), then there is a directed edge (v i f j,k ) E S for exactly one k {1,...,d fj }. While the edges directed from variable nodes to check nodes are completely constrained, the edges originating at the check nodes are not. It is this freedom that allows one to design a specific T S that is sufficient for describing the structure of low-weight deviations on the computation tree. To make decisions regarding which variable node each check node s outgoing edge is connected to, a rank is applied to each of the variable nodes in the set S. Each variable node is assigned a unique rank between 1 and S, givenbyr, where the rank of a variable node v i is denoted R(v i ). Consider a ranking strategy where ranks 1 through A are given to the variable nodes in the absorbing set, and ranks A +1 through S are given to remaining variable nodes in S. The ranks are used to decide between multiple options concerning the outgoing edge connections of each check node in T S. For example, when there are two options for outgoing edges given by (f j,k v 1 ) and (f j,k v 2 ), edge (f j,k v 1 ) is chosen if R(v 1 ) < R(v 2 ), and edge (f j,k v 2 ) is chosen otherwise. The following example illustrates the process of generating a directed-edge graph (4, 2) Stopping Set S (2, 2) Absorbing Set A v 1 v 2 v 3 v 4 f 1 f 2 f 3 f 4 f 5 Figure 2. Example of a (4, 2) stopping set S that contains a (2, 2) absorbing set A. v 1 v 2 v 3 v 4 f 1,1 f 1,2 f 2,1 f 2,2 f 2,3 f 3,1 f 3,2 f 3,3 f 4,1 f 4,2 f 5,1 f 5,2 Figure 3. Directed-edge graph T S generated from the (4, 2) stopping set S. from a stopping set and a given rank. Example 5: Consider a stopping set S and the absorbing set A given in Fig. 2. This stopping set can be used to generate the directed-edge graph T S given in Fig. 3. Note that each check node in T S is connected to the graph via a single incoming edge and a single outgoing edge. After assigning ranks R(v 1,v 2,v 3,v 4 ) = (1, 2, 3, 4) to variable nodes v 1,...,v 4, decisions are made regarding which variable node each check node s outgoing edge is connected to. For example, check node f 2,1 could be assigned one of two outgoing edges, (f 2,1 v 2 ) or (f 2,1 v 4 ). Because R(v 2 ) < R(v 4 ), variable node v 2 has higher priority and the check node f 2,1 is connected to v 2 via the outgoing edge (f 2,1 v 2 ). Finally, Fig. 4 shows the deviation created using T S after l = 2 iterations. If each of the variable nodes in S were assigned to a binary 1, and every remaining variable node in the computation tree were assigned a binary 0, the resulting deviation exists on the computation tree as a valid configuration satisfying all check nodes. figures/psota4.eps Figure 4. Deviation on the computation tree constructed using the directed graph T S after l =2iterations. C. Computing the Weights of Deviations While it is easy to assign a rank to each of the variable nodes such that all variable nodes in A are ranked above those not in

6 3572 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 12, DECEMBER 2012 A, this simple strategy will not always result in a minimumweight deviation. It is now necessary to introduce the concept of deviation weight and how the weight of deviations is calculated after a large number of iterations. The weight of any valid configuration on the computation tree can be calculated using the number of different variable nodes in the valid configuration, and their respective multiplicities. It is the proportion of each variable node that determines the weight of the valid configuration. Deviations are simply minimal valid configurations on the computation tree that include the root node. Thus, their weight can also be determined from the proportions of variable nodes that they contain. Recall that the method presented in this paper creates deviations using nodes in a stopping set S. If a i is the multiplicity of variable node v i Swithin a deviation δ S,then the LLR cost of δ S can be modeled by the normal distribution ( ) N a vi, (a vi σ) 2 v i S v i S where σ 2 is the variance of the AWGN. This cost can be rescaled to have distribution given by (( v N a ) 2 i S v i ( v v (a i S v i ) 2, a ) 2 ) i S v i v (a i S v i ) 2 σ2. (2) The mean of the distribution is the weight of the deviation, denoted by ( v w(δ S )= a ) 2 i S v i v (a i S v i ) 2. (3) This particular formulation of deviation weight is analogous to Hamming weight, in the sense that it makes it possible to directly compare the weight of deviations to the weight of codewords. Thus, this formulation lends itself well to minimum distance approximations and asymptotic error rate analysis, as will be demonstrated later in Section IV. To calculate the weight of a deviation using Equation (3) after any number of iterations, it is necessary to obtain the multiplicity of variable nodes within the deviation. Because it is computationally intractable to construct deviations even after a relatively small number of iterations, an alternative method is presented for recursively calculating the multiplicities of variable nodes using the directed-edge graph. In [21], Frey et al. provided a method for computing the number and multiplicity of variable nodes in a computation tree using an adjacency matrix derived from the Tanner graph. A modification of this approach is used with the directed-edge graph T S to derive the multiplicity of variable nodes in the resulting deviation. The recursive method begins with a vector m 0 of length V S + F S. Each vector m i contains the number of nodes at level i in the deviation. The order of node multiplicities in m i is (v 1,...,v VS,f 1,...,f FS ).Leveli =0corresponds to the top level, containing only the root node. Thus, m 0 contains a 1 in the position of the root node and 0s in all other positions. The following recursive operations used to determine the number of nodes on each level i>0 of the deviation are given by m i = { Am i 1 if i is even Am i 1 Bm i 2 if i is odd, (4) where the adjacency matrix of T S is given by A, andb is a matrix used to remove specific parent check nodes from the check node levels in the deviation. The following example demonstrates how to construct A and B from a small directededge graph. Example 6: Consider the directed-edge graph T S shown in Fig. 3. Using a standard node ordering of (v 1,v 2,...,v S,f 1,1,...,f 1,df1,...,f S,1,...,f S,df S ) for the vector of multiplicities m i, the adjacency matrix of T S is given by A = Note that a binary 1 in position (j, k) of A indicates that there is an edge directed from the node at column j to the node at row k. The matrix B is used to remove parent check nodes from the multiplicity vector m in order to enforce the condition that no computation tree node is connected to multiple copies of another. This property is necessarily maintained so that nodes are locally identical to their copies in the Tanner graph. For example, consider the computation tree in Fig. 4. In this example m 1 = [0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0], so level i = 1 contains one copy of f 1,1, f 2,1, and f 4,1. Then, Am 1 = m 2 =[0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], meaning that level i = 2 contains two copies of v 2 and one copy of v 3. Next, the recursive method of Equation (4) requires two operations to determine m 3. First, Am 2 = [0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 0, 0, 2, 1, 0], which includes some nodes that are not in the deviation at level m 3, including f 1,2. The node f 1,2 appears because v 3 has an edge directed toward f 1,2 in T S. However, since f 1,1 is the parent node of v 3 in level i =2of the computation tree, it is not allowed for v 3 to be connected to another copy of f 1. Thus, the matrix B is used to remove copies of the parent node that would otherwise appear in the vector m 3 after multiplying by the adjacency matrix. The matrix B corresponding to the directed-edge graph T S

7 PSOTA et al.: A DEVIATION-BASED CONDITIONAL UPPER BOUND ON THE ERROR FLOOR PERFORMANCE FOR MIN-SUM DECODING of Fig. 3 is given by 0 B = The total number of nodes in a deviation with I levels is given by I M # = m i. (5) i=0 To obtain the proportion of each variable node in the deviation, it is necessary to divide M # by the total number of variable nodes in the deviation. Thus, the proportion of variable nodes in the deviation with I levels is given by M # M % = S j=1 M (6) #,j It should be noted that some deviations will experience fluctuations in M % as I changes. This undesirable behavior is the result of the particular cycle structure of the stopping set and, in terms of decoding, this is often mitigated by using normalized MS decoding [22] to reduce the disproportional effects of the leaf nodes. In order to remove the potential fluctuation, it is necessary to average M % over a range of different I. Empirical evidence suggests that averaging M % over a range of I =82to I = 100 is sufficient to obtain a reliable proportion. D. Computing a Conditional Upper Bound on the Probability of Error The pseudocode in Algorithm 2, comprised of the methods presented in Subsections III-B, and III-C, describes the method used for generating a conditional upper bound on the probability of error for min-sum decoding of low-density parity-check codes. The proposed method is essentially a greedy algorithm that swaps variable node rankings a fixed number of times, max#swaps, with the goal of constructing a directed edge subgraph that produces the minimum-weight deviation. The flow diagram given in Fig. 5 illustrates the various steps and iterations performed within the algorithm. It should be noted that, in order to obtain w(δ S ) in Algorithm 2, it is necessary to substitute the numbers {M %,1,M %,2,...,M %, S } in for {a v1,a v2,...,a v S } in Equation (3). E. Computational Complexity Algorithm 1 begins by enumerating all the short cycles in the Tanner graph by constructing computation trees rooted at each variable node, and pruning variable nodes from the Algorithm 2 Upper Bounding Min-Sum using Dominant Trapping Sets for Directed-Edge Deviation Construction Using the method given in Algorithm 1, enumerate a complete set {(A i,j, S j ) A i,j S j } of minimal, dominant absorbing sets A i,j and stopping sets S j that contain them. UnionBound for each A i,j do Generate random rank vector R for variable nodes in S j, such that R(v k ) < R(v l ) for all v k A i,j and v l / A i,j. w min (δ Sj ) swaps 0 while swaps max#swaps do for all variable node pairs {v k,v l } do Swap R(v k ) and R(v l ). Find the directed-edge Tanner graph T Sj using R. Compute A and B, andfindw(δ Sj ) using Eq. (4). if w(δ Sj ) w min (δ Sj) then w min (δ Sj ) w(δ Sj ) else Swap R(v k ) and R(v l ). end if swaps swaps +1 end for end while UnionBound UnionBound w min (δ S ) end for tree if they appear on a higher layer. By limiting the analysis to cycles of length G +4,where G is the girth of the Tanner graph, the complexity of cycle enumeration is reduced to O(N(d V d F ) (G+4)/2 ) for (d V,d F )-regular LDPC codes. Let the number of short cycles considered by Algorithm 1 be denoted by C. The complexity of Algorithm 1 is O(CN 2 d V l), as the complexity of MS decoding for l iterations is O(Nd V l) and the number of adding variable nodes to a particular cycle is bounded by N. Let the number of set pairs considered by Algorithm 2 be P. For each pair, an adjacency matrix of size ( V S + F S ) ( V S + F S ) is formed, and the recursive operation of Equation (4) is performed I times after each swap. The dimensions of the adjacency matrix are upper bounded by V S + F S (1 + d V )N Thus, the complexity of Algorithm 2isO(P(d V N) 2 I max#swaps). IV. RESULTS The accuracy of the conditional upper bound generated by Algorithm 2 is examined in this section using a variety of short low-density parity-check codes. First, the linear code defined by the parity-check matrix H [20,10] =

8 3574 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 12, DECEMBER 2012 For all trapping sets Undo swap no Max. # swaps made? yes no Parity-check matrix Enumerate cycles Identify trapping sets and superset stopping sets Initialize directed edge subgraph Swap variable node rankings pairwise Lower- or equal-weight deviation? Is unique deviation? yes no Store deviation weight yes Disregard deviation Figure 5. Flow diagram of the proposed method, used to compute an upper bound on the error rate performance of MS decoding of LDPC codes. is used to demonstrate the proposed upper-bounding algorithm. The method given in [8] was used to find all the dominant, minimal absorbing sets with less than four nodes. No absorbing sets with 4 or more nodes were considered, because the minimum distance of the code is equal to 4, and we are interested in evaluating the probability of error associated with non-codeword errors. Table I shows absorbing sets, corresponding superset stopping sets, and the deviation weights found for H [20,10] using the complete approach described by Algorithm 2. Notice that the smallest deviation weights come from subsets of the absorbing set (v 6,v 12,v 15 ), which is also a stopping set. Simulation results given in Fig. 6 show the performance of normalized min-sum decoding of H [20,10] along with the conditional upper bound found using Algorithm 2. Once the Table I DOMINANT ABSORBING SETS OF H [20,10] WITH THEIR CORRESPONDING MINIMUM WEIGHT DEVIATIONS Absorbing Set Stopping Set Min Dev. Weight (2, 2): 017 (5, 3): (2, 2): 316 (5, 3): (2, 2): 119 (5, 3): (2, 2): 214 (5, 3): (2, 2): 318 (4, 0): (2, 2): 417 (5, 3): (2, 2): 615 (3, 1): (2, 2): 717 (3, 1): (2, 2): (3, 1): (2, 2): 811 (5, 3): (2, 2): 819 (4, 0): (2, 2): (4, 0): (2, 2): 914 (5, 3): (2, 2): (5, 3): (2, 2): (4, 0): (2, 2): (5, 3): (3, 1): (5, 3): (3, 1): 3716 (5, 3): (3, 1): 1419 (5, 3): (3, 1): 2514 (5, 3): (3, 1): (5, 3): (3, 1): (3, 1): (3, 1): 6717 (5, 3): (3, 1): (5, 3): (3, 1): (5, 3): (3, 1): (5, 3): (3, 1): (5, 1): (3, 1): 2318 (7, 1): (3, 1): (6, 2): (3, 1): (6, 2): (3, 1): (6, 2): (3, 1): (6, 2): (3, 1): 3811 (5, 3): SNR surpasses 6.5 db, the conditional upper bound is greater than the simulated performance and remains in this position. An encouraging sign for any bound is that its slope matches that of the simulated performance. This indicates that the weights of the minimum-weight errors have been accurately captured in the bound. Algorithm 2 was also used to compute a conditional upper bound on the well-known N = 155, K =64, d min =20, (3, 5)-regular Tanner LDPC code [23]. It has been observed through simulations that the (8, 2) absorbing sets are the most likely cause of errors as the SNR increases [19]. Simulation results given in Table II and Fig. 7 also suggest that the (8, 2) absorbing sets are the most likely cause of error at high SNR,

9 PSOTA et al.: A DEVIATION-BASED CONDITIONAL UPPER BOUND ON THE ERROR FLOOR PERFORMANCE FOR MIN-SUM DECODING [20,10]: Conditional U.B. [20,10]: MS Word Error Rate P w % of Total Errors (5,3) (6,4) (7,3) (8,2) (9,3) (10,2) Other SNR E b /N 0 (db) Figure 6. Probability of word error for normalized MS decoding of an N = 20, K = 10, d min = 4, (3, 6)-regular linear code along with a deviation-based conditional upper bound. Table II DISTRIBUTION OF ABSORBING SET ERRORS ENCOUNTERED DURING NORMALIZED MIN-SUM DECODING OF THE N = 155, K =64, d min =20, (3, 5)-REGULAR TANNER LDPC CODE [23] (a, w a) 4.0dB 4.5dB 5.0dB 5.5dB 6.0dB (5, 3) (6, 4) (7, 3) (8, 2) (9, 3) (10, 2) Other Total as the percentage of errors caused by (8, 2) absorbing sets increases from 24% at 4.0 db to 52% at 5.0 db and 73% at 6.0 db. The method given in [8] was used to find all the dominant, minimal absorbing sets with less than nine variable nodes. The sets in Table III were identified along with the stopping sets that contain them. Using Algorithm 2, a total of 501 unique deviations with weight w(δ S ) < 12.11, and the minimum deviation weight was w(δ S )= The histogram of the deviation weights is given in Fig. 8. It should be noted that different absorbing sets often resulted in the same deviation. Thus, if two deviations contain essentially the same proportion of variable nodes, they are only counted as one deviation in the union bound. The performance of normalized min-sum decoding of the Tanner code is given in Fig. 9. The conditional upper bound crosses the simulated performance at 4.5 db and remains slightly above until the last simulated point at 6.25 db. The error floor region of this code appears to begin at 5.0 db, which is also the point where the conditional upper bound begins hovering just above the simulated frame error rate. The convergence of Algorithm 2 is shown in Fig. 10 for up E b /N 0 (db) Figure 7. Percentages of absorbing set types observed during normalized MS decoding of the N = 155, K =64, d min =20, (3, 5)-regular Tanner LDPC code [23]. Table III DOMINANT ABSORBING SETS FOUND USING ALGORITHM 1 ON THE N = 155, K =64, d min =20, (3, 5)-REGULAR TANNER LDPC CODE Absorbing Set (a, w a) Multiplicity (4, 4) 465 (5, 3) 155 (6, 4) 930 (7, 3) 930 (8, 2) 465 to 4000 swaps when operating on the length N = 155 Tanner code. The algorithm does not have guaranteed convergence, however, the exponential decrease in the minimum deviation weight is a strong indicator of convergent behavior. While the mean output decreases from to (17.9%) between swaps 1 to 100, it only decreases from to (0.0045%) between swaps 2000 and 4000, indicating that an increased number of swaps will not significantly change the output. The conditional upper bound was also generated for two randomly generated low-density parity-check codes: a (3, 6)- regular code with length N = 1008 and dimension K = 504, anda(3, 5)-regular code with length N = 1000 and dimension K = 400. It is, in general, difficult to observe the error floor using computer simulations on codes with these block lengths. Therefore, each of these particular codes was chosen from a set of ten randomly generated LDPC codes due to their relatively high error floors. Simulation results and conditional upper bounds are shown in Fig. 11. Both codes appear to reach an error floor at 3 db, from which point on the conditional upper bound generated by Algorithm 2 lies directly above the simulated word error rate with a similar slope. The similarity of the slopes indicates that the minimumweight deviation found by Algorithm 2 accurately predicts the effect of the most probable error event on the word error rate of MS decoding. Simulation results are also given in Fig. 11 for a girth

10 3576 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 12, DECEMBER [155,64]: Conditional U.B. [155,64]: MS Word Error Rate Multiplicity P w Minimum Deviation Weights SNR E b /N 0 (db) Figure 8. Minimum deviation weights of the absorbing sets of the N = 155, K =64, d min =20, (3, 5)-regular Tanner LDPC code [23]. Figure 9. Probability of word error for normalized MS decoding of the N = 155, K =64, d min =20, (3, 5)-regular Tanner LDPC code [23] along with a deviation-based conditional upper bound. G = 10 (3, 6)-regular low density parity-check code with length N = 1008 and dimension K = 504. This code was generated using the algorithm in [24], and has been shown to significantly outperform randomly-generated LDPC codes at low error rates. As shown in the figure, simulation results have been obtained for this code down to a word error rate of approximately P w =10 8, at which point there does not appear to be an error floor. Further simulation results at higher SNRs are prohibited by an exponential increase in the time required to simulate lower word error rates. One can however compute a conditional upper bound, also shown in Fig. 11, using the method described here. Based on this upper bound, it is reasonable to speculate that an error floor will be reached at a word error rate of approximately P w =10 12 and that its slope will be similar to that of the bound. This reasoning leads to the estimate of the code s performance at lower word error rates shown in Fig. 11 and demonstrates the utility of the conditional upper bound when simulation results are not obtainable. The convergence behavior of the Algorithm 2 is shown in Fig. 12 for the two randomly generated low-density paritycheck codes and the G =10code. A total of max#swaps = 20,000 was used to ensure that the algorithm had reached a sufficient level of convergence. For the N = 1008, K = 504, (3, 6)-regular random LDPC code, the mean output decreases from to (62.3%) between swaps 1 to 2000, and the mean output decreases from to (0.93%) between swaps 10,000 and 20,000. For the N = 1000, K = 400, (3, 5)-regular random LDPC code, the mean output decreases from to (66.3%) between swaps 1 to 2000, and the mean output decreases from to (3.08%) between swaps 10,000 and 20,000. For the N = 1008, K = 504, (3, 6)-regular G =10LDPC code, the mean output decreases from to (46.1%) between swaps 1 to 2000, and the mean output decreases from to (0.23%) between swaps 10,000 and 20,000. These results demonstrate that the algorithm produces negligible decreases in the minimum deviation weight after a Min Deviation Weight [155,64]: Trapping Set Samples [155,64]: Mean Deviation Weight Number of Swaps Figure 10. Convergence behavior of Algorithm 2 for the N = 155, K = 64, d min =20, (3, 5)-regular Tanner LDPC code [23]. sufficient number of rank swaps. A common concern with greedy algorithms that perform iterative minimization of some function is that they are getting stuck at local minima. To test whether or not this was occurring, a single (10, 2) dominant absorbing set was chosen from the N = 1000 LDPC code, and Algorithm 2 operated on that particular set using 100 randomly assigned initialization ranks. The results, given in Fig. 13, show that the algorithm converges to nearly the same minimum deviation weight in each of the 100 trials after a maximum of 12,300 swaps. It is worth noting that, in each of the trials, the resulting minimumweight deviation consisted of nearly identical portions of the same variable nodes. Thus, the deviations obtained from the algorithm were the same in each trial. V. CONCLUSION Analytical methods are necessary to predict the performance of low-density parity-check codes with iterative decoding beyond the reach of computer simulations. Absorbing sets have emerged as the most likely candidates for the cause of errors at high SNR [1] [2]. However, the precise probability of errors caused by absorbing sets is often estimated using

11 PSOTA et al.: A DEVIATION-BASED CONDITIONAL UPPER BOUND ON THE ERROR FLOOR PERFORMANCE FOR MIN-SUM DECODING [1008,504]Rand: Conditional U.B [1008,504]Rand: MS Word Error Rate [1000,400]Rand: Conditional U.B [1000,400]Rand: MS Word Error Rate [1008,504]G=10: Conditional U.B [1008,504]G=10: MS Word Error Rate [1008,504]G=10: Estimated WER Min Deviation Weight [1008,504]Rand: Trapping Set Samples [1008,504]Rand: Mean Deviation Weight 20 P w Min Deviation Weight Number of Swaps x (a) Random: N = 1008, K = 504, (3, 6)-regular [1000,400]Rand: Trapping Set Samples [1000,400]Rand: Mean Deviation Weight SNR E b /N 0 (db) Figure 11. Normalized MS decoding performance and deviation-based conditional upper bound for two randomly generated LDPC codes with parameters N = 1008 K = 504 (3, 6)-regular and N = 1000 K = 400 (3, 5)-regular, and a G =10N = 1008 K = 504 (3, 6)-regular LDPC code generated using the algorithm in [24]. An estimate of the word error rate is provided for the G =10LDPC code using the conditional upper bound. importance sampling [12], or by making assumptions about the messages passed during iterative decoding [2]. Here, a new method was presented for generating a conditional upper bound on the performance of iterative decoding of short low-density parity-check codes. This method draws on the results of both absorbing sets and deviations [6]. Specifically, the proposed method makes the common assumption that errors at high SNR are caused by absorbing sets, and then upper bounds the probability of error associated with each absorbing set by computing the minimum weight of the resulting deviation. Finally, a union upper bound is derived from the collection of minimum deviation weights to predict the performance of the LDPC code with iterative decoding in the error floor region. Simulations show that the conditional upper bound does indeed tightly bound the decoding performance in the error floor region. Work is currently being done on a more thorough analysis of the algorithm for finding the minimum-weight deviation for each absorbing set. While the algorithm appears to converge on a minimum weight deviation after a finite number of iterations, its convergence is not guaranteed. VI. ACKNOWLEDGEMENTS This work was supported in part by NASA EPSCoR Grant NNX09AQ08A, AFOSR grant FA , and Department of Education grant P200A REFERENCES [1] Z. Zhang, L. Dolecek, B. Nikolić, V. Anantharam, and M. J. Wainwright, Design of LDPC decoders for improved low error rate performance: Min Deviation Weight Number of Swaps x (b) Random: N = 1000, K = 400, (3, 5)-regular [1008,504]G=10: Trapping Set Samples [1008,504]G=10: Mean Deviation Weight Number of Swaps x 10 4 (c) G =10: N = 1008, K = 504, (3, 6)-regular Figure 12. Convergence behavior of Algorithm 2 using two randomly generated LDPC codes and the G = 10 LDPC code generated using the algorithm in [24]. Min Deviation Weight [1000,400]: Trapping Set Samples [1000,400]: Mean Deviation Weight Number of Swaps x 10 4 Figure 13. Convergence behavior of Algorithm 2 using 100 different random rank R initializations on a (10, 2) dominant absorbing set. The (10, 2) dominant absorbing set produces the minimum weight deviation for a randomly generated LDPC codes N = 1000 K = 400 (3, 5)-regular. quantization and algorithm choices, IEEE Trans. Commun., vol. 57, pp. 1 12, Nov [2] C. Schlegel and S. Zhang, On the dynamics of the error floor behavior in (regular) LDPC codes, IEEE Trans. Inf. Theory, vol. 56, pp , July [3] D. J. C. MacKay and R. M. Neal, Near Shannon limit performance of low-density parity check codes, IEE Electron. Lett., vol. 32, pp , Aug

12 3578 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 12, DECEMBER 2012 [4] G. D. Forney, Jr., R. Koetter, F. R. Kschischang, and A. Reznik, On the effective weights of pseudocodewords for codes defined on graphs with cycles, in Codes, Systems, and Graphical Models (Minneapolis, MN, 1999), vol. 123 of IMA Vol. Math. Appl., pp Springer, [5] D. Changyan, D. Proetti, I. E. Telatar, T. J. Richardson, and R. L. Urbanke, Finite length analysis of low-density parity-check codes on the binary erasure channel, IEEE Trans. Inf. Theory, vol. 48, pp , June [6] N. Wiberg, Codes and decoding on general graphs, Ph.D. thesis, Linköping University, Linköping, Sweden, [7] E. Psota and L. C. Pérez, LDPC decoding and code design on extrinsic trees, in Proc Int. Symp. Inf. Theory. [8] E. Psota and L. C. Pérez, The manifestation of stopping sets and absorbing sets as deviations on the computation trees of LDPC codes, J. Electrical Computer Eng., vol. 2010, Article ID , p. 17, [9] R. M. Tanner, A recursive approach to low complexity codes, IEEE Trans. Inf. Theory, vol. 27, pp , Sep [10] J. Forney, Codes on graphs: normal realizations, IEEE Trans. Inf. Theory, vol. 47, pp , Feb [11] D. J. C. MacKay and M. S. Postol, Weaknesses of Margulis and Ramanujan-Margulis low-density parity-check codes, in Electronic Notes in Theoretical Computer Science, p Elsevier, [12] T. J. Richardson, Error floors of LDPC codes, in Proc Allerton Conf. Commun., Control, Computing. [13] H. Xiao and A. Banihashemi, Error rate estimation of finite-length low-density parity-check codes decoded by soft-decision iterative algorithms, in Proc IEEE Int. Symp. Inf. Theory, pp [14] M. Stepanov and M. Chertkov, Instanton analysis of low-density paritycheck codes in the error-floor regime, in Proc IEEE Int. Symp. Inf. Theory, pp [15] M. Chertkov and M. Stepanov, Pseudo-codeword landscape, in Proc IEEE Int. Symp. Inf. Theory, pp [16] S. Chilappagari, M. Chertkov, M. Stepanov, and B. Vasic, Instantonbased techniques for analysis and reduction of error floors of LDPC codes, IEEE J. Sel. Areas Commun., vol. 27, pp , Aug [17] C. Cole, S. Wilson, E. Hall, and T. Giallorenzi, A general method for finding low error rates of LDPC codes, in arxiv.org/abs/cs/ , [18] C.-C. Wang, S. Kulkarni, and H. Poor, Finding all small errorprone substructures in LDPC codes, IEEE Trans. Inf. Theory, vol. 55, pp , May [19] M. Dehkordi and A. Banihashemi, An efficient algorithm for finding dominant trapping sets of LDPC codes, in Proc Int. Symp. Turbo Codes Iterative Inf. Processing, pp [20] T. Tian, C. Jones, J. Villasenor, and R. Wesel, Selective avoidance of cycles in irregular LDPC code construction, IEEE Trans. Commun., vol. 52, pp , Aug [21] B. J. Frey, R. Koetter, and A. Vardy, Signal-space characterization of iterative decoding, IEEE Trans. Inf. Theory, vol. 47, pp , Feb [22] J. Chen, A. Dholakia, E. Eleftheriou, M. Fossorier, and X.-Y. Hu, Reduced-complexity decoding of LDPC codes, IEEE Trans. Commun., vol. 53, pp , Aug [23] R. Tanner, D. Sridhara, A. Sridharan, T. Fuja, and J. Costello, LDPC block and convolutional codes based on circulant matrices, IEEE Trans. Inf. Theory, vol. 50, pp , Dec [24] E. Psota and L. C. Pérez, Iterative construction of regular LDPC codes from independent tree-based minimum distance bounds, IEEE Commun. Lett., vol. 15, pp , Mar Eric T. Psota is a Research Assistant Professor in the Department of Electrical Engineering at the University of Nebraska-Lincoln (UNL). He received his BS, MS, and PhD degrees in electrical engineering from the University of Nebraska-Lincoln in 2004, 2006, and 2010, respectively. His research interests include multi-view computer vision, biomedical image processing, and the study of iterative decoders for error control coding. He is a member of the IEEE and Eta Kappa Nu. Jędrzej Kowalczuk is a Research Assistant in the Department of Electrical Engineering at the University of Nebraska-Lincoln (UNL). He received his BS and MS degrees in 2007, both in computer science, from the Technical University of Szczecin, Poland. Currently, he is pursuing a PhD degree in electrical engineering at the University of Nebraska-Lincoln. His research interests include multi-view computer vision, image processing, and high-performance computing. He is a student member of the IEEE. Lance C. Pérez has been a faculty member in the Department of Electrical Engineering at the University of Nebraska-Lincoln (UNL) since August 1996, where he is currently a Professor of electrical engineering. His research interests are in the areas of error control coding, information, image processing, and engineering education. He received his BS degree in electrical engineering from the University of Virginia, and the MS and PhD degrees in electrical engineering from the University of Notre Dame. He is a senior member of the IEEE and a member of ASEE, Tau Beta Pi, and Eta Kappa Nu.