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IEEE Information Theory Workshop (ITW)

Characterization of graph-cover pseudocodewords of codes over F3

Vitaly Skachek

Published Aug 1, 2010• 8 citations
Publisher

Abstract

Linear-programming pseudocodewords play a pivotal role in our understanding of the linear-programming decoding algorithms. These pseudocodewords are known to be equivalent to the graph-cover pseudocodewords. The latter pseudocodewords, when viewed as points in the multidimensional Euclidean space, lie inside a fundamental cone. This fundamental cone depends on the choice of a parity-check matrix of a code, rather than on the choice of the code itself. The cone does not depend on the channel, over which the code is employed. The knowledge of the boundaries of the fundamental cone could help in studying various properties of the pseudocodewords, such as their minimum pseudoweight, pseudoredundancy of the codes, etc. For the binary codes, the full characterization of the fundamental cone was derived by Koetter et al. However, if the underlying alphabet is large, such characterization becomes more involved. In this work, a characterization of the fundamental cone for codes over F3is discussed.

DOI
10.1109/cig.2010.5592884

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Linear-programming pseudocodewords play a pivotal role in our understanding of the linear-programming decoding algorithms. These pseudocodewords are known to be equivalent to the graph-cover pseudocodewords. The latter pseudocodewords, when viewed as points in the multidimensional Euclidean space, lie inside a fundamental cone. This fundamental cone depends on the choice of a parity-check matrix of a code, rather than on the choice of the code itself. The cone does not depend on the channel, over which the code is employed. The knowledge of the boundaries of the fundamental cone could help in studying various properties of the pseudocodewords, such as their minimum pseudoweight, pseudoredundancy of the codes, etc. For the binary codes, the full characterization of the fundamental cone was derived by Koetter et al. However, if the underlying alphabet is large, such characterization becomes more involved. In this work, a characterization of the fundamental cone for codes over F3is discussed.
Characterization of Graph-cover Pseudocodewords of Codes over F3F_3F3​
Linear-programming pseudocodewords play a pivotal role in our understanding of the linear-programming decoding algorithms. These pseudocodewords are known to be equivalent to the graph-cover pseudocodewords. The latter pseudocodewords, when viewed as points in the multidimensional Euclidean space, lie inside a fundamental cone. This fundamental cone depends on the choice of a parity-check matrix of a code, rather than on the choice of the code itself. The cone does not depend on the channel, over which the code is employed. The knowledge of the boundaries of the fundamental cone could help in studying various properties of the pseudocodewords, such as their minimum pseudoweight, pseudoredundancy of the codes, etc. For the binary codes, the full characterization of the fundamental cone was derived by Koetter et al. However, if the underlying alphabet is large, such characterization becom is more involved. In this work, a characterization of the fundamental cone for codes over F3F_3F3​ is discussed.
The success of modern algorithms for the decoding problem such as message-passing iterative decoding and linear programming decoding lies in their local nature. This feature allows the algorithms to be extremely fast and capable of correcting more errors than guaranteed by the classical minimum distance of the code. Nonetheless, the performance of these decoders depends crucially on the Tanner graph representation of the code. In order to understand this choice of representation, we need to analyze the pseudocodewords of the Tanner graph of a code. These pseudocodewords are outputs of local decoding algorithms which may not be legitimate codewords. In this dissertation, we introduce a lifted fundamental cone and show that there is a one-to-one correspondence between graph cover pseudocodewords of a binary code and integer points in the lifted fundamental cone. We use this fact to prove the rationality of the generating function of the pseudocodewords for a general binary parity-check code. Our approach also yields algorithms for producing this generating function and provides tools for studying the irreducible pseudocodewords. Understanding irreducible pseudocodewords is crucial to determining the best representation of a code. Moreover, combining these techniques with the recent characterization of fundamental cone over F_3, we can analyze ternary parity-check codes. Finally, we make progress in the study of more general nonbinary codes by determining constraints satisfied by all pseudocodewords of a code over F_p where p is prime.
This paper considers pseudocodewords of lowdensity parity-check codes over alphabets with prime cardinality p for use over the p-ary symmetric channel. Pseudocodewords are decoding algorithm outputs that may not be legitimate codewords. Here, we consider pseudocodewords arising from graph cover decoding and linear programming decoding. For codes over the binary alphabet, such pseudocodewords correspond to rational points of the fundamental polytope. They can be characterized via the fundamental cone, which is the conic hull of the fundamental polytope; the pseudocodewords are precisely those integer vectors within the fundamental cone that reduce modulo 2 to a codeword. In this paper, we determine a set of conditions that pseudocodewords of codes over Fp, the finite field of prime cardinality p, must satisfy. To do so, we introduce a class of critical multisets and a mapping, which associates a real number to each pseudocodeword over Fp. The real numbers associated with pseudocodewords are subject to lower bounds imposed by the critical multisets. The inequalities are given in terms of the parity-check matrix entries and critical multisets. This gives a necessary and sufficient condition for pseudocodewords of codes over F2and F3and a necessary condition for those over larger alphabets. In addition, irreducible pseudocodewords of codes over F3are found as a Hilbert basis for the lifted fundamental cone.
An important property of high-performance, low complexity codes is the existence of highly efficient algorithms for their decoding. Many of the most efficient, recent graph-based algorithms, e.g. message passing algorithms and decoding based on linear programming, crucially depend on the efficient representation of a code in a graphical model. In order to understand the performance of these algorithms, we argue for the characterization of codes in terms of a so called fundamental cone in Euclidean space which is a function of a given parity check matrix of a code, rather than of the code itself. We give a number of properties of this fundamental cone derived from its connection to unramified covers of the graphical models on which the decoding algorithms operate. For the class of cycle codes, these developments naturally lead to a characterization of the fundamental polytope as the Newton polytope of the Hashimoto edge zeta function of the underlying graph.
While low-density parity-check (LDPC) codes are near capacity-achieving when paired with iterative decoders, these decoders may not output a codeword due to the existence of pseudocodewords. Thus, pseudocodewords have been studied to give insight into the performance of modern decoders including iterative and linear programming decoders. These pseudocodewords are found to be dependent on the parity-check matrix of the code and the particular decoding algorithm used. In this paper, we consider LP decoding, which has been linked to graph cover decoding, providing functions which capture these pseudocodewords. In particular, we analyze the underlying structure of pseudocodewords from quantum stabilizer codes that arise from LP decoding, quasi-cyclic LDPC codes, and spatially-coupled LDPC codes.
The performance of linear programming decoding is determined by the set of nonzero linear programming pseudocodewords. The minimum pseudoweight of this set of linear programming pseudocodewords is also generally accepted as a good predictor of the performance of iterative message-passing decoding. Since the linear programming decoder has a natural description based on the Tanner graph of a code, linear programming pseudocodewords can be analyzed from a graphical viewpoint. In this paper, two graphical characterizations of linear programming pseudocodewords of cycle codes are provided: one for the entire set of linear programming pseudocodewords, and one for the set of minimal linear programming pseudocodewords. The first of these characterizations is used to determine a formula for the minimum degree of a graph cover necessary to realize a linear programming pseudocodeword of a cycle code, and the second is used to prove a result concerning the asymptotic performance of the linear programming decoder when considering transmission over either the additive white Gaussian noise channel or the binary symmetric channel. Finally, a discussion contrasting these two characterizations both with each other and with Horn's characterization of bad pseudocycles is provided.