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Permutation-block LDPC codes: influence of Tanner-graph structure on minimum hamming distance

Benjamin Michael Thorn

Published Jun 28, 2011• 0 citations

Abstract

Only since the mid-1990s, with the increase in computational power, has the\nimportance been realized regarding low-density parity-check (LDPC) codes as leading error\ncorrecting codes. The work in this thesis primarily studies LDPC codes that are de???ned by a\nparity-check matrix comprised of r _ r permutation matrix blocks, that is, permutation-block\nLDPC codes. More specifically, we study their minimum Hamming distance, as this is a\nparameter that affects the performance of linear codes, and thus LDPC codes.\nIt has been proven that there are upper bounds on the minimum Hamming distance of\nquasi-cyclic codes, de???ned by a parity-check matrix comprised of r _ r circulant matrix\nblocks, regardless of how large we allow r. However, we show that if general permutation\nmatrices are used instead of circulant matrices, we are able to exceed the bounds allowed by\nquasi-cyclic codes. Graphical representations of the codes, called Tanner-graphs, are used in\nproving many useful results regarding permutation-block LDPC codes and their minimum\nHamming distance. We show a characterization of the minimum Hamming distance of these\ncodes and we produce an algorithm for constructing permutation-block LDPC codes with\nlarge minimum Hamming distance.

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Only since the mid-1990s, with the increase in computational power, has the\nimportance been realized regarding low-density parity-check (LDPC) codes as leading error\ncorrecting codes. The work in this thesis primarily studies LDPC codes that are de???ned by a\nparity-check matrix comprised of r _ r permutation matrix blocks, that is, permutation-block\nLDPC codes. More specifically, we study their minimum Hamming distance, as this is a\nparameter that affects the performance of linear codes, and thus LDPC codes.\nIt has been proven that there are upper bounds on the minimum Hamming distance of\nquasi-cyclic codes, de???ned by a parity-check matrix comprised of r _ r circulant matrix\nblocks, regardless of how large we allow r. However, we show that if general permutation\nmatrices are used instead of circulant matrices, we are able to exceed the bounds allowed by\nquasi-cyclic codes. Graphical representations of the codes, called Tanner-graphs, are used in\nproving many useful results regarding permutation-block LDPC codes and their minimum\nHamming distance. We show a characterization of the minimum Hamming distance of these\ncodes and we produce an algorithm for constructing permutation-block LDPC codes with\nlarge minimum Hamming distance.

Quasi-cyclic (QC) low-density parity-check (LDPC) codes are an important instance of proto-graph-based LDPC codes. In this paper we present upper bounds on the minimum Hamming distance of QC LDPC codes and study how these upper bounds depend on graph structure parameters (like variable degrees, check node degrees, girth) of the Tanner graph and of the underlying proto-graph. Moreover, for several classes of proto-graphs we present explicit QC LDPC code constructions that achieve (or come close) to the respective minimum Hamming distance upper bounds.

In this correspondence, the construction of low-density parity-check (LDPC) codes from circulant permutation matrices is investigated. It is shown that such codes cannot have a Tanner graph representation with girth larger than , and a relatively mild necessary and sufficient condition for the code to have a girth of or is derived. These results suggest that families of LDPC codes with such girth values are relatively easy to obtain and, consequently, additional parameters such as the minimum distance or the number of redundant check sums should be considered. To this end, a necessary condition for the codes investigated to reach their maximum possible minimum Hamming distance is proposed.

In this correspondence, the construction of low-density parity-check (LDPC) codes from circulant permutation matrices is investigated. It is shown that such codes cannot have a Tanner graph representation with girth larger than 12, and a relatively mild necessary and sufficient condition for the code to have a girth of 6, 8,10, or 12 is derived. These results suggest that families of LDPC codes with such girth values are relatively easy to obtain and, consequently, additional parameters such as the minimum distance or the number of redundant check sums should be considered. To this end, a necessary condition for the codes investigated to reach their maximum possible minimum Hamming distance is proposed.

Quasi-cyclic (QC) low-density parity-check (LDPC) codes are an important instance of proto-graph-based LDPC codes. In this paper we present upper bounds on the minimum Hamming distance of QC LDPC codes and study how these upper bounds depend on graph structure parameters (like variable degrees, check node degrees, girth) of the Tanner graph and of the underlying proto-graph. Moreover, for several classes of proto-graphs we present explicit QC LDPC code constructions that achieve (or come close to) the respective minimum Hamming distance upper bounds. Because of the tight algebraic connection between QC codes and convolutional codes, we can state similar results for the free Hamming distance of convolutional codes. In fact, some QC code statements are established by first proving the corresponding convolutional code statements and then using a result by Tanner that says that the minimum Hamming distance of a QC code is upper bounded by the free Hamming distance of the convolutional code that is obtained by “unwrapping” the QC code.

To reduce the cycles in the Tanner graph which may affect the code performance,a recursive method to construct low-density parity-check(LDPC) codes was proposed.The method used a parity-check matrix of a short LDPC code as its mother matrix upon which a long code was constructed with circulant permutation matrices.By choosing the circulant permutation matrix parameters properly,the number of cycles having a given length in the Tanner graph of the constructed long codes were equal to or less than that of the short code.Either regular or irregular LDPC codes were constructed.Simulation results show that the LDPC codes constructed by this method have very low error-floor and almost no performance degradation due to their special structure as compared with random LDPC codes.

A construction is proposed for low density parity check codes using quadratic permutation polynomials over finite integer rings. Graph isomorphisms and automorphisms are identified and used in an efficient search for good codes. Graphs with girth as large as 12 were found. Upper bounds on the minimum Hamming distance are found algorithmically. The bounds indicate that the minimum distance grows with block length. One of the new codes has a similar error performance as the best known PEG LDPC code. Finally, the new codes are quasi-cyclic