Strong Secrecy for Erasure Wiretap Channels

We show that duals of certain low-density parity-check (LDPC) codes, when used in a standard coset coding scheme, provide strong secrecy over the binary erasure wiretap channel (BEWC). This result hinges on a stopping set analysis of ensembles of LDPC codes with block length $n$ and girth $\geq 2k$, for some $k \geq 2$. We show that if the minimum left degree of the ensemble is $l_\mathrm{min}$, the expected probability of block error is \calO(\frac{1}{n^{\lceil l_\mathrm{min} k /2 \rceil - k}}) when the erasure probability $\epsilon < \epsilon_\mathrm{ef}$, where $\epsilon_\mathrm{ef}$ depends on the degree distribution of the ensemble. As long as $l_\mathrm{min} > 2$ and $k > 2$, the dual of this LDPC code provides strong secrecy over a BEWC of erasure probability greater than $1 - \epsilon_\mathrm{ef}$.Comment: Submitted to the Information Theory Workship (ITW) 2010, Dubli

Growth Rate of the Weight Distribution of Doubly-Generalized LDPC Codes: General Case and Efficient Evaluation

The growth rate of the weight distribution of irregular doubly-generalized LDPC (D-GLDPC) codes is developed and in the process, a new efficient numerical technique for its evaluation is presented. The solution involves simultaneous solution of a 4 x 4 system of polynomial equations. This represents the first efficient numerical technique for exact evaluation of the growth rate, even for LDPC codes. The technique is applied to two example D-GLDPC code ensembles.Comment: 6 pages, 1 figure. Proc. IEEE Globecom 2009, Hawaii, USA, November 30 - December 4, 200

Weight Distribution for Non-binary Cluster LDPC Code Ensemble

In this paper, we derive the average weight distributions for the irregular non-binary cluster low-density parity-check (LDPC) code ensembles. Moreover, we give the exponential growth rate of the average weight distribution in the limit of large code length. We show that there exist $(2,d_c)$-regular non-binary cluster LDPC code ensembles whose normalized typical minimum distances are strictly positive.Comment: 12pages, 6 figures, To be presented in ISIT2013, Submitted to IEICE Trans. Fundamental

Capacity-Achieving Codes with Bounded Graphical Complexity on Noisy Channels

We introduce a new family of concatenated codes with an outer low-density parity-check (LDPC) code and an inner low-density generator matrix (LDGM) code, and prove that these codes can achieve capacity under any memoryless binary-input output-symmetric (MBIOS) channel using maximum-likelihood (ML) decoding with bounded graphical complexity, i.e., the number of edges per information bit in their graphical representation is bounded. In particular, we also show that these codes can achieve capacity on the binary erasure channel (BEC) under belief propagation (BP) decoding with bounded decoding complexity per information bit per iteration for all erasure probabilities in (0, 1). By deriving and analyzing the average weight distribution (AWD) and the corresponding asymptotic growth rate of these codes with a rate-1 inner LDGM code, we also show that these codes achieve the Gilbert-Varshamov bound with asymptotically high probability. This result can be attributed to the presence of the inner rate-1 LDGM code, which is demonstrated to help eliminate high weight codewords in the LDPC code while maintaining a vanishingly small amount of low weight codewords.Comment: 17 pages, 2 figures. This paper is to be presented in the 43rd Annual Allerton Conference on Communication, Control and Computing, Monticello, IL, USA, Sept. 28-30, 200

On the Growth Rate of the Weight Distribution of Irregular Doubly-Generalized LDPC Codes

In this paper, an expression for the asymptotic growth rate of the number of small linear-weight codewords of irregular doubly-generalized LDPC (D-GLDPC) codes is derived. The expression is compact and generalizes existing results for LDPC and generalized LDPC (GLDPC) codes. Assuming that there exist check and variable nodes with minimum distance 2, it is shown that the growth rate depends only on these nodes. An important connection between this new result and the stability condition of D-GLDPC codes over the BEC is highlighted. Such a connection, previously observed for LDPC and GLDPC codes, is now extended to the case of D-GLDPC codes.Comment: 10 pages, 1 figure, presented at the 46th Annual Allerton Conference on Communication, Control and Computing (this version includes additional appendix

Second-Order Weight Distributions

A fundamental property of codes, the second-order weight distribution, is proposed to solve the problems such as computing second moments of weight distributions of linear code ensembles. A series of results, parallel to those for weight distributions, is established for second-order weight distributions. In particular, an analogue of MacWilliams identities is proved. The second-order weight distributions of regular LDPC code ensembles are then computed. As easy consequences, the second moments of weight distributions of regular LDPC code ensembles are obtained. Furthermore, the application of second-order weight distributions in random coding approach is discussed. The second-order weight distributions of the ensembles generated by a so-called 2-good random generator or parity-check matrix are computed, where a 2-good random matrix is a kind of generalization of the uniformly distributed random matrix over a finite filed and is very useful for solving problems that involve pairwise or triple-wise properties of sequences. It is shown that the 2-good property is reflected in the second-order weight distribution, which thus plays a fundamental role in some well-known problems in coding theory and combinatorics. An example of linear intersecting codes is finally provided to illustrate this fact.Comment: 10 pages, accepted for publication in IEEE Transactions on Information Theory, May 201