Abstract S. Kim et al. have been analyzed the girth of some algebraically structured quasi-cyclic (QC) low-density parity-check (LDPC) Codes, i.e. Tanner (3; 5) of length 5p, where p is a prime of the form 15m+1. In this paper, by extension this method to Tanner (3; 7) Codes of length 7p, where p is a prime of the form 21m+1, the girth values of Tanner (3; 7) Codes will be derived. As an advantage, the rate of Tanner (3; 7) Codes is about 0: 17 more than the rate of Tanner (3; 5) Codes.

In this paper, a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6. For a J ´ L exponent matrix E, the lower bound Q (E) is obtained explicitly such that (J, L) QC LDPC Codes with girth at least 6 exist for any circulant permutation matrix (CPM) size m ³ Q (E). The results show that the exponent matrices constructed with our recursive algorithm have smaller lower-bound than the ones proposed recently with girth 6.

In [30] a class of quasi-cyclic LDPC Codes has been proposed, whose information rate is 1/2. We generalize that construction to arbitrary rates l-1/l and we provide a Grobner basis for their dual Codes, in some generic cases.

In this paper, we define a structure to obtain exponent matrices of girth-8 QC-LDPC Codes with column weight 3. Using the difference matrices introduced by Amirzade et al., we investigate necessary and sufficient conditions which result in a Tanner graph with girth 8. Our proposed method contributes to reduce the search space in recognizing the elements of an exponent matrix. In fact, in this method we only search to obtain one row of an exponent matrix. The other rows are multiplications of that row.

This paper proposes an efficient joint secret key encryption-channel coding cryptosystem, based on regular Extended Difference Family Quasi-Cyclic Low-Density Parity-Check Codes. The key length of the proposed cryptosystem decreases up to 85 percent using a new efficient compression algorithm.Cryptanalytic methods show that the improved cryptosystem has a significant security advantage over Rao-Nam cryptosystem against chosen plaintext attacks, benefiting from an improvement on the structure of the Rao-Nam cryptosystem and proper choices of code parameters. Moreover, the proposed cryptosystem benefits from the highest code rate and a proper error performance.

Joint encryption encoding schemes have been released to fulfill both reliability and security desires in a single step. Using Low Density Parity-Check (LDPC) Codes in joint encryption encoding schemes, as an alternative to classical linear Codes, would shorten the key size as well as improving error correction capability. In this article, a joint encryption encoding scheme using Quasi-Cyclic Low Density Parity-Check (QC-LDPC) Codes based on finite geometry is presented. It is observed that our proposed scheme not only outperforms its predecessors in key size and transmission rate, but also remains secure against all known cryptanalyses of code-based secret key cryptosystems. In this paper, we have proposed an idea to make QC-LDPC based cryptosystems secure against reaction attacks. It is subsequently shown that our scheme benefits from low computational complexity. By taking the advantage of QC-LDPC Codes based on finite geometry, the key size of our scheme is very close to its target security level. In addition, using the proposed scheme, a wide range of desirable transmission rates are achievable. This variety of Codes makes our cryptosystem suitable for a number of different communication and cryptographic standards such as wireless personal area networks (WPAN) and digital video broadcasting (DVB).

In this paper, we consider a class of column-weight two quasi-cyclic low-density parity check Codes in which the girth can be large enough, as an arbitrary multiple of 8. Then we devote a convolutional form to these Codes, such that their generator matrix can be obtained by elementary row and column operations on the parity-check matrix. Finally, we show that the free distance of the convolutional Codes is equal to the minimum distance of their block counterparts.