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.

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.

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.

An (a; b) elementary trapping set (ETS), where a and b denote the size and the number of unsatis ed check nodes in the ETS, in uences the performance of low-density parity-check (LDPC) CODES. The smallest size of an ETS in LDPC CODES with column weight 3 and girth 6 is 4. In this paper, we concentrate on a well-known algebraic-based construction of girth-6 QC-LDPC CODES based on powers of a primitive element in a  nite  eld Fq. For this structure, we provide the su cient conditions to obtain 3 n submatrices of an exponent matrix in constructing girth-6 QC-LDPC CODES whose ETSs have the size of at least 5. For structures on  nite  eld Fq, where q is a power of 2, all non-isomorphic 3  n submatrices of the exponent matrix which yield QC-LDPC CODES free of small ETSs are presented.

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 [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.