This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly LDPC codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descenda...

In this article, new regular incidence structures are presented. They arise from sets of conics in the affine plane blown–up at its rational points. The LDPC codes given by these incidence matrices are studied. These sparse incidence matrices turn out to be redundant, which means that their number of rows exceeds their rank. Such a feature is absent from random LDPC codes and is in general inte...

Ahstrrrct-A class of convolutional codes called cross parity check (CPC) codes, useful for the protection of data stored on magnetic tape, is described and analyzed. CPC codes are first explained geometrically; their construction is described in terms of constraining data written onto a tape in such a way that, when lines of varying slope are drawn across the tape, the bits falling on those lin...

In this paper, the notion of trades over finite fields is introduced. In particular, trades over GF (3) (ternary trades) are studied. By considering the incidence matrix of t-subsets vs. k-subsets of a v-set as a parity check matrix of a ternary code, we obtain a new family of codes in which every codeword is a ternary trade. The spectrum of weights of these codes is discussed; a simple and fas...

An algorithm for encoding low-density parity check (LDPC) codes is investigated. The algorithm computes parity check symbols by solving a set of sparse equations, and the triangular factorization is employed to solve the equations efficiently. It is shown analytically and experimentally that the proposed algorithm is more efficient than the Richardson’s encoding algorithm if the code has a smal...

—We present the Low Density Parity Check (LDPC) forward error correction algorithm adapted for the Quantum Key Distribution (QKD) protocol in a form readily applied by developers. A sparse parity check matrix is required for the LDPC algorithm and we suggest using some that have been defined by the IEEE and ETSI standards organizations for use in various communication protocols. We evaluate the...

First unequal error protection (UEP) proposals date back to the 1960’s (Masnick and Wolf; 1967), but now with the introduction of scalable video, UEP develops to a key concept for the transport of multimedia data. The paper presents an overview of some new approaches realizing UEP properties in physical transport, especially multicarrier modulation, or with LDPC and Turbo codes. For multicarrie...

Recently, lattice theory has been considered as useful tool for information theory. It has been also shown that lattice codes can be a strong competitor to traditional capacity achieving coding schemes such as LDPC or Turbo codes. Lattice codes are also attractive due to the fact that code space and signal space are equal, instead of traditional codes, where these spaces are different. However,...

The performance of maximum-likelihood (ML) decoding on the binary erasure channel for finite-length low-density paritycheck (LDPC) codes from two random ensembles is studied. The theoretical average spectrum of the Gallager ensemble is computed by using a recurrent procedure and compared to the empirically found average spectrum for the same ensemble as well as to the empirical average spectrum...

A class of maximum-girth geometrically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes with columnweight J 3 is presented. The method is based on the slope concept between two circulant permutation matrices and the concept of slope matrices. A LDPC code presented by a mv ×ml parity-check matrix H , consisting of m ×m matrices each of which is either a circulant permutation ma...