Spectral Graph Analysis of Quasi-Cyclic Codes

IEEE Transactions on Information Theory, 2000

We present a new class of sparse and easily invertible circulant matrices that can have a sparse inverse though not being permutation matrices. Their study is useful in the design of quasi-cyclic low-density generator matrix codes, that are able to join the inner structure of quasi-cyclic codes with sparse generator matrices, so limiting the number of elementary operations needed for encoding. Circulant matrices of the proposed class permit to hit both targets without resorting to identity or permutation matrices that may penalize the code minimum distance and often cause significant error floors.

2021

Computing Research Repository - CORR, 2010

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 descendant codes are developed, including the characterizations of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite geometry LDPC codes using the results developed in first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping sets of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite geometry and finite field cyclic and qua...

2010 Information Theory and Applications Workshop (ITA), 2010

Quasi-cyclic codes are the most promising class of structured LDPC codes due to their ease of implementation and excellent performance over noisy channels when decoded with message-passing algorithms as extensive simulation studies have shown. An approach for constructing quasi-cyclic LDPC codes based on Latin squares over finite fields is presented. By analyzing the parity-check matrices of these codes, expressions for their ranks are derived. Experimental results show that, with iterative decoding algorithms, the constructed codes perform very well over the AWGN and the binary erasure channels. Consider the Galois field GF( ). Let ¡ be a primitive element of GF( ). Then, the powers of ¡ ¦ ¢¡ ¤£ ¦¥ ¨ § © ¦ ¡ § ¦ ¡ ¦ ¢¡ ¦ ¦ ¢¡ "! #£ $ , give all the elements of GF( ) and Let ' be a © )( 10 © 2( circulant permutation matrix (CPM) whose top row is given by the over GF(2) where the components are labeled from 0 to ¤( 54 and the single 1-component is located at the 1st position. Then ' consists of the ( %( )-tuple © © © 6 7© and its © %( 84 right cyclic shifts as rows. For @9 BA DC E , let ' GF § H' I0 P' I0 Q #0 P' be the product of ' with itself A times, called the A th power of ' . Then, ' @F is also a © R( ¤0 © R( CPM whose top row has a single 1-component at the A th position. For A 1 § S 2( , . Then the set a § cb d' 2 ¦ ' ¦ ¢' @ ¦ ¦ ' P! T£ ¦ fe of CPMs forms a cyclic group of order G( under matrix multiplication over GF(4 ) with ' G! #£ % #£ $F as the multiplicative inverse of ' @F and ' 2 as the identity element. For © g9 hA PC i G( , we represent the nonzero element ¡ pF of GF( ) by the © 6( q0 © 6( CPM ' PF . This matrix representation is referred to as the © U( -fold binary matrix dispersion (or simply binary matrix dispersion) of ¡ F . Since there are $( nonzero elements in GF( ) and there are exactly r( different CPMs over GF(4 ) of size © r( s0 © r( , there is a one-to-one correspondence between a nonzero element of GF( ) and a CPM of size © 8( R0 © P( . For a nonzero element t in GF( ), we use the notation u © t to denote its binary matrix dispersion. If t P § H¡ pF , then u © t D § v' @F . For the 0-element of GF( ), its binary matrix dispersion is defined as the © R( w0 © U( zero matrix (ZM), denoted by ' £ %¥ . Consider a x 60 y matrix over GF( ), © 2( &0 © 2( CPM u © F p k WX qu © A ¦ r over GF(2) and each 0-entry into a © 1( s0 © 1( ZM, we obtain the following x t0 ty array of CPMs and/or ZMs over GF(2) of size © )( W0 © )( : £ u § wv u F p © 4 ( $0 © 4 d ( identity matrix. For © g9 iA GC 4 ( , applying and 2£ % to the © 4 ( 10 © 4 H( CPM ' @F over GF(2), we obtain the following © 4 d n( W0 © 4 v( matrix over GF(4 d ):

Finite Fields and Their Applications, 2017

Generalized quasi-cyclic (GQC) codes form a natural generalization of quasi-cyclic (QC) codes. They are viewed here as mixed alphabet codes over a family of ring alphabets. Decomposing these rings into local rings by the Chinese Remainder Theorem yields a decomposition of GQC codes into a sum of concatenated codes. This decomposition leads to a trace formula, a minimum distance bound, and to a criteria for the GQC code to be self-dual or to be linear complementary dual (LCD). Explicit long GQC codes that are LCD, but not QC, are exhibited.

2018

Low Density Parity Check Codes (LDPC) are a class of linear error-correcting codes which have shown ability to approach or even to reach the capacity of the transmission channel. This class of code approaches asymptotically the fundamental limit of information theory more than the Turbo Convolutional codes. It’s ideal for long distance transmission satellite, mobile communications and it’s also used in storage systems. In this paper, a new method for constructing quasicyclic low density parity-check (QC-LDPC) codes derived from cyclic codes is presented. The proposed method reduces the incidence vectors, by eliminating the conjugates lines in parity-check matrix of the derived cyclic code to construct circulant shifting sub-matrices. In the end, this method produces a large class of regular LDPC codes of quasi-cyclic structure having very low density, high coding rates and Tanner graphs which have no short cycles with girth of at least 6. Performance with computer simulations are al...

Advances in Mathematics of Communications, 2012

A list decoding algorithm for matrix-product codes is provided when C1,. .. , Cs are nested linear codes and A is a non-singular by columns matrix. We estimate the probability of getting more than one codeword as output when the constituent codes are Reed-Solomon codes. We extend this list decoding algorithm for matrix-product codes with polynomial units, which are quasi-cyclic codes. Furthermore, it allows us to consider unique decoding for matrix-product codes with polynomial units.

IEEE Transactions on Information Theory, 2004

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.

Information Theory and Applications, 2012

A construction of binary and non-binary quasi-cyclic (QC)-LDPC codes based on partitions of finite fields of characteristic 2 is proposed. The construction is carried out in the Fourier transform domain. The parity-check matrices of these QC-LDPC codes are arrays of circulant permutation matrices. The ranks of these arrays are analyzed and combinatorial expressions are derived. Example codes are given and

IEEE Transactions on Communications, 2015

This paper presents two simple and very flexible methods for constructing non-binary (NB) quasi-cyclic (QC) LDPC codes. The proposed construction methods have several known ingredients including base array, masking, binary to nonbinary replacement and matrix-dispersion. By proper choice and combination of these ingredients, NB-QC-LDPC codes with excellent performance can be constructed. The constructed codes can be decoded with a reduced-complexity iterative decoding scheme which significantly reduces the hardware implementation complexity. I. INTRODUCTION L DPC CODES, discovered in 1962 [1] and rediscovered in late 1990's [2], [3], are currently the most promising coding technique for error control in communication and data storage systems due to their capacity-approaching performances and practically implementable decoding algorithms. Since their rediscovery, a great deal of research effort has been expended in design, analysis, decoding, generalizations and applications of these amazing codes. However, most of the research effort has been focused only on binary LDPC codes. Research effort expended in non-binary (NB) LDPC codes is far less than that devoted to their binary counterparts. This lack of enthusiasm in NB-LDPC codes may be due to the concern of their decoding complexity in both computation and hardware implementation. NB-LDPC codes do have advantages over their binary counterparts for communication and data storage channels where both random and burst errors occur simultaneously. Furthermore, for using high-order modulations with large signal constellations for communication, it is very natural to use NB-LDPC codes. For all of these reasons, NB-LDPC codes deserve more attention and research effort. There are various types of LDPC codes. Among them, the most preferred type of LDPC codes for practical applications in communication and storage systems are LDPC codes with quasi-cyclic (QC) structure, called QC-LDPC codes [4], [5]. A QC-LDPC code is given by the null space of an array H of sparse circulant matrices of the same size over a finite field, binary or non-binary. In most of the constructions of QC-LDPC codes, the sparse circulant matrices in the paritycheck array H of a QC-LDPC code are circulant permutation matrices (CPMs). Such a parity-check array H of a QC-LDPC