Spectral Graph Analysis of Quasi-Cyclic Codes

📝 Abstract

In this paper we analyze the bound on the additive white Gaussian noise channel (AWGNC) pseudo-weight of a (c,d)-regular linear block code based on the two largest eigenvalues of H^T H. In particular, we analyze (c,d)-regular quasi-cyclic (QC) codes of length rL described by J x L block parity-check matrices with circulant block entries of size r x r. We proceed by showing how the problem of computing the eigenvalues of the rL x rL matrix H^T H can be reduced to the problem of computing eigenvalues for r matrices of size L x L. We also give a necessary condition for the bound to be attained for a circulant matrix H and show a few classes of cyclic codes satisfying this criterion.

💡 Analysis

In this paper we analyze the bound on the additive white Gaussian noise channel (AWGNC) pseudo-weight of a (c,d)-regular linear block code based on the two largest eigenvalues of H^T H. In particular, we analyze (c,d)-regular quasi-cyclic (QC) codes of length rL described by J x L block parity-check matrices with circulant block entries of size r x r. We proceed by showing how the problem of computing the eigenvalues of the rL x rL matrix H^T H can be reduced to the problem of computing eigenvalues for r matrices of size L x L. We also give a necessary condition for the bound to be attained for a circulant matrix H and show a few classes of cyclic codes satisfying this criterion.

📄 Content

arXiv:0908.1966v2 [cs.IT] 16 Aug 2009 Spectral Graph Analysis of Quasi-Cyclic Codes Roxana Smarandache Department of Mathematics and Statistics San Diego State University San Diego, CA 92182, USA Email: rsmarand@sciences.sdsu.edu Mark F. Flanagan Department of Electronic Engineering University College Dublin Belfield, Dublin 4, Ireland Email: mark.flanagan@ieee.org Abstract—In this paper we analyze the bound on the additive white Gaussian noise channel (AWGNC) pseudo-weight of a (c, d)-regular linear block code based on the two largest values λ1 > λ2 of the eigenvalues of HTH: wmin p (H) ⩾n 2c−λ2 λ1−λ2 . [6]. In particular, we analyze (c, d)-regular quasi-cyclic (QC) codes of length rL described by J × L block parity-check matrices with circulant block entries of size r × r. We proceed by showing how the problem of computing the eigenvalues of the rL × rL matrix HTH can be reduced to the problem of computing eigenvalues for r matrices of size L × L. We also give a necessary condition for the bound to be attained for a circulant matrix H and show a few classes of cyclic codes satisfying this criterion. Index Terms—Low-density parity-check codes, pseudo- codewords, pseudo-weights, eigenvalues, eigenvectors. I. INTRODUCTION Low-density parity-check (LDPC) codes offer excellent tradeoffs between performance and complexity for error cor- rection in communication systems. Quasi-cyclic (QC) LDPC codes in particular have proved extremely attractive due to their implementation advantages, both in encoding and decod- ing [1], [2], [3]. Many analyses of QC-LDPC codes have been carried out based on optimization of parameters such as the minimum Hamming distance of the code or the girth of the Tanner graph. However, it has been shown that an excellent first-order measure of performance over the AWGNC is the minimum pseudo-weight of the code [4]. So far, few results exist in the literature on the minimum pseudo-weight of QC- LDPC and related codes. Spectral graph analysis was used in [5], and more recently, in [6], to obtain bounds on the minimum Hamming weight, and minimum AWGNC pseudo-weight, respectively, of a length-n (c, d)-regular code C over the binary field F2: dmin ⩾wmin p (H) ⩾n 2c −λ2 λ1 −λ2 ; dmin ⩾n2 d 2c + d −2 −λ2 λ1 −λ2 , with λ1 > · · · > λs being the distinct ordered eigenvalues of HTH ∈Rn×n (where H is viewed as a matrix in Rm×n). These bounds are, for most codes, loose. However, in particular cases, like the projective geometry codes [7], [8], [9], they are attained. A current problem with these bounds is that for most LDPC codes, it is not practical to evaluate the eigenvalues λ1, λ2 due to the size of the matrix HTH. In this paper we show how to compute the AWGN pseudo- weight lower bound for quasi-cyclic (QC) and related codes by utilizing the A-submodule structure of quasi-cyclic codes, A = R[X]/(Xr −1) [10], [11], [12]. In particular, we begin by showing how the polynomial parity-check matrix that describes a cyclic code can be used to compute the required eigenvalues, and then generalize this approach to compute the required eigenvalues for QC codes. We also define the class of “nested circulant” matrices, and show that these have eigenvalues which are given by evaluating a multivariate associated polynomial at points whose coordinates are particular roots of unity. Finally, we give a necessary condition for the pseudo-weight lower bound to be attained when H is circulant and show a few classes of cyclic codes satisfying this criterion. II. BASIC NOTATION AND DEFINITIONS All codes in this paper will be binary linear codes of a certain length n specified through a (scalar) parity-check matrix H = (hj,i) ∈Fm×n 2 as the set of all vectors c ∈Fn 2 such that H · cT = 0T, where T denotes transposition. The minimum Hamming distance of a code C will be denoted by dmin(C). The fundamental cone K(H) of H is the set of all vectors ω ∈Rn that satisfy ωi ⩾0 for all i ∈I(H) , (1) ωi ⩽ X i′∈Ij(H)\i ωi′ for all j ∈J (H), i ∈Ij(H) , (2) where J (H) and I(H) denote the sets of row and column indices of H respectively, and Ij(H) ≜{i ∈I | hj,i = 1} for each j ∈J (H). A vector ω ∈K(H) is called a pseudo- codeword. The AWGNC pseudo-weight of a pseudo-codeword ω is defined to be wp(ω) = wAWGNC p (ω) ≜∥ω∥2 1/∥ω∥2 2. (For a motivation of these definitions, see [14], [15]). The minimum of the AWGNC pseudo-weight over all nonzero pseudo-codewords is called the minimum AWGNC pseudo- weight and is denoted by wmin p (H). For any integer s ≥1, let Rs = {exp(ı2πr/s) : 0 ≤ r < s} denote the set of complex s-th roots of unity, and let R− s = Rs{1}. The symbol ∗denotes complex conjugation. Also, an r × r circulant matrix B, whose entries are square L × L matrices, will be called an L-block circulant matrix; we shall denote this by B = circ(b0, b1, · · · , br−1) where the (square L × L matrix) entries in the first column of B are b0, b1, … , br−1 respectively. Finally, Z, R, C, and F2 will be the ring of integers, the field of real num

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