Algebraic Constructions of Graph-Based Nested Codes from Protographs Christine A. Kelley Jorg¨ Kliewer Department of Mathematics Klipsch School of Electrical and Computer Engineering University of Nebraska-Lincoln New Mexico State University Lincoln, NE 68588, USA Las Cruces, NM 88003, USA Email: [email protected] Email: [email protected] Abstract—Nested codes have been employed in a large number on a separation of channel and network coding for non-ergodic of communication applications as a specific case of superposition discrete-input fading channels. In these applications we require codes, for example to implement binning schemes in the presence the subcodes to be better in threshold and/or in error-floor than of noise, in joint network-channel coding, or in physical-layer secrecy. Whereas nested lattice codes have been proposed recently the global code. for continuous-input channels, in this paper we focus on the In this paper we focus on array-code type constructions [8], construction of nested linear codes for joint channel-network [9] and propose an algebraic design of nested linear codes coding problems based on algebraic protograph LDPC codes. In based on protograph LDPC codes [10], [11]. In particular, particular, over the past few years several constructions of codes in [12], [13] a lifting technique based on voltage graphs has have been proposed that are based on random lifts of suitably chosen base graphs. More recently, an algebraic analog of this been proposed which has been shown to provide a large girth approach was introduced using the theory of voltage graphs. In of the code graph and thus a good error-floor performance. this paper we illustrate how these methods can be used in the In contrast to previous approaches based on concatenated and construction of nested codes from algebraic lifts of graphs. random LDGM codes [5], [14] and also to constructions based on random LDPC codes we show that the advantage of the I. INTRODUCTION above algebraic constructions in the error floor regime also Nested codes have been widely used to implement binning carries over to the nested code setting. schemes based on coset codes in the presence of noise for II. PRELIMINARIES numerous scenarios, for example for the noisy Wyner-Ziv problem [1] and the dual Gel’fand-Pinsker problem [2]. In A. Nested codes particular, for the case with continuous-input channels, binning Consider M different information vectors i! of length K!, schemes based on nested lattice codes have been proposed ! =1,...,M,whichwewanttoencodejointlyinsuchaway in [3]. Recently, in [4] the authors consider discrete-input that each information vector is associated with a codeword channels and present compound LDGM/LDPC constructions from a different subcode. The overall codeword c is generated which are optimal under ML decoding. by multiplying the concatenation of all information vectors While nested codes in these contexts are related to joint with a generator matrix G of the global code C according to source-channel coding problems, the class of algebraic nested G1 codes we will address in this paper are defined based on a joint channel and network coding scenario. Such nested cT =[iT ,iT ...iT ] . =[iT ,iT ...iT ]G = 1 2 M . 1 2 M codes have been originally proposed in [5] for the generalized broadcast relay problem, where a relay node broadcasts N GM T T T packets to several destination nodes, which already know some i1 G1 ⊕ i2 G2 ⊕···⊕iM GM , (1) of the packets apriori.Arelatedconceptwasusedin[6] where each of the subcodes C with generator G of rate R = in the context of two-way relaying. The idea is that instead ! ! ! K /N is associated with the corresponding information vector of information words, codewords of different subcodes C , ! ! i and ⊕ represents a bitwise XOR. The goal is now to find 1 ≤ ! ≤ N,arealgebraicallysuperimposedviaabitwise ! general systematic design strategies where the subcodes, any XOR. In contrast to nested codes for the joint source-channel combination of subcodes, and the global code C have good coding scenario described above, here each subcode and any threshold and/or error floor properties. arbitrary combination of the subcodes is intended to form a For the sake of simplicity we focus on M =2and the good channel code. In particular, this also holds for the linear binary case in the following. Our aim is to design an LDPC combination of all subcodes, the global code C.Ithasbeen code such that its generator matrix G satisfies (1), where H ∈ shown in [7] for a broadcast scenario with side information {0, 1}(N−K1−K2)×N represents a corresponding parity check that such a construction is able to outperform a scheme based matrix. If G is not rank deficient, the null space of H of T T This work has been supported in part by NSF grant CCF-0830666 and in dimension (N − K1 − K2) contains the codewords c1 ⊕ c2 = T T part by NSF grant EPS-0701892. i1 G1 ⊕ i2 G2. 1 Likewise, the columns of the parity check matrices H1, 2 i H2 associated with G1, G2 each form a basis for their null i 31 2 spaces of dimensions (N − K1) and (N − K2),respectively. i i 1 3 AnecessaryconditiontopreventG from having a rank 2 (13)(2) smaller than K1 + K2 is that H1, H2 cannot have more than 3 1 2 (N − K1 − K2) linear independent parity check equations 1 3 in common. Based on these considerations, we propose the (123) 2 following design strategy. First, randomly generate a matrix i=(1)(2)(3) 3 M ∈{0, 1}N×N of full rank N,accordingtoagivenrow Fig. 1. A permutation voltage graph G is shown on the left and its derived ˆ ˆ and column degree distribution. This matrix is then partitioned graph G on the right, where G serves as a protograph for G.Thedarkeredges correspond to the connections between the clouds of verticesincidentwith into three submatrices the nontrivial labeled edges. T (N×N) (N×K2) (N×K1) (N×(N−K1−K2)) Gα,calledthe(right) derived graph,isa|G|-degree lift of G M = 'M1 M2 M3 ( . (2) and has vertex set V × G and edge set E × G,whereif(u, v) Next, the individual parity check matrices for the nested is a positively oriented edge in G with voltage b,then(u, a) is α code are obtained as connected to (v,ab) in G .Alternatively,anotherconstruction takes the voltage group to be the symmetric group S on n (N×(N−K1−K2)) T n H =[M3 ] , elements and has α map the positively-oriented edges of G into − × × − − T H((N K1) N) = M N×K2 M (N (N K1 K2)) , Sn.Thisyieldsapermutation voltage graph.Thepermutation 1 ' 1 3 ( derived graph Gα is a degree n lift (instead of n!)withvertices − × × − − T H((N K2) N) = M N×K1 M (N (N K1 K2)) . V ×{1,...,n} and edges E ×{1,...,n}.Ifπ ∈ Sn is a 2 ' 2 3 ( permutation voltage on the edge e =(u, v) of G,thenthere α Thus, both H1 and H2 are guaranteed to have a null space is an edge from (u, i) to (v,π(i)) in G for i =1, 2,...,n. of dimensions (N − K1) and (N − K2),respectively,andH We will represent each vertex (v,i) and each edge (e, i) in has (N − K1 − K2) parity check equations that are satisfied the derived graph by vi and ei,respectively.Inbothcases,the by C1 and C2. labeled base graph (i.e. voltage graph) algebraically determines Proposition 1. The nested code property in (1) holds also if aspecificliftofthegraph.Fig.1showsapermutationvoltage graph G = K , with two nontrivial permutation voltages on M and thus one or more of the matrices H, H1,andH2 are 2 3 (row) rank deficient. For a rank deficit r of the check matrix its edges to the group S3,andthecorrespondingdegree3 H the rate loss for the global code C is given as ∆R ≤ r/N. permutation derived graph. Henceforth, derived (lifted) graphs will be denoted by Gˆ Proof: Denote the rank deficit for the matrices M , M 1 2 since the voltage assignment α should be clear from context. as r ≥ 0, r ≥ 0,respectively.ThismeansthatG has 1 2 1 In this paper we will focus on permutation voltage graphs for now a rank of at least K + r + r ,andG arankofat 1 1 2 designing nested codes. least K2 + r + r2,resp.,whichleadstoanoverallrankof at least K1 + K2 + r + r1 + r2 for the generator matrix G. III. NESTED CODES FROM PROTOGRAPHS Since both subcodes have at most N − K − K − r check 1 2 We now describe a simple method to construct nested equations in common the row rank of G must not be smaller codes from protographs in which the base Tanner graphs than K + K + r to ensure the nested code property which 1 2 corresponding to small parity-check matrices H ,H ,andH is satisfied for any r ≥ 0, r ≥ 0.BysettingR# + R# = 1 2 1 2 1 2 are lifted to obtain Tanner graphs with corresponding parity- (K + K + r)/N where R# and R# denote the new rates for 1 2 1 2 check matrices Hˆ , Hˆ ,andHˆ .Thesimplicityofourmethod the subcodes C and C ,aratelossof∆R ≤ r/N for the 1 2 1 2 is that it involves just one lifting of the base graph G code C is obtained. M corresponding to M. Note that an extension of the above design strategy to M> We start with a small bipartite base graph G with n left 2 can be obtained in a straightforward way by modifying the M vertices, denoted by the set L,andn right vertices, denoted partitioning and construction of M in (2).