Design of efficiently encodable moderate-length high-rate irregular LDPC codes

This paper presents a new class of irregular low-density parity-check (LDPC) codes of moderate length (10/sup 3//spl les/n/spl les/10/sup 4/) and high rate (R/spl ges/3/4). Codes in this class admit low-complexity encoding and have lower error-rate floors than other irregular LDPC code-design approaches. It is also shown that this class of LDPC codes is equivalent to a class of systematic serial turbo codes and is an extension of irregular repeat-accumulate codes. A code design algorithm based on the combination of density evolution and differential evolution optimization with a modified cost function is presented. Moderate-length, high-rate codes with no error-rate floors down to a bit-error rate of 10/sup -9/ are presented. Although our focus is on moderate-length, high-rate codes, the proposed coding scheme is applicable to irregular LDPC codes with other lengths and rates.     

Stopping set distribution of LDPC code ensembles

Stopping sets determine the performance of low-density parity-check (LDPC) codes under iterative decoding over erasure channels. We derive several results on the asymptotic behavior of stopping sets in Tanner-graph ensembles, including the following. An expression for the normalized average stopping set distribution, yielding, in particular, a critical fraction of the block length above which codes have exponentially many stopping sets of that size. A relation between the degree distribution and the likely size of the smallest nonempty stopping set, showing that for a /spl radic/1-/spl lambda/'(0)/spl rho/'(1) fraction of codes with /spl lambda/'(0)/spl rho/'(1)<1, and in particular for almost all codes with smallest variable degree >2, the smallest nonempty stopping set is linear in the block length. Bounds on the average block error probability as a function of the erasure probability /spl epsi/, showing in particular that for codes with lowest variable degree 2, if /spl epsi/ is below a certain threshold, the asymptotic average block error probability is 1-/spl radic/1-/spl lambda/'(0)/spl rho/'(1)/spl epsi/.     

A new binary sequence family with low correlation and large size

For odd n=2l+1 and an integer /spl rho/ with 1/spl les//spl rho//spl les/l, a new family S/sub o/(/spl rho/) of binary sequences of period 2/sup n/-1 is constructed. For a given /spl rho/, S/sub o/(/spl rho/) has maximum correlation 1+2/sup n+2/spl rho/-1/2/, family size 2/sup n/spl rho//, and maximum linear span n(n+1)/2. Similarly, a new family of S/sub e/(/spl rho/) of binary sequences of period 2/sup n/-1 is also presented for even n=2l and an integer /spl rho/ with 1/spl les//spl rho/相似文献   

The inequalities of quantum information theory

Let /spl rho/ denote the density matrix of a quantum state having n parts 1, ..., n. For I/spl sube/N={1, ..., n}, let /spl rho//sub I/=Tr/sub N/spl bsol/I/(/spl rho/) denote the density matrix of the state comprising those parts i such that i/spl isin/I, and let S(/spl rho//sub I/) denote the von Neumann (1927) entropy of the state /spl rho//sub I/. The collection of /spl nu/=2/sup n/ numbers {S(/spl rho//sub I/)}/sub I/spl sube/N/ may be regarded as a point, called the allocation of entropy for /spl rho/, in the vector space R/sup /spl nu//. Let A/sub n/ denote the set of points in R/sup /spl nu// that are allocations of entropy for n-part quantum states. We show that A~/sub n/~ (the topological closure of A/sub n/) is a closed convex cone in R/sup /spl nu//. This implies that the approximate achievability of a point as an allocation of entropy is determined by the linear inequalities that it satisfies. Lieb and Ruskai (1973) have established a number of inequalities for multipartite quantum states (strong subadditivity and weak monotonicity). We give a finite set of instances of these inequalities that is complete (in the sense that any valid linear inequality for allocations of entropy can be deduced from them by taking positive linear combinations) and independent (in the sense that none of them can be deduced from the others by taking positive linear combinations). Let B/sub n/ denote the polyhedral cone in R/sup /spl nu// determined by these inequalities. We show that A~/sub n/~=B/sub n/ for n/spl les/3. The status of this equality is open for n/spl ges/4. We also consider a symmetric version of this situation, in which S(/spl rho//sub I/) depends on I only through the number i=/spl ne/I of indexes in I and can thus be denoted S(/spl rho//sub i/). In this case, we give for each n a finite complete and independent set of inequalities governing the symmetric allocations of entropy {S(/spl rho//sub i/)}/sub 0/spl les/i/spl les/n/ in R/sup n+1/.     

A 640-Mb/s 2048-bit programmable LDPC decoder chip

A 14.3-mm/sup 2/ code-programmable and code-rate tunable decoder chip for 2048-bit low-density parity-check (LDPC) codes is presented. The chip implements the turbo-decoding message-passing (TDMP) algorithm for architecture-aware (AA-)LDPC codes which has a faster convergence rate and hence a throughput advantage over the standard decoding algorithm. It employs a reduced complexity message computation mechanism free of lookup tables, and features a programmable network for message interleaving based on the code structure. The chip decodes any mix of 2048-bit rate-1/2 (3,6)-regular AA-LDPC codes in standard mode by programming the network, and attains a throughput of 640 Mb/s at 125 MHz for 10 TDMP-decoding iterations. In augmented mode, the code rate can be tuned up to 14/16 in steps of 1/16 by augmenting the code. The chip is fabricated in 0.18-/spl mu/m six-metal-layer CMOS technology, operates at a peak clock frequency of 125 MHz at 1.8 V (nominal), and dissipates an average power of 787 mW.     

Several properties of short LDPC codes

In this paper, we present several properties on minimum distance(d/sub min/) and girth(G/sub min/) in Tanner graphs for low-density parity-check (LDPC) codes with small left degrees. We show that the distance growth of (2, 4) LDPC codes is too slow to achieve the desired performance. We further give a tight upper bound on the maximum possible girth. The numerical results show that codes with large G/sub min/ could outperform the average performance of regular ensembles of the LDPC codes over binary symmetric channels. The same codes perform about 1.5 dB away from the sphere-packing bound on additive white Gaussian noise channels.     

Joint (3,k)-regular LDPC code and decoder/encoder design

Recently, low-density parity-check (LDPC) codes have attracted a lot of attention in the coding theory community. However, their real-world applications are still problematic mainly due to the lack of effective decoder/encoder hardware design approaches. In this paper, we present a joint (3,k)-regular LDPC code and decoder/encoder design technique to construct a class of (3,k)-regular LDPC codes that not only have very good error-correcting capability but also exactly fit to high-speed partly parallel decoder and low-complexity encoder implementations. We also develop two techniques to further modify this joint design scheme to achieve more flexible tradeoffs between decoder hardware complexity and decoding speed.     

Evaluation of the Green's function for the mixed potential integral equation (MPIE) method in the time domain for layered media

In the use of the time-domain integral equation (TDIE) method for the analysis of layered media, it is important to have the time-domain layered medium Green's function computed for many source-to-field distances /spl rho/ and time instants t. In this paper, a numerical method is used that computes the mixed potential Green's functions G/sub v/(/spl rho/,t) and G/sub A/(/spl rho/,t) for a multilayered medium for many /spl rho/'s and t's simultaneously. The method is applicable to multilayered media and for lossless or lossy dispersive media. Salient features of the method are: 1) the use of complex /spl omega/ so that the surface wave poles are lifted off the real k/sub /spl rho// axis such that pole extractions are not required; 2) the use of half-space extraction so that the integrand for the Sommerfeld integral decays exponentially along the k/sub /spl rho// axis to obtain fast convergence of the integral; and 3) the use of the fast Hankel transform so that the Green's function is calculated for many values of /spl rho/ simultaneously. For a four-layer medium, we illustrate the numerical results by a three-dimensional plot of /spl rho/G/sub v/(/spl rho/,t) versus /spl rho/ and t and demonstrate the space-time evolution of these Green's functions. For a maximum frequency range of 8 GHz, the method requires only a few CPU minutes to compute a table of 100 (points in /spl rho/) /spl times/ 168 (points in t) uniformly spaced values of G/sub v/(/spl rho/,t) on an 867-MHz Pentium PC.     

List decoding from erasures: bounds and code constructions

We consider the problem of list decoding from erasures. We establish lower and upper bounds on the rate of a (binary linear) code that can be list decoded with list size L when up to a fraction p of its symbols are adversarially erased. Such bounds already exist in the literature, albeit under the label of generalized Hamming weights, and we make their connection to list decoding from erasures explicit. Our bounds show that in the limit of large L, the rate of such a code approaches the "capacity" (1 - p) of the erasure channel. Such nicely list decodable codes are then used as inner codes in a suitable concatenation scheme to give a uniformly constructive family of asymptotically good binary linear codes of rate /spl Omega/(/spl epsiv//sup 2//log(1//spl epsiv/)) that can be efficiently list-decoded using lists of size O(1//spl epsiv/) when an adversarially chosen (1 - /spl epsiv/) fraction of symbols are erased, for arbitrary /spl epsiv/ > 0. This improves previous results in this vein, which achieved a rate of /spl Omega/(/spl epsiv//sup 3/log(1//spl epsiv/)).     

A new family of Grassmann space-time codes for non-coherent MIMO systems

A family of space-time codes suited for noncoherent multi-input multi-output (MIMO) systems is presented. These codes use all the complex degrees of freedom of the system, i.e. M/spl times/(1-(M/T)) symbols per channel use. They are constructed as codes on the Grassmann manifold G/sub T,M/(/spl Copf/) where T is the temporal codelength and M is the number of transmit antennas.     

New class of nonlinear systematic error detecting codes

A code C detects error e with probability 1-Q(e),ifQ(e) is a fraction of codewords y such that y, y+e/spl isin/C. We present a class of optimal nonlinear q-ary systematic (n, q/sup k/)-codes (robust codes) minimizing over all (n, q/sup k/)-codes the maximum of Q(e) for nonzero e. We also show that any linear (n, q/sup k/)-code V with n /spl les/2k can be modified into a nonlinear (n, q/sup k/)-code C/sub v/ with simple encoding and decoding procedures, such that the set E={e|Q(e)=1} of undetected errors for C/sub v/ is a (k-r)-dimensional subspace of V (|E|=q/sup k-r/ instead of q/sup k/ for V). For the remaining q/sup n/-q/sup k-r/ nonzero errors, Q(e)/spl les/q/sup -r/for q/spl ges/3 and Q(e)/spl les/ 2/sup -r+1/ for q=2.     

Girth-8 (3,L)-规则QC-LDPC码的一种确定性构造方法

对于围长(girth)至少为8的低密度奇偶校验(LDPC)码,目前的绝大多数构造方法都需要借助于计算机搜索。受贪婪构造算法启发,该文利用完全确定的方式构造出一类围长为8的(3, L)- 规则QC-LDPC码。这类QC-LDPC码的校验矩阵由3L个PP的循环置换矩阵构成。对于任意整数P3L2/4,这类校验矩阵的围长均为8。     

Explicit construction of families of LDPC codes with no 4-cycles

Low-density parity-check (LDPC) codes are serious contenders to turbo codes in terms of decoding performance. One of the main problems is to give an explicit construction of such codes whose Tanner graphs have known girth. For a prime power q and m/spl ges/2, Lazebnik and Ustimenko construct a q-regular bipartite graph D(m,q) on 2q/sup m/ vertices, which has girth at least 2/spl lceil/m/2/spl rceil/+4. We regard these graphs as Tanner graphs of binary codes LU(m,q). We can determine the dimension and minimum weight of LU(2,q), and show that the weight of its minimum stopping set is at least q+2 for q odd and exactly q+2 for q even. We know that D(2,q) has girth 6 and diameter 4, whereas D(3,q) has girth 8 and diameter 6. We prove that for an odd prime p, LU(3,p) is a [p/sup 3/,k] code with k/spl ges/(p/sup 3/-2p/sup 2/+3p-2)/2. We show that the minimum weight and the weight of the minimum stopping set of LU(3,q) are at least 2q and they are exactly 2q for many LU(3,q) codes. We find some interesting LDPC codes by our partial row construction. We also give simulation results for some of our codes.     

Exact thresholds and optimal codes for the binary-symmetric channel and Gallager's decoding algorithm A

We show that for the case of the binary-symmetric channel and Gallager's decoding algorithm A the threshold can, in many cases, be determined analytically. More precisely, we show that the threshold is always upper-bounded by the minimum of (1-/spl lambda//sub 2//spl rho/'(1))/(/spl lambda/'(1)/spl rho/'(1)-/spl lambda//sub 2//spl rho/'(1)) and the smallest positive real root /spl tau/ of a specific polynomial p(x) and we observe that for most cases this bound is tight, i.e., it determines the threshold exactly. We also present optimal degree distributions for a large range of rates. In the case of rate one-half codes, for example, the threshold x/sub 0//sup */ of the optimal degree distribution is given by x/sup *//sub 0//spl sim/0.0513663. Finally, we outline how thresholds of more complicated decoders might be determined analytically.     

Lossless and near-lossless source coding for multiple access networks

A multiple access source code (MASC) is a source code designed for the following network configuration: a pair of correlated information sequences {X/sub i/}/sub i=1//sup /spl infin// and {Y/sub i/}/sub i=1//sup /spl infin// is drawn independent and identically distributed (i.i.d.) according to joint probability mass function (p.m.f.) p(x,y); the encoder for each source operates without knowledge of the other source; the decoder jointly decodes the encoded bit streams from both sources. The work of Slepian and Wolf describes all rates achievable by MASCs of infinite coding dimension (n/spl rarr//spl infin/) and asymptotically negligible error probabilities (P/sub e//sup (n)//spl rarr/0). In this paper, we consider the properties of optimal instantaneous MASCs with finite coding dimension (n相似文献   

The empirical distribution of rate-constrained source codes

Let X = (X/sub 1/,...) be a stationary ergodic finite-alphabet source, X/sup n/ denote its first n symbols, and Y/sup n/ be the codeword assigned to X/sup n/ by a lossy source code. The empirical kth-order joint distribution Q/spl circ//sup k/[X/sup n/,Y/sup n//spl rceil/(x/sup k/,y/sup k/) is defined as the frequency of appearances of pairs of k-strings (x/sup k/,y/sup k/) along the pair (X/sup n/,Y/sup n/). Our main interest is in the sample behavior of this (random) distribution. Letting I(Q/sup k/) denote the mutual information I(X/sup k/;Y/sup k/) when (X/sup k/,Y/sup k/)/spl sim/Q/sup k/ we show that for any (sequence of) lossy source code(s) of rate /spl les/R lim sup/sub n/spl rarr//spl infin//(1/k)I(Q/spl circ//sup k/[X/sup n/,Y/sup n//spl rfloor/) /spl les/R+(1/k)H (X/sub 1//sup k/)-H~(X) a.s. where H~(X) denotes the entropy rate of X. This is shown to imply, for a large class of sources including all independent and identically distributed (i.i.d.). sources and all sources satisfying the Shannon lower bound with equality, that for any sequence of codes which is good in the sense of asymptotically attaining a point on the rate distortion curve Q/spl circ//sup k/[X/sup n/,Y/sup n//spl rfloor//spl rArr//sup d/P(X/sup k/,Y~/sup k/) a.s. whenever P(/sub X//sup k//sub ,Y//sup k/) is the unique distribution attaining the minimum in the definition of the kth-order rate distortion function. Consequences of these results include a new proof of Kieffer's sample converse to lossy source coding, as well as performance bounds for compression-based denoisers.     

New construction of multiwavelength optical orthogonal codes

We investigate multiwavelength optical orthogonal codes (MWOOCs) for optical code-division multiple access. Particularly, we present a new construction method for (mn,/spl lambda/+2,/spl lambda/) MWOOCs with the number of available wavelengths m, codeword length n, and constant Hamming weight /spl lambda/+2 that have autocorrelation and cross-correlation values not exceeding /spl lambda/. In the proposed scheme, there is no constraint on the relationship between the number of available wavelengths and the codeword length, and it is also possible to use an arbitrary /spl lambda/. We show that the constructed code is optimal, especially for /spl lambda/=1. Finally, we analyze the bit error rate of the new code and compare it with that of other optical codes.     

More results on the weight enumerator of product codes

We consider the product code C/sub p/ of q-ary linear codes with minimum distances d/sub c/ and d/sub r/. The words in C/sub p/ of weight less than d/sub r/d/sub c/+max(d/sub r//spl lceil/(d/sub c//g)/spl rceil/,d/sub c//spl lceil/(d/sub r//q)/spl rceil/) are characterized, and their number is expressed in the number of low-weight words of the constituent codes. For binary product codes, we give an upper bound on the number of words in C/sub p/ of weightless than min(d/sub r/(d/sub c/+/spl lceil/(d/sub c//2)/spl rceil/+1)), d/sub c/(d/sub r/+/spl lceil/(d/sub r//2)/spl rceil/+1) that is met with equality if C/sub c/ and C/sub r/ are (extended) perfect codes.     

Efficient encoding of low-density parity-check codes

Low-density parity-check (LDPC) codes can be considered serious competitors to turbo codes in terms of performance and complexity and they are based on a similar philosophy: constrained random code ensembles and iterative decoding algorithms. We consider the encoding problem for LDPC codes. More generally we consider the encoding problem for codes specified by sparse parity-check matrices. We show how to exploit the sparseness of the parity-check matrix to obtain efficient encoders. For the (3,6)-regular LDPC code, for example, the complexity of encoding is essentially quadratic in the block length. However, we show that the associated coefficient can be made quite small, so that encoding codes even of length n≃100000 is still quite practical. More importantly, we show that “optimized” codes actually admit linear time encoding