Family Of Error-correcting Codes Research Articles

Optimized decoder for low-density parity check codes based on genetic algorithms

Low-density parity check (LDPC) codes, are a family of error-correcting codes, their performances close to the Shannon limit make them very attractive solutions for digital communication systems. There are several algorithms for decoding LDPC codes that show great diversity in terms of performance related to error correction. Also, very recently, many research papers involved the genetic algorithm (GA) in coding theory, in particular, in the decoding linear block codes case, which has heavily contributed to reducing the bit error rate (BER). In this paper, an efficient method based on the GA is proposed and it is used to improve the power of correction in terms of BER and the frame error rate (FER) of LDPC codes. Subsequently, the proposed algorithm can independently decide the most suitable moment to stop the decoding process, moreover, it does not require channel information (CSI) making it adaptable for all types of channels with different noise or intensity. The simulations show that the proposed algorithm is more efficient in terms of BER compared to other LDPC code decoders.

Implementation of High Speed and Area Efficient Polar Encoder

Abstract: Polar codes were introduced by E.Arıkan in 2008. They are the first family of error-correcting codes that attain the capacity of binary memoryless and symmetric channels with efficient encoding, decoding, and construction algorithms. In this paper, implementation of high speed and area efficient polar encoder for systematic polar codes is presented. According to an iterative property of the generator matrix and particular lower triangular structure of the matrix, the number of XOR computations are reduced. In this implementation, total area of the encoder is decreased by 33.69%, delay is decreased by 54.6% and power consumption is decreased by 39.44%.

High-performance hard-input LDPC decoding on multi-core devices for optical space links

LDPC codes are a family of error-correcting codes that are present in most space communication standards. Thanks to their large processing power and their parallelization capabilities, prevailing multicore devices facilitate real-time implementations of digital communication systems, which were previously implemented into dedicated hardware devices. Previous works were done over the last decade on the implementation of Gbps decoders on programmable devices. However, these works focus on the soft input LDPC decoding algorithms. But, hard-input LDPC decoders are also required to design and prototype for instance next optical-based satellite communication systems. These systems should provide high throughput (≥10 Gbps) and low latency (≤1ms) internet links. In this article, the first software-based implementation of a hard-input multi-Gbps LDPC decoder is detailed. Thanks to different parallelization strategies and deeply optimized SIMD codes, throughput up to 7.5 Gbps is achieved when 10 Gallager-E iterations are executed onto an INTEL Xeon device making possible the design of software base station systems providing throughputs of tens of Gbps for optical system evaluation or base station design.

Goppa codes over the p-adic integers and integers modulo pe

Goppa codes were defined by Valery D. Goppa in 1970. In 1978, Robert J. McEliece used this family of error-correcting codes in his cryptosystem, which has gained popularity in the last decade due to its resistance to attacks from quantum computers. In this paper, we present Goppa codes over the p-adic integers and integers modulo pe. This allows the creation of chains of Goppa codes over different rings. We show some of their properties, such as parity-check matrices and minimum distance, and suggest their cryptographic application, following McEliece's scheme.

Partial Order-Based Decoding of Rate-1 Nodes in Fast Simplified Successive-Cancellation List Decoders for Polar Codes

Polar codes are the first family of error-correcting codes that can achieve channel capacity. Among the known decoding algorithms, Successive-Cancellation List (SCL) decoding supported by a Cyclic Redundancy Check (CRC) shows the best error-correction performance at the cost of a high decoding complexity. The decoding of Rate-1 nodes belongs to the most complex tasks in SCL decoding. In this paper, we present a new algorithm that largely reduces the number of considered candidates in a Rate-1 node and generate all required candidates in parallel. For this purpose, we use a partial order of the candidate paths to prove that only a specified number of candidates needs to be considered. Further complexity reductions are achieved by an extended threshold-based path exclusion scheme at the cost of negligible error-correction performance loss. We present detailed Application-Specific Integrated Circuit (ASIC) implementation data on a 28 nm Fully Depleted Silicon on Insulator (FD-SOI) Complementary Metal-Oxide-Semiconductor (CMOS) technology for decoders with code length 128. We show that the new decoders outperform state-of-the-art reference decoders. For list size 8, improvements of up to 158.8% and 62.5% in area and energy efficiency are observed, respectively.

Bounds for List-Decoding and List-Recovery of Random Linear Codes

A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$ that is independent of the code length. It is said to be list-recoverable for input $\ell$ if for every sufficiently large subset of codewords (of $L$ or more), there is a coordinate where the codewords take more than $\ell$ values. The parameter $L$ is said to be the list size in either case. The capacity, i.e., the largest possible rate for these notions as the $L \to \infty$, is known to be $1-h_q(p)$ for list-decoding, and $1-\log_q \ell$ for list-recovery, where $q$ is the alphabet of the code family. In this work, we study the of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below, $\epsilon > 0$ is the gap to capacity). (1) A random linear code of rate $1 - \log_q(\ell) - \epsilon$ requires $L \ge \ell^{\Omega(1/\epsilon)}$ for list-recovery from input $\ell$. This is surprisingly in contrast to completely random codes, where $L = O(\ell/\epsilon)$ suffices w.h.p. (2) A random linear code of rate $1 - h_q(p) - \epsilon$ requires $L \ge \lfloor h_q(p)/\epsilon+0.99 \rfloor$ for list-decoding from error fraction $p$, when $\epsilon$ is sufficiently small. (3) A random binary linear code of rate $1 - h_2(p) - \epsilon$ is list-decodable from average error fraction $p$ with with $L \leq \lfloor h_2(p)/\epsilon \rfloor + 2$. The second and third results together precisely pin down the sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values.

ONE-HOMOGENEOUS WEIGHT CODES OVER FINITE CHAIN RINGS

Abstract. This paper determines the structures of one-homogeneousweight codes over finite chain rings and studies the algebraic propertiesof these codes. We present explicit constructions of one-homogeneousweight codes over finite chain rings. By taking advantage of the distance-preserving Gray map defined in [7] from the finite chain ring to its residuefield, we obtain a family of optimal one-Hamming weight codes over theresidue field. Further, we propose a generalized method that also includesthe examples of optimal codes obtained by Shi et al. in [17]. 1. IntroductionConstant-weight codes represent an important class of codes within the fam-ily of error-correcting codes [11]. A linear code having constant-weight meansthat every nonzero codeword has the same weight. In the literature there aremany papers on binary constant-weight codes which have several applicationssuch as the design of demultiplexers for nano-scale memories [8] and the con-struction of frequency hopping lists for use in GSM networks [12]. Especially,considerable research has been done on the central problem regarding constant-weight codes which is the determination of A(n,d,w), the largest possible sizeof a constant-weight code of length n, Hamming distance at least d, and con-stant weight w. Due to the difficulty in finding good constant-weight codes,various upper and lower bounds on A(n,d,w) have been developed [1, 3, 15, 18].Moreover, there are further studies in this direction including nonbinary finitefields in [2, 10]. It has been shown that there exists a unique one-weight binarylinear code of dimension k such that any two columns in its generator matrixare linearly independent for every positive integer k. Later, this result has beenextended to the ring Z

Performance of parallel concatenated zigzag codes for different segment length

General turbo codes have an excellent bit-error-rate (BER) performance at low signal-to-noise ratios (SNR) in additive white Gaussian noise (AWGN) channels. However, the standard turbo decoding algorithm is highly complex. To reduce its decoding computational complexity, zigzag codes are currently used. A zigzag code is a family of error-correcting codes similar to the single parity check codes. It is an important issue, how the segment length called J and the interleaver can affect concatenated zigzag code performance. In this study, the segment size and interleaver structure are investigated in order to obtain a high bit-error-rate performance for the concatenated zigzag code with four encoders and three random interleavers. Realized computer simulation results are compared with/without interleaver for various values of J over a noisy channel.

MACWILLIAMS IDENTITIES FOR WEIGHT ENUMERATORS WITH RESPECT TO THE RT METRIC

Linear codes constitute an important family of error-correcting codes and have a rich algebraic structure. Initially, these codes were studied with respect to the Hamming metric; while for the past few years, they are also studied with respect to a non-Hamming metric, known as the Rosenbloom–Tsfasman metric (also known as RT metric or ρ metric). In this paper, we introduce and study the split ρ weight enumerator of a linear code in the R-module Mn×s(R) of all n × s matrices over R, where R is a finite Frobenius commutative ring with unity. We also define the Lee complete ρ weight enumerator of a linear code in Mn×s(ℤk), where ℤk is the ring of integers modulo k ≥ 2. We also derive the MacWilliams identities for each of these ρ weight enumerators.

2-Transitivity is Insufficient for Local Testability

A basic goal in property testing is to identify a minimal set of features that make a property testable. For the case when the property to be tested is membership in a binary linear error-correcting code, Alon et al. (Trans Inf Theory, 51(11):4032---4039, 2005) had conjectured that the presence of a single low-weight codeword in the dual, and "2-transitivity" of the code (i.e., the code being invariant under a 2-transitive group of permutations on the coordinates of the code) suffice to get local testability. We refute this conjecture by giving a family of error-correcting codes where the coordinates of the codewords form a large field of characteristic two, and the code is invariant under affine transformations of the domain. This class of properties was introduced by Kaufman & Sudan (STOC, 2008) as a setting where many results in algebraic property testing generalize. Our result shows a complementary virtue: This family also can be useful in producing counterexamples to natural conjectures.

Predictable Three-Parameter Design of Network Covert Communication Systems

This paper presents a predictable and quantifiable approach to designing a covert communication system capable of effectively exploiting covert channels found in the various layers of network protocols. Three metrics are developed that characterize the overall system. A measure of probability of detection is derived using statistical inference techniques. A system efficiency measure is developed based upon the noiseless capacity of the covert channel. A measure of reliability is developed as the bit-error rate of the combined noisy channel and an appropriate error-correcting code. To support reliable communication, a family of error-correcting codes are developed that handle the high symbol insertion rates found in these covert channels. The system metrics are each shown to be a function of the covert channel signal-to-noise ratio, and as such can be used to perform system level design trade-offs. Validation of the system design methodology is provided by means of an experiment using real network traffic data.

An Ensemble of Iteratively Decodable Codes Constructed Based on a Superposition Method

In this paper, the construction of new families of error-correcting codes adapted from Boutros and Lentmaier's generalized low-density (GLD) codes, which we called generalized irregular low-density (GILD) codes, is investigated. By introducing irregular base matrices, significant improvements on Boutros and Lentmaier's results are achieved. Two cases are considered; in the first case, the same constituent code is used, and in the second case, different constituent codes are used. It is proved by an ensemble performance argument that these codes exist and are asymptotically good in the sense of the minimum-distance criterion. It is also proved that the performance of GILD codes approaches the binary symmetric channel capacity limit. Iterative decoding of GILD codes for communication over an additive white Gaussian noise channel is also studied. The high flexibility in selecting the parameters of GILD codes and their better performance and higher rate make them more attractive than GLD codes, and hence, suitable for small and medium block lengths forward-error-correcting schemes. Comparisons of simulation results between some GILD codes and some good low-density parity-check codes show very close performances

New near shannon limit codes: Zigzag codes and concatenated zigzag codes

This paper introduces a family of error-correcting codes called zigzag codes. A zigzag code is described by a highly structured zigzag graph. Due to the structural properties of the graph, very low-complexity soft-in, soft-out decoding rules can be implemented. We present a decoding rule, based on the Max-Log-APP(MLA) formulation, which requires a total of only 20 addition-equivalent-operations per information bit per iteration. Simulation of a rate-1/2 concatenated zigzag code with four constituent encoders with interleaver length 65536 yields a bit error rate (BER) and of 10−5 at 0.9 dB and 1.4 dB away from the Shannon limit by optimal (APP) and low-cost sub-optimal (MLA) decoders, respectively.

A new lower bound for the minimum distance of binary Goppa codes

Binary Goppa codes are a large and powerful family of error-correcting codes. But how to find the true minimum distance of binary Goppa codes is not solved yet. In this paper a new lower bound for the minimum distance of binary Goppa codes is shown. This new lower bound improves the results in Y. Sugiyama (1976) and Feng Guiliang's (1983) papers. The method in this paper can be generalized to other Goppa codes easily.