Calculation of error probabilities for distance-invariant nonlinear error control codes

Expressions are given for the distance distribution of some nonlinear codes which enable error probabilities to be calculated using methods commonly associated with linear error control codes.<>     

New class of codes suitable for computer error control

A new concept of generalised orthogonality is proposed and a class of generalised orthogonal codes based on SBIBD theory is developed     

Maximum runlength-limited codes with error control capabilities

New methods are presented to protect maximum runlength-limited sequences against random and burst errors and to avoid error propagation. The methods employ parallel conversion techniques and enumerative coding algorithms that transform binary user information into constrained codewords. The new schemes have a low complexity and are very efficient. The approach can be used for modulation coding in recording systems and for synchronization and line coding in communication systems. The schemes enable the usage of high-rate constrained codes, as error control can be provided with similar capabilities as for unconstrained sequences     

Turbo-codes: the ultimate error control codes?

Turbo-codes have attracted a great deal of interest since their discovery in 1993. This paper reviews the reasons for this, in particular their attainment of the ultimate limits of the capacity of a communication channel. The paper describes the two fundamental concepts on which they are based: concatenated coding and iterative decoding. This latter is the real `turbo-principle', which is the real secret of their remarkable performance. The paper also reviews the direction of research in this area since 1993, and shows that, far from bringing coding research to an end, turbo-codes have led to a renaissance. In particular, other applications of the `turbo-principle' have emerged, and these are discussed, along with the practical applications of turbo-codes that have appeared, from mobile radio to deep-space exploration     

Even-weighted codes for error detection

For a binary symmetric channel, a code V with only evenweighted words performs better than a corresponding code V? with both odd- and even-weighted words, from the point of the probability of undetected errors. We derive an estimate of the improvement in the performance.     

Watermark detection based on the properties of error control codes

Watermark detection is a topic which is seldom addressed in the watermarking literature. Most authors concentrate on developing novel watermarking algorithms. In a practical watermarking system, however, one must be able to distinguish between watermarked and unwatermarked documents. Many existing systems belong to the class of so called 'yes/no' watermarks, where the detector correlates the candidate image with some known sequence to determine whether a mark is present. Unfortunately, these watermarks often carry no extra information and are not very useful. On the other hand, multi-bit watermarking schemes typically use a separate reference watermark and the payload of the watermark is decoded only when this reference watermark is successfully detected in the received image. It is shown that it is not necessary to use a reference watermark for detection purposes if the watermark payload is encoded with an error control code. One can then put all the energy into the payload watermark and increase its robustness. The turbo code is used as an example of error control codes in the work presented, and simulation results using an algorithm based on the authors' previous work verifies their theory.     

Multilevel codes for unequal error protection

Two combined unequal error protection (UEP) coding and modulation schemes are proposed. The first method multiplexes different coded signal constellations, with each coded constellation providing a different level of error protection. In this method, a codeword specifies the multiplexing rule and the choice of the codeword from a fixed codebook is used to convey additional important information. The decoder determines the multiplexing rule before decoding the rest of the data. The second method is based on partitioning a signal constellation into disjoint subsets in which the most important data sequence is encoded, using most of the available redundancy, to specify a sequence of subsets. The partitioning and code construction is done to maximize the minimum Euclidean distance between two different valid subset sequences. This leads to ways of partitioning the signal constellations into subsets. The less important data selects a sequence of signal points to be transmitted from the subsets. A side benefit of the proposed set partitioning procedure is a reduction in the number of nearest neighbors, sometimes even over the uncoded signal constellation     

Unequal error protection for convolutional codes

In this correspondence, unequal error-correcting capabilities of convolutional codes are studied. For errors in the information symbols and code symbols, the free input- and output-distances, respectively, serve as "unequal" counterparts to the free distance. When communication takes place close to or above the channel capacity the error bursts tend to be long and the free distance is not any longer useful as the measure of the error correcting capability. Thus, the active burst distance for a given output and the active burst distance for a given input are introduced as "unequal" counterparts to the active burst distance and improved estimates of the unequal error-correcting capabilities of convolutional codes are obtained and illustrated by examples. Finally, it is shown how to obtain unequal error protection for both information and code symbols using woven convolutional codes.     

Sparse-graph codes for quantum error correction

Sparse-graph codes appropriate for use in quantum error-correction are presented. Quantum error-correcting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparse-graph codes keep the number of quantum interactions associated with the quantum error-correction process small: a constant number per quantum bit, independent of the block length. Third, sparse-graph codes often offer great flexibility with respect to block length and rate. We believe some of the codes we present are unsurpassed by previously published quantum error-correcting codes.     

Poor error correction codes are poor error detection codes (Corresp.)

We show that binary group codes that do not satisfy the asymptotic Varshamov-Gilbert bound have an undesirable characteristic when used as error detection codes for transmission over the binary symmetric channel.     

Parallel error correcting codes

We introduce the concept of "parallel error correcting" codes, the error correcting codes for parallel channels. Here, a parallel channel is a set of channels such that the additive error over a finite field occurs in one of its members at time T if the same error occurs in all members at the same time. The set of codewords of a parallel error correcting code has to be a product set, if the messages transmitted are from independent information sources. We present a simple construction of optimal parallel error correcting codes based on ordinary optimal error correcting codes and a construction of optimal linear parallel codes for independent sources based on optimal ordinary linear error correcting codes. The decoding algorithms for these codes are provided as well     

Reed-Solomon codes for correcting phased error bursts

A code structure is introduced that represents a Reed-Solomon (RS) code in two-dimensional format. Based on this structure, a novel approach to multiple error burst correction using RS codes is proposed. For a model of phased error bursts, where each burst can affect one of the columns in a two-dimensional transmitted word, it is shown that the bursts can be corrected using a known multisequence shift-register synthesis algorithm. It is further shown that the resulting codes posses nearly optimal burst correction capability, under certain probability of decoding failure. Finally, low-complexity systematic encoding and syndrome computation algorithms for these codes are discussed. The proposed scheme may also find use in decoding of different coding schemes based on RS codes, such as product or concatenated codes.     

Weighted sum codes for error detection and their comparison withexisting codes

Describes a new family of error detection codes called weighted sum codes. These codes are preferred over four existing codes (CRC, Fletcher checksum, Internet checksum, and XTP CXOR), because they combine powerful error detection properties (as good as the CRC) with attractive implementation properties. One variant, WSC-1, has efficient software and hardware implementations; while a second variant, WSC-2, is almost as efficient in software (still significantly better than CRC) and offers commutative processing (that enables efficient out-of-order, parallel, and incremental update processing)     

Reduced-redundancy product codes for burst error correction

In a typical burst error correction application of a product code of n_v×n_h arrays, one uses an [n_h, n_h-r_h] code C_h that detects corrupted rows, and an [n_v, n_v-r_v] code C_v that is applied to the columns while regarding the detected corrupted rows as erasures. Although this conventional product code scheme offers very good error protection, it contains excessive redundancy, due to the fact that the code C_h provides the code C_v with information on many error patterns that exceed the correction capability of C_v. A coding scheme is proposed in which this excess redundancy is eliminated, resulting in significant savings in the overall redundancy compared to the conventional case, while offering the same error protection. The redundancy of the proposed scheme is n_hr_v+r_h(lnr_v+O(1))+r_v , where the parameters r_h and r_v are close in value to their counterparts in the conventional case, which has redundancy n_hr_v+n_vr_h-r_h r_v. In particular, when the codes C_h and C _v have the same rate and r_h≪n_h, the redundancy of the proposed scheme is close to one-half of that of the conventional product code counterpart. Variants of the scheme are presented for channels that are mostly bursty, and for channels with a combination of random errors and burst errors     

Concatenated Reed-Muller codes for unequal error protection

We present a new coding scheme that combines the advantages of a product-like concatenation of Reed-Muller codes with so-called iterative “turbo” decoding and provides powerful unequal error protection abilities. It is shown that various levels of error protection can be realized using a sophisticated encoding scheme for Reed-Muller codes. A discussion of this code construction, the resulting distance profile between the different levels and the iterative decoding scheme is given. The results are very promising and impressively confirm the unequal error protection capabilities of the presented coding scheme     

QPSK block-modulation codes for unequal error protection

Unequal error protection (UEP) codes find applications in broadcast channels, as well as in other digital communication systems, where messages have different degrees of importance. Binary linear UEP (LUEP) codes combined with a Gray mapped QPSK signal set are used to obtain new efficient QPSK block-modulation codes for unequal error protection. Several examples of QPSK modulation codes that have the same minimum squared Euclidean distance as the best QPSK modulation codes, of the same rate and length, are given. In the new constructions of QPSK block-modulation codes, even-length binary LUEP codes are used. Good even-length binary LUEP codes are obtained when shorter binary linear codes are combined using either the well-known |u¯|u¯+v¯|-construction or the so-called construction X. Both constructions have the advantage of resulting in optimal or near-optimal binary LUEP codes of short to moderate lengths, using very simple linear codes, and may be used as constituent codes in the new constructions. LUEP codes lend themselves quite naturally to multistage decoding up to their minimum distance, using the decoding of component subcodes. A new suboptimal two-stage soft-decision decoding of LUEP codes is presented and its application to QPSK block-modulation codes for UEP illustrated     

On testing for improper error detection codes

A family of tests for improper codes is given. These tests can be used in cases where the complete weight distribution of the code is unknown. It was found that knowledge of the number of minimum weight codewords can be used to greatly increase the effectiveness of the asymptotic Varshamov-Gilbert test. Further improvement is possible as more is known about the number of other weight codewords     

On adaptive hybrid error control in wireless networks using Reed-Solomon codes

This letter models and analyzes an adaptive radio link level error control protocol using Reed-Solomon codes for wireless networks. Results show that the proposed dynamic rate adaptive strategy provides a much improved throughput relative to a conventional type-I and type-II hybrid-automatic repeat request (ARQ) protocols.