Patrick G. Farrell

A survey of array error control codes

Array error control codes are linear block or convolutional codes which are constructed from several single parity check or other component codes, assembled in two or more geometrical dimensions or directions, with emphasis on simple component codes and low complexity methods of decoding. This survey attempts to introduce, relate and compare all known types of array codes and their applications. After an introduction to the basic properties and uses of array codes, the second section of the paper describes binary array code constructions for random, burst and cluster error control. The third section describes a number of binary convolutional array codes, for both random and burst error control. Non-binary and byte-oriented array codes, both block and convolutional, are covered in the fourth section, which mentions some quite powerful and yet practical constructions. The fifth section discusses various enhancements which can be applied to array codes. The paper ends with conclusions, open problems, and an extensive set of references.

Resisting the Bergen-Hogan Attack on Adaptive Arithmetic Coding

Arithmetic coding is an optimal data compression algorithm in terms of entropy. As it is a shortest length coding, it is well recognized that the randomness of the output of arithmetic coding is very good. Based on arithmetic coding, we propose a novel encryption algorithm. The algorithm can resist existing attacks on arithmetic coding encryption algorithms. The statistical properties of the system are very good. General approach to attack this algorithm is difficult. The algorithm is easily programmed. The algorithm on its own is also an effective data compression algorithm. The compression ratio is only 2% worse than the original arithmetic coding algorithm.

Constructions for Variable-Length Error-Correcting Codes

Two construction techniques for variable-length error-correcting (VLEC) codes are given. The first uses fixed-length linear codes and anticodes to build new VLEC codes, whereas the second uses a heuristic algorithm to perform a computer search for good VLEC codes. VLEC codes may be used for combined source and channel coding. It is shown that over an additive white Gaussian noise channel the codes so constructed can perform better than standard cascaded source and channel codes with similar parameters.

On variable-length error-correcting codes

A different viewpoint on variable-length error correcting (VLEC) codes is presented, as compared to that found in the literature. Consequently, a maximum likelihood decoding algorithm for binary VLEC codes over the binary symmetric channel (BSC) is derived. It is shown that this algorithm achieves significant coding gain over the /spl alpha/-prompt decoding introduced by Hartnett et. Al. (1990), at the expense of increased complexity.<<ETX>>

Erasure Correction Performance of Linear Block Codes

We estimate the probability of incorrect decoding of a linear block code, used over an erasure channel, via its weight spectrum, and define the weight spectra that allow us to achieve the capacity of the channel and the random coding exponent. We derive the erasure correcting capacity of long binary BCH codes with slowly growing distance and their duals. Concatenated codes of growing length n→∞ and polynomial decoding complexity O(n2), achieving the capacity of the erasure channel (or any other discrete memoryless channel), are considered.

Coded modulation with convolutional codes over rings

The objective of this work is to present a multilevel convolutional coding method based on rings of integers modulo-m (generally m is a power of 2) which is suitable for Coded Modulation schemes. This new method of coded modulation is proposed because of the similarities between an m-ary phase shift keying modulation and the structure of a ring of integers modulo-m. Tables of multilevel transparent systematic convolutional codes with some of their characteristics are also presented. Finally, curves of decoding performance are shown for some convolutional codes suitable for coded 4-PSK and 8-PSK.

New approaches to reduced-complexity decoding

Abstract We examine new approaches to the problem of decoding general linear codes under the strategies of full or bounded hard decoding and bounded soft decoding. The objective is to derive enhanced new algorithms that take advantage of the major features of existing algorithms to reduce decoding complexity. We derive a wide range of results on the complexity of many existing algorithms. We suggest a new algorithm for cyclic codes, and show how it exploits all the main features of the existing algorithms. Finally, we propose a new approach to the problem of bounded soft decoding, and show that its asymptotic complexity is significantly lower than that of any other currently known general algorithm. In addition, we give a characterization of the weight distribution of the average linear code and thus show that the Gilbert-Varshamov bound is tight for virtually all linear codes over any symbol field.

Finite field transforms and symmetry groups

Abstract Decoding methods for error-correcting codes which are based on syndrome look-up tables are of limited use due to the rapidly increasing amount of storage that they require as the number of check digits of the code increases. A method is described which uses shortened syndrome look-up tables in an efficient way, thus providing an improvement with respect to classical syndrome decoding methods. The algorithm can be characterised in general as a type of permutation decoding which uses transform domain information, with the interesting variation that permutations not preserving the code are also allowed.

Multi-Dimensional Ring TCM Codes for Fading Channels

A multi-dimensional non-binary ring trellis coded modulation (R-TCM) scheme suitable for fading channels which is superior to conventional binary TCM is considered.

Bounds On Parity Checks Based Array Codes

A class of general linear block array codes called ParityCheck-Based (PCB) array codes, or v-order array codes, is introduced. The lower bound for the code minimum distance d and the upper bound for the number of information symbols k are provided. For two special cases, v=l and v=2, the lower bounds of code length n versus d and code efficiency R are given, respectively. It turns out, through parameters analysis and computer simulation, that PCB array codes have some attractive advantages both in performance and in practical uses. SUMMARY