Sergo Shavgulidze

Woven codes with outer warp: variations, design, and distance properties

We consider convolutional and block encoding schemes which are variations of woven codes with outer warp. We propose methods to evaluate the distance characteristics of the considered codes on the basis of the active distances of the component codes. With this analytical bounding technique, we derived lower bounds on the minimum (or free) distance of woven convolutional codes, woven block codes, serially concatenated codes, and woven turbo codes. Next, we show that the lower bound on the minimum distance can be improved if we use designed interleaving with unique permutation functions in each row of the warp of the woven encoder. Finally, with the help of simulations, we get upper bounds on the minimum distance for some particular codes and then investigate their performance in the Gaussian channel. Throughout this paper, we compare all considered encoding schemes by means of examples, which illustrate their distance properties.

An Introduction to Generalized Concatenated Codes

This paper presents an introduction to the theory of generalized concatenated codes. We consider the encoding and decoding procedures of both ordinary concatenated codes, and generalized concatenated codes where block codes are used as inner codes. We give the main parameters of these codes, and by means of examples, show that under the same conditions the latter outperforms the former with respect to minimum code distance. We also heuristically describe the generalized concatenated decoding procedure. Then we show how the generalized concatenated coding ideas can be applied to encoded memoryless modulation. Such construction is often referred to as a multilevel code in literature. We consider the multilevel codes from the standpoint of generalized concatenated codes and with the help of a simple example, show how encoding and decoding procedures can be carried out. Ordinary concatenated and generalized concatenated coding schemes are considered next, using inner convolutional codes or inner modulation with memory. With the help of examples, we analyze the distance properties of such constructions.

On the error exponent for woven convolutional codes with outer warp

In this correspondence, the error exponents and decoding complexity of binary woven convolutional codes with outer and inner warp are studied. It is shown that for both constructions an error probability that is exponentially decreasing with the memory of the woven convolutional codes can be achieved with a nonexponentially increasing decoding complexity. Furthermore, the error exponent for woven convolutional codes with inner warp is larger than the one for woven convolutional codes with outer warp.

Generalized concatenation of convolutional codes

Generalized concatenation of inner and outer convolutional codes is studied. The essential problem is the partitioning of the inner code. A construction procedure to obtain a suitable equivalent description for a given code is derived, which gives the best possible distances in the partitioned subcodes. The construction rule is illustrated with two examples. A small code is studied to emphasize the ideas of the construction and a large code, namely the satellite standard code, demonstrates the considerable gains in decoding performance when using generalized concatenation. Furthermore, the possibilities for iterative decoding are pointed out.

Partially concatenated convolutional codes

We present a new concatenated code construction. The resulting codes can be viewed as intermediate between parallel and serially concatenated convolutional codes. Proper partitioning of the outer code sequence provides a new degree of freedom for code design. Various methods are considered to analyze code properties.

New Coding Techniques for Codes over Gaussian Integers

This work presents block codes over Gaussian integers. We introduce Gaussian integer rings which extend the number of possible signal constellations over Gaussian integer fields. Many well-known code constructions can be used for codes over Gaussian integer rings, e.g., the Plotkin construction or product codes. These codes enable low complexity decoding in the complex domain. Furthermore, we demonstrate that the concept of set partitioning can be applied to Gaussian integers. This enables multilevel code constructions. In addition to the code constructions, we present a low complexity soft-input decoding algorithm for one Mannheim error correcting codes. The presented decoding method is based on list decoding, where the list of candidate codewords is obtained by decomposing the syndrome into two sub-syndromes. Considering all decompositions of the syndrome we construct lists of all possible errors of Mannheim weight two. In the last decoding step the squared Euclidean distance is used to select the best codeword from the list. Simulation results for the additive white Gaussian noise channel demonstrate that the proposed decoding method achieves a significant coding gain compared with hard-input decoding.

Woven coded continuous phase frequency shift keying

We introduce woven coded continuous phase frequency shift keying (WCCPFSK) as a serial concatenation of several outer convolutional codes and inner continuous phase frequency shift keying (CPFSK). WCCPFSK is a generalization of the serially concatenated CPFSK modulation scheme which uses only one outer convolutional code. We compute the active burst distances of CPFSK and estimate the free distance of systems with one outer code and of WCCPFSK with several outer codes. We show that WCCPFSK with several outer codes achieves the larger free distance. This results in better performance at low bit error rates, which we demonstrate in simulations.

Generalized concatenation of encoded tamed frequency modulation

A novel construction for encoded tamed frequency modulation (TFM) is introduced which is based on the principles of generalized concatenation. The inner TFM is partitioned into nested subsystems which increases the free Euclidean distances. In order to obtain a large distance among the nested TFM subsystems, the scrambler matrices have to be computed which transfer the original TFM into the equivalent TFM with better partitioning properties. Then outer convolutional codes with different error-correcting capabilities are used to protect the partitioning. The new concatenated and generalized concatenated constructions were simulated in an additive white Gaussian noise channel. A multistep decoding algorithm based on soft-output demodulation was used. We present various simulation results which show a significant coding gain in comparison with the best known trellis codes having the same trellis state complexity.

A Soft Input Decoding Algorithm for Generalized Concatenated Codes

This paper proposes a soft input decoding algorithm and a decoder architecture for generalized concatenated (GC) codes. The GC codes are constructed from inner nested binary Bose-Chaudhuri-Hocquenghem (BCH) codes and outer Reed-Solomon codes. In order to enable soft input decoding for the inner BCH block codes, a sequential stack decoding algorithm is used. Ordinary stack decoding of binary block codes requires the complete trellis of the code. In this paper, a representation of the block codes based on the trellises of supercodes is proposed in order to reduce the memory requirements for the representation of the BCH codes. This enables an efficient hardware implementation. The results for the decoding performance of the overall GC code are presented. Furthermore, a hardware architecture of the GC decoder is proposed. The proposed decoder is well suited for applications that require very low residual error rates.

Generalized concatenated codes for correcting two-dimensional clusters of errors and independent errors

Correction of two-dimensional error clusters is required in many storage systems. Most known error correction systems for such errors are based on two-dimensional interleaving and interleaved codes. However, interleaved codes are not well suited to correct additional independent errors. This work proposes a novel combination of generalized concatenated codes with two-dimensional interleaving to correct two-dimensional error clusters and independent errors.