Family Of Convolutional Codes Research Articles

A Construction of Maximum Distance Profile Convolutional Codes With Small Alphabet Sizes

Convolutional codes are essential in a wide range of practical applications due to their efficient non-algebraic decoding algorithms. In this paper, we first propose a new family of matrices over finite fields by combining Vandermonde and Moore matrices. Using favourable properties of the matrices in this new family enables us to construct a new family of convolutional codes with memory 1 and maximum distance profile. It is notable that the alphabet sizes of this new family of convolutional codes with maximum distance profile can be kept significantly smaller than those in the literature. Keeping the code rate to a constant, the alphabet size is roughly the square root of the previously best-known value.

A Decoding Algorithm for Convolutional Codes

It is shown how the decoding algorithms of Pellikaan and Rosenthal can be coupled to produce a decoding algorithm for convolutional codes. Bounds for the computational cost per decoded codeword are also computed. As a case study, our results are applied to a family of convolutional codes constructed by Rosenthal–Schumacher–York and, in this situation, the previous bounds turn out to be polynomial on the degree of the code.

On the existence and construction of maximum distance profile convolutional codes

In this paper, we study the conditions for a convolutional code to be MDP in terms of the size of the base field Fq as well as the openness of the MDP property in a given family of convolutional codes. Given (n,k,δ), our main result is an explicit bound depending on (n,k,δ) such that if q is greater than this bound, there exists a (n,k,δ) MDP convolutional code. A similar result is also offered for complete MDP convolutional codes. We show that these bounds are much lower than that those appeared so far in the literature. Finally, we show an explicit and simple construction procedure for MDP convolutional Goppa codes of dimension one.

Classical and Quantum Convolutional Codes Derived From Algebraic Geometry Codes

In this paper, we construct new families of convolutional codes. Such codes are obtained by means of algebraic geometry codes. Additionally, more families of convolutional codes are constructed by means of puncturing, extending, expanding and by the direct product code construction applied to algebraic geometry codes. The parameters of the new convolutional codes are better than or comparable to the ones available in literature. In particular, a family of almost near MDS codes is presented.

Feedback equivalence of convolutional codes over finite rings

Abstract The approach to convolutional codes from the linear systems point of view provides us with effective tools in order to construct convolutional codes with adequate properties that let us use them in many applications. In this work, we have generalized feedback equivalence between families of convolutional codes and linear systems over certain rings, and we show that every locally Brunovsky linear system may be considered as a representation of a code under feedback convolutional equivalence.

Convolutional Codes With Maximum Column Sum Rank for Network Streaming

The column Hamming distance of a convolutional code determines the error correction capability when streaming over a class of packet erasure channels. We introduce a metric known as the column sum rank that parallels the column Hamming distance when streaming over a network with link failures. We prove the rank analogues of several known column Hamming distance properties and introduce a new family of convolutional codes that maximize the column sum rank up to the code memory. Our construction involves finding a class of super-regular matrices that preserve this property after multiplication with non-singular block diagonal matrices in the ground field.

On cyclotomic cosets and code constructions

New properties of q-ary cyclotomic cosets modulo n=qm−1, where q≥3 is a prime power, are investigated in this paper. Based on these properties, the dimension as well as bounds for the designed distance of some families of classical cyclic codes can be computed. As an application, new families of nonbinary Calderbank–Shor–Steane (CSS) quantum codes as well as new families of convolutional codes are constructed in this work. These new CSS codes have parameters better than the ones available in the literature. The convolutional codes constructed here have free distance greater than the ones available in the literature.

CONVOLUTIONAL CODES DERIVED FROM MELAS CODES

A family of convolutional codes is constructed algebraically in this note. These new convolutional codes are derived from Melas codes. The proposed codes have basic generator matrices, consequently, they are non catas- trophic.

Flexible Convolutional Codes: Variable Rate and Complexity

In this study, a method is presented for constructing convolutional codes of variable rate and decoding complexity. Starting with an (n,1,m) mother code, the techniques of puncturing and path pruning are utilized in order to construct large families of convolutional codes of various code rates and complexity. Decoding is performed using the trellis of the mother code.

Maximum Slope Convolutional Codes

The slope is an important distance parameter for a convolutional code. It can be used to obtain a lower bound on the active burst distance and in this respect essentially determines the error-correcting capability of the code. An upper bound on the slope of rate R=b/c convolutional codes is derived. A new family of convolutional codes, called the maximum slope (MS) code family, is introduced. Tables for rate R=1/2 MS codes with memory 1/spl les/m/spl les/6 are presented. Additionally, some new rate R=(c-1)/c, 3/spl les/c/spl les/6, punctured convolutional codes with rate R=1/2 optimum free distance (OFD) and MS mother codes are presented. Simulation results for the bit error performance of serially concatenated turbo codes with MS component codes are presented.

Improvements in Turbo Decoding

In recent years iterative concatenated decoding has regained popularity starting with the remarkable results presented in a paper by a group of French researchers. They introduced a new family of convolutional codes, nicknamed “Turbo codes” due to its resemblance with the turbo engine. A turbo code is built from parallel concatenation of two recursive systematic codes linked together by non-uniform interleaving. Two maximum-a-posteriori probability decoders do decoding iteratively, each using the decoding results from other one. For sufficiently large interleaver size, the error performance seems to be close to Shannon limit.We have extended the implementation principle of turbo codes and have used an encoder, which uses one additional recursive systematic convolutional (RSC) encoder and an interleaver. We show that this multiple turbo code performs better than the standard turbo code. We have also proposed a dynamic decoding algorithm, which uses variable number of iterations. We show that dynamic algorithm requires less time as compared to the other algorithm for achieving the same error performance and that the extended turbo code can achieve better results with small interleaver size.

Near optimum error correcting coding and decoding: turbo-codes

This paper presents a new family of convolutional codes, nicknamed turbo-codes, built from a particular concatenation of two recursive systematic codes, linked together by nonuniform interleaving. Decoding calls on iterative processing in which each component decoder takes advantage of the work of the other at the previous step, with the aid of the original concept of extrinsic information. For sufficiently large interleaving sizes, the correcting performance of turbo-codes, investigated by simulation, appears to be close to the theoretical limit predicted by Shannon.