2D Convolutional Codes Research Articles

Optimal Construction for Decoding 2D Convolutional Codes over an Erasure Channel

In general, the problem of building optimal convolutional codes under a certain criteria is hard, especially when size field restrictions are applied. In this paper, we confront the challenge of constructing an optimal 2D convolutional code when communicating over an erasure channel. We propose a general construction method for these codes. Specifically, we provide an optimal construction where the decoding method presented in the bibliography is considered.

A decoding algorithm for 2D convolutional codes over the erasure channel

<p style='text-indent:20px;'>Two-dimensional (2D) convolutional codes are a generalization of (1D) convolutional codes, which are suitable for transmission over an erasure channel. In this paper, we present a decoding algorithm for 2D convolutional codes over such a channel by reducing the decoding process to several decoding steps applied to 1D convolutional codes. Moreover, we provide constructions of 2D convolutional codes that are specially tailored to our decoding algorithm.</p>

Multidimensional quasi-twisted codes: equivalent characterizations and their relation to multidimensional convolutional codes

We study multidimensional analogues of quasi-twisted codes from different points of view. Their concatenated structure allows us to characterize self-dual and complementary-dual classes of such codes as well as to show that multidimensional quasi-twisted (QT) codes are asymptotically good, together with their self-dual and complementary-dual subclasses. They are naturally related to nD convolutional codes as well. It is known that the minimum distance of quasi-cyclic codes provides a lower bound on the free distance of convolutional codes. An analogous result was shown for certain 1-generator 2D convolutional codes by using quasi-2D-cyclic codes. We prove a similar relation between convolutional codes and the related QT codes first, and then generalize the relation further to certain product convolutional codes and the related product QT codes, which improves the previous result in terms of dimension and number of generators. We also provide two-dimensional ternary and binary codes of modest lengths which yield good parameters.

Serial concatenation of a block code and a 2D convolutional code

In this paper we study two different concatenation schemes of twodimensional (2D) convolutional codes. We consider Fornasini–Marchesini state space representation of 2D linear systems to describe our concatenated codes. Also we present upper and lower bounds on the distance of the proposed concatenated codes.

Series concatenation of 2D convolutional codes by means of input-state-output representations

ABSTRACTIn this paper, we investigate the properties of two-dimensional (2D) convolutional codes which are obtained from series concatenation of two 2D convolutional codes. For this purpose, we confine ourselves to dealing with finite-support 2D convolutional codes and make use of the Fornasini–Marchesini input-state-output (ISO) model representations. Within these ISO representations, we study when the structural properties of modal reachability and modal observability of the two given ISO representations carry over to the resulting 2D convolutional code. Moreover, we provide necessary conditions for obtaining a systematic concatenated convolutional code. Finally, we present a lower bound on its free distance.

MDS 2D convolutional codes with optimal 1D horizontal projections

Two dimensional (2D) convolutional codes is a class of codes that generalizes standard one-dimensional (1D) convolutional codes in order to treat two dimensional data. In this paper we present a novel and concrete construction of 2D convolutional codes with the particular property that their projection onto the horizontal lines yield optimal [in the sense of Almeida et al. (Linear Algebra Appl 499:1–25, 2016)] 1D convolutional codes with a certain rate and certain Forney indices. Moreover, using this property we show that the proposed constructions are indeed maximum distance separable, i.e., are 2D convolutional codes having the maximum possible distance among all 2D convolutional codes with the same parameters. The key idea is to use a particular type of superregular matrices to build the generator matrix.

A minimum distance bound for quasi-nD-cyclic codes

We provide a new concatenated structure for multidimensional quasi-cyclic (QnDC) codes over Fq and we give a trace representation for their codewords, which extends the known representations of QC and nD cyclic codes. Based on these results, we obtain a minimum distance bound for QnDC cyclic codes. Since QnDC codes are naturally related to nD convolutional codes, this bound also applies to a special class of 1-generator 2D convolutional codes.

Maximum Distance Separable 2D Convolutional Codes

Maximum distance separable (MDS) block codes and MDS 1D convolutional codes are the most robust codes for error correction within the class of block codes of a fixed rate and 1D convolutional codes of a certain rate and degree, respectively. In this paper, we generalize this concept to the class of 2D convolutional codes. For that, we introduce a natural bound on the distance of a 2D convolutional code of rate $k/n$ and degree $\delta $ , which generalizes the Singleton bound for block codes and the generalized Singleton bound for 1D convolutional codes. Then, we prove the existence of 2D convolutional codes of rate $k/n$ and degree $\delta $ that reach such bound when $n \geq k (({(\lfloor ({\delta }/{k}) \rfloor + 2)(\lfloor ({\delta }/{k}) \rfloor + 3)})/{2})$ if $k {\nmid } \delta $ , or $n \geq k (({(({\delta }/{k}) + 1)(({\delta }/{k}) + 2)})/{2})$ if $k \mid \delta $ , by presenting a concrete constructive procedure.

Realization of 2D convolutional codes of rate $$\frac{1}{n}$$ by separable Roesser models

In this paper, two-dimensional convolutional codes constituted by sequences in $$(\mathbb F ^n)^{\mathbb{Z }^{2}}$$ where $$\mathbb F $$ is a finite field, are considered. In particular, we restrict to codes with rate $$\frac{1}{n}$$ and we investigate the problem of minimal dimension for realizations of such codes by separable Roesser models. The encoders which allow to obtain such minimal realizations, called R-minimal encoders, are characterized.

A construction of MDS 2D convolutional codes of rate [formula omitted] based on superregular matrices

In this paper two-dimensional convolutional codes with finite support are considered, i.e., convolutional codes whose codewords have compact support indexed in N2 and take values in Fn, where F is a finite field. The main goal of this work is to analyze the (free) distance properties of this type of codes of rate 1/n and degree δ. We first establish an upper bound on the maximum possible distance for these codes. We then present particular constructions of two-dimensional convolutional codes with finite support of rate 1/n and degree δ that attain such a bound and therefore have the maximum distance among all two-dimensional convolutional codes with finite support with the same rate and degree. We call such codes maximum distance separable two-dimensional convolutional codes.

Input-state-output representations and constructions of finite support 2D convolutional codes

Two-dimensional convolutional codes are considered, with codewords having compact support indexed in $\mathbb N$2 and taking values in $\mathbb F$n, where $\mathbb F$ is a finite field. Input-state-output representations of these codes are introduced and several aspects of such representations are discussed. Constructive procedures of such codes with a designed distance are also presented.