On the weight distribution of convolutional codes

Detailed information about the weight distribution of a convolutional code is given by the adjacency matrix of the state diagram associated with a controller canonical form of the code. We will show that this matrix is an invariant of the code. Moreover, it will be proven that codes with the same adjacency matrix have the same dimension and the same Forney indices and finally that for one-dimensional binary convolutional codes the adjacency matrix determines the code uniquely up to monomial equivalence

Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound

We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show relations between them. A slightly improved version of the so-called linkage construction is presented which is e.g. used to construct constant dimension codes with subspace distance $d=4$, dimension $k=3$ of the codewords for all field sizes $q$, and sufficiently large dimensions $v$ of the ambient space, that exceed the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.Comment: 30 pages, 3 table

Column Rank Distances of Rank Metric Convolutional Codes

In this paper, we deal with the so-called multi-shot network coding, meaning that the network is used several times (shots) to propagate the information. The framework we present is slightly more general than the one which can be found in the literature. We study and introduce the notion of column rank distance of rank metric convolutional codes for any given rate and finite field. Within this new framework we generalize previous results on column distances of Hamming and rank metric convolutional codes [3, 8]. This contribution can be considered as a continuation follow-up of the work presented in [10]

Decoding of 2D convolutional codes over an erasure channel

In this paper we address the problem of decoding 2D convolutional codes over an erasure channel. To this end we introduce the notion of neighbors around a set of erasures which can be considered an analogue of the notion of sliding window in the context of 1D convolutional codes. The main idea is to reduce the decoding problem of 2D convolutional codes to a problem of decoding a set of associated 1D convolutional codes. We first show how to recover sets of erasures that are distributed on vertical, horizontal and diagonal lines. Finally we outline some ideas to treat any set of erasures distributed randomly on the 2D plane. © 2016 AIMS

A state space approach to periodic convolutional codes

In this paper we study periodically time-varying convolutional codes by means of input-state-output representations. Using these representations we investigate under which conditions a given time-invariant convolutional code can be transformed into an equivalent periodic time-varying one. The relation between these two classes of convolutional codes is studied for period 2. We illustrate the ideas presented in this paper by constructing a periodic time-varying convolutional code from a time-invariant one. The resulting periodic code has larger free distance than any time-invariant convolutional code with equivalent parameters

Periodic state-space representations of periodic convolutional codes

In this paper we study the representation of periodically time-varying convolutional codes by means of periodic input-state-output models. In particular, we focus on period two and investigate under which conditions a given two-periodic convolutional code (obtained by alternating two time-invariant encoders) can be represented by a periodic input-state-output system. We first show that one cannot expect, in general, to obtain a periodic input-state-output representation of a periodic convolutional code by means of the individual realizations of each of the associated time-invariant codes. We, however, provide sufficient conditions for this to hold in terms of the column degrees of the associated column reduced generator matrices. Moreover, we derive a sufficient condition to obtain a periodic state-space realization that is minimal. Finally, examples to illustrate the results are presented.publishe

Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be published in Lecture Notes in Computer Scienc