In this work we study a variant of the Niederreiter cryptosystem. Instead of hiding the parity-check matrix of the secret code with constant matrices, we propose to use polynomial matrices. The public key is then the parity-check matrix of a convolutional code. The plaintext is a sequence of vectors of low weight and the ciphertext is the syndrome of this sequence. We propose the use of a Generalized Reed-Solomon code as the secret code and construct the polynomial matrices to obtain a secure scheme. We analyze its security and show that this approach can be of interest to obtain smaller public keys when compared to the NIST proposal Classic McEliece.

The notion of Generalized Hamming weights of block codes has been investigated since the nineties due to its significant role in coding theory and cryptography. In this paper we extend this concept to the context of convolutional codes. In particular, we focus on column distances and introduce the novel notion of generalized column distances (GCD). We first show that the hierarchy of GCD is strictly increasing. We then provide characterizations of such distances in terms of the truncated parity-check matrix of the code, that will allow us to determine their values. Finally, the case in which the parity-check matrix is in systematic form is treated.

Locally repairable convolutional codes (LRCCs) for distributed storage systems (DSSs) are introduced in this work. They enable local repair, for a single node erasure (or more generally, ∂ - 1 erasures per local group), and sliding-window global repair, which can correct erasure patterns with up to djc-1 erasures in every window of j + 1 consecutive blocks of n nodes, where djcis the jth column distance of the code. The parameter j can be adjusted, for a fixed LRCC, according to different catastrophic erasure patterns, requiring only to contact n(j + 1) - djc+1 nodes, plus less than μn other nodes, in the storage system, where μ is the memory of the code. A Singleton-type bound is provided for dcj. If it attains such a bound, an LRCC can correct the same number of catastrophic erasures in a window of length n(j + 1) as an optimal locally repairable block code of the same rate and locality, and with block length n(j + 1). In addition, the LRCC is able to perform the flexible and somehow local sliding-window repair by adjusting j. Furthermore, by adjusting and/or sliding the window, the LRCC can potentially correct more erasures in the original window of n(j + 1) nodes than an optimal locally repairable block code of the same rate and locality, and length n(j+1). Finally, the concept of partial maximum distance profile (partial MDP) codes is introduced. Partial MDP codes can correct all information-theoretically correctable erasure patterns for a given locality, local distance and information rate. An explicit construction of partial MDP codes whose column distances attain the provided Singleton-type bound, up to certain parameter j = L, is obtained based on known maximum sum-rank distance convolutional codes.

In this work we introduce the concept of Unrestricted Generalized Column Distance (UGCD) for convolutional codes. This is the concept equivalent to the generalized Hamming weights for block codes. We show that the hierarchy of UGCD is strictly increasing and show how to compute it of parity-check matrix in general form. We also provide a way to compute it out of a systematic parity-check matrix.

Locally repairable convolutional codes (LRCCs) for distributed storage systems (DSSs) are introduced in this work. They enable local repair, for a single node erasure, and sliding-window global repair, which can correct up to djc-1 node erasures in a window of j+1 consecutive blocks of n nodes, where djcis the jth column distance of the code. The parameter j can be adjusted, for a fixed LRCC, according to different catastrophic erasure patterns, requiring only to contact n(j+1)-djc+1 nodes, plus less than μn other nodes, in the storage system, where μ is the memory of the code. A Singleton-type bound is provided for djc. If it attains such a bound, an LRCC can correct the same number of catastrophic erasures in a window of length n(j + 1) as an optimal locally repairable block code of the same rate and locality, and with block length n(j + 1), but being able to perform the flexible and somehow local sliding-window repair by adjusting j. Furthermore, by sliding the window to consider previous or consequent nodes without erasures, or by increasing the window size, the LRCC can potentially correct more erasures in the original window of n(j + 1) nodes than the optimal locally repairable block code. Finally, an explicit construction of LRCCs whose column distances attain the provided Singleton-type bound, up to certain parameter j = L, is obtained based on known maximum sum-rank distance convolutional codes.

Maximum distance profile codes over finite nonbinary fields have been introduced and thoroughly studied in the last decade. These codes have the property that their column distances are maximal among all codes of the same rate and degree. In this paper, we aim at studying this fundamental concept in the context of convolutional codes over a finite ring. We extensively use the concept of p-encoder to establish the theoretical framework and derive several bounds on the column distances. In particular, a method for constructing (not necessarily free) maximum distance profile convolutional codes over Zpr is presented.

In this work, we adapt the notion of generalized Hamming weight of block codes to introduce the novel concept of generalized column distances for convolutional codes. This can be considered as an extension of the work done in [18] on the generalized Hamming weights for free distance of convolutional codes. We also introduce the concept of Almost-MDP and Near-MDP convolutional code. The problem of constructing convolutional codes with design generalized column distances remains an interesting open problem that requires further research.

So far, in the area of Random Linear Network Coding, attention has been given to the so-called one-shot network coding, meaning that the network is used just once to propagate the information. In contrast, one can use the network more than once to spread redundancy over different shots. In this paper, we propose rank metric convolutional codes for this purpose. The framework we present is slightly more general than the one which can be found in the literature. We introduce a rank distance, which is suitable for convolutional codes, and derive a new Singleton-like upper bound. Codes achieving this bound are called Maximum Rank Distance (MRD) convolutional codes. Finally, we prove that this bound is optimal by showing a concrete construction of a family of MRD 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 δ, 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 δ that reach such bound when n k(((I(δ/k)J + 2)(I(δ/k)J + 3))/2) if k f δ, or n k((((δ/k) + 1)((δ/k) + 2))/2) if k | δ, by presenting a concrete constructive procedure.