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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, $\partial - 1$ erasures per local group), and sliding-window global repair, which can correct erasure patterns with up to ${\mathrm{d}}_{j}^{c} - 1$ erasures in every window of $j+1$ consecutive blocks of $n$ nodes, where ${\mathrm{d}}_{j}^{c} - 1$ is the $j$ th 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 $n(j+1) - {\mathrm{d}}_{j}^{c} + 1$ nodes, plus less than $\mu n$ other nodes, in the storage system, where $\mu$ is the memory of the code. A Singleton-type bound is provided for ${\mathrm{d}}_{j}^{c} - 1$. 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. [ABSTRACT FROM AUTHOR]

Maximum distance profile codes over finite non-binary 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 ${\mathbb Z}_{p^{r}}$ is presented. [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]