In graph theory and its applications to networking, such as telecommunications or transportation, path-finding is a central problem. While single-path algorithms are well established, methods for handling sets of multiple paths are less developed. A companion paper introduced a formal model for defining attributes over sets of paths based on their structural properties; this paper addresses that model's practical implementation. We present an optimized algorithm for computing cut sets of a path set, a nontrivial task that can be infeasible without efficient methods, and validate its performance via systematic benchmarks on network simulations of varying complexity. Additionally, we introduce a vectorized computational framework that expresses property calculations as matrix operations, enabling concise implementations in array-oriented languages. Together, these contributions establish practical foundations for the companion model, demonstrating that its implementation is both feasible and characterized by predictable, acceptable execution times.

In graph theory and its practical networking applications, e.g., telecommunications and transportation, the problem of finding paths has particular importance. Selecting paths requires giving scores to the alternative solutions to drive a choice. While previous studies have provided comprehensive evaluation of single-path solutions, the same level of detail is lacking when considering sets of paths. This paper emphasizes that the path characterization strongly depends on the properties under consideration. While property-based characterization is also valid for single paths, it becomes crucial to analyse multiple path sets. From the above consideration, this paper proposes a mathematical approach, defining a functional model that lends itself well to characterizing the path set in its general formulation. The paper shows how the functional model contextualizes specific attributes.

Fault tree analysis is a system malfunction hazard evaluation quantitative and qualitative procedure. The method is well-known and widely used, especially in the safety systems domain, where it is a mandatory integral part of the so-called "Hazard Evaluation" documentation. This paper proposes an alternative or complementary deductive fault analysis method: it uses system topology to build a hypergraph representation of the system to identify component criticality and support loss of functionality probability evaluation. Once automated, the proposed method seems promising when the system engineers explore the different architectures. They may have indication about architecture's reliability without continuous feedback from the system safety team. The system safety team must check the solution once the engineers select the final architecture. They can also use the proposed method to validate the correctness of the fault tree analysis. Copyright (c) 2024 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

In this paper, first steps are taken towards characterising lattices of cyclic flats $\mathcal{Z}(M)$ that belong to matroids $M$ that can be represented over a prescribed finite field $\mathbb{F}_q$. Two natural maps from $\mathcal{Z}(M)$ to the lattice of cyclic flats of a minor of $M$ are given. Binary matroids are characterised via their lattice of cyclic flats. It is shown that the lattice of cyclic flats of a simple binary matroid without isthmuses is atomic.

In this letter, locally recoverable codes with maximal recoverability are studied with a focus on identifying the MDS codes resulting from puncturing and shortening. By using matroid theory and the relation between MDS codes and uniform minors, the list of all the possible uniform minors is derived. This list is used to improve the known non-asymptotic lower bound on the required field size of a maximally recoverable code.

Locally repairable codes (LRCs) have gained significant interest for the design of large distributed storage systems as they allow a small number of erased nodes to be recovered by accessing only a few others. Several works have thus been carried out to understand the optimal rate-distance tradeoff, but only recently the size of the alphabet has been taken into account. In this paper, a novel definition of locality is proposed to keep track of the precise number of nodes required for a local repair when the repair sets do not yield MDS codes. Then, a new alphabet-dependent bound is derived, which applies both to the new definition and the initial definition of locality. The new bound is based on consecutive residual codes and intrinsically uses the Griesmer bound. A special case of the bound yields both the extension of the Cadambe-Mazumdar bound and the Singleton-type bound for codes with locality (r, δ), implying that the new bound is at least as good as these bounds. Furthermore, an upper bound on the asymptotic rate-distance tradeoff of LRCs is derived, and yields the tightest known upper bound for large relative minimum distances. Achievability results are also provided by deriving the locality of the family of Simplex codes together with a few examples of optimal codes.

Recent research on distributed storage systems (DSSs) has revealed interesting connections between matroid theory and locally repairable codes (LRCs). The goal of this chapter is to introduce the reader to matroids and polymatroids, and illustrate their relation to distributed storage systems. While many of the results are rather technical in nature, effort is made to increase accessibility via simple examples. The chapter embeds all the essential features of LRCs, namely locality, availability, and hierarchy alongside with related generalised Singleton bounds.

Recent research on distributed storage systems (DSSs) has revealed interesting connections between locally repairable codes (LRCs) and their associated matroids and polymatroids. In this paper we define L-polymatroids — polymatroids with an added length function — in order to consider completely general LRCs in that they are defined as subsets of A = A1×· · ·×An, where each Ai is some arbitrary finite set. Earlier research in this area has only considered codes over non-mixed alphabets, i.e., A1 = · · · = An. We generalize the notions of locality and availability to Lpolymatroids, and a Singleton-type bound for L-polymatroids is given. This result implies a corresponding bound on LRCs and generalizes earlier Singleton-type bounds given on LRCs. Moreover, the necessary structural conditions are given for L-polymatroids achieving the bound, yielding also the corresponding necessary conditions for LRCs. Finally, implications of our results for quasi-uniform codes and in particular quasiuniform codes from a construction built on cosets of groups are examined.

The concept of parity check matrices of linear binary codes has been extended by Heden [9] to parity check systems of nonlinear binary codes. In the present paper we extend this concept to parity check systems of nonlinear codes over finite commutative Frobenius rings. Using parity check systems, results on how to get some fundamental properties of the codes are given. Moreover, parity check systems and its connection to characters is investigated and a MacWilliams type theorem on the distance distribution is given.

Recently, locally repairable codes has gained significant interest for their potential applications in distributed storage systems. However, most constructions in existence are over fields with size that grows with the number of servers, which makes the systems computationally expensive and difficult to maintain. Here, we study linear locally repairable codes over the binary field, tolerating multiple local erasures. We derive bounds on the minimum distance on such codes, and give examples of LRCs achieving these bounds. Our main technical tools come from matroid theory, and as a byproduct of our proofs, we show that the lattice of cyclic flats of a simple binary matroid is atomic.

Locally repairable codes for distributed storage systems have gained a lot of interest recently, and various constructions can be found in the literature. However, most of the constructions result in either large field sizes and hence too high computational complexity for practical implementation, or in low rates translating into waste of the available storage space. In this paper we address this issue by developing theory towards code existence and design over a given field. This is done via exploiting recently established connections between linear locally repairable codes and matroids, and using matroid-theoretic characterisations of linearity over small fields. In particular, nonexistence can be shown by finding certain forbidden uniform minors within the lattice of cyclic flats. It is shown that the lattice of cyclic flats of binary matroids have additional structure that significantly restricts the possible locality properties of 𝔽_2 -linear storage codes. Moreover, a collection of criteria for detecting uniform minors from the lattice of cyclic flats of a given matroid is given, which is interesting in its own right.

Locally repairable codes (LRCs) are error correcting codes used in distributed data storage. Besides a global level, they enable errors to be corrected locally, reducing the need for communication between storage nodes. There is a close connection between almost affine LRCs and matroid theory which can be utilized to construct good LRCs and derive bounds on their performance. This article presents two improvements to such results in [T. Westerb\ack et al., On the Combinatorics of Locally Repairable Codes, Arxiv: 1501.00153]: The class of parameters $(n,k,d,r,\delta)$ for which there exists a matroid achieving the generalized Singleton bound for LRCs, is expanded. Also, an improved lower bound is given for $d_{\rm{max}}(n,k,r,\delta)$, the maximal achievable minimum distance $d$ that a matroid with parameters $(n,k,r,\delta)$ can have. This bound is proved to be optimal for the main class of matroids used to derive the existence bounds in [T. Westerb\ack et al., On the Combinatorics of Locally Repairable Codes, Arxiv: 1501.00153] and in this article. The results obtained directly translate to similar results on LRCs using the connection between them and matroids.

We consider the problem of fast zeta and Mobius transforms in finite posets, particularly in lattices. It has previously been shown that for a certain family of lattices, zeta and Mobius transforms can be computed in O(e) elementary arithmetic operations, where e denotes the size of the covering relation. We show that this family is exactly that of geometric lattices. We also extend the algorithms so that they work in e operations for all semimodular lattices, including chains and divisor lattices. Finally, for both transforms, we provide a more general algorithm that works in e operations for all ER-labelable posets.

This paper provides a link between matroid theory and locally repairable codes (LRCs) that are either linear or more generally almost affine. Using this link, new results on both LRCs and matroid theory are derived. The parameters (n, k, d, r, δ) of LRCs are generalized to matroids, and the matroid analog of the generalized singleton bound by Gopalan et al. for linear LRCs is given for matroids. It is shown that the given bound is not tight for certain classes of parameters, implying a nonexistence result for the corresponding locally repairable almost affine codes that are coined perfect in this paper. Constructions of classes of matroids with a large span of the parameters (n, k, d, r, δ) and the corresponding local repair sets are given. Using these matroid constructions, new LRCs are constructed with prescribed parameters. The existence results on linear LRCs and the nonexistence results on almost affine LRCs given in this paper strengthen the nonexistence and existence results on perfect linear LRCs given by Song et al.

Typically, locally repairable codes (LRCs) and regenerating codes have been studied independently of each other, and it has not been clear how the parameters of one relate to those of the other. In this paper, a novel connection between locally repairable codes and exact regenerating codes is established. Via this connection, locally repairable codes are interpreted as exact regenerating codes. Further, some of these codes are shown to perform better than time-sharing codes between minimum bandwidth regenerating and minimum storage regenerating codes.

In this paper, locally repairable codes with all-symbol locality are studied. Methods to modify already existing codes are presented. It is also shown that, with high probability, a random matrix with a few extra columns guaranteeing the locality property is a generator matrix for a locally repairable code with a good minimum distance. The proof of the result provides a constructive method to find locally repairable codes. Finally, constructions of three infinite classes of optimal vector-linear locally repairable codes over a small alphabet independent of the code size are given.

Locally repairable codes (LRCs) are error correcting codes used in distributed data storage. Besides a global level, they enable errors to be corrected locally, reducing the need for communication between storage nodes. There is a close connection between almost affine LRCs and matroid theory which can be utilized to construct good LRCs and derive bounds on their performance. A generalized Singleton bound for linear LRCs with parameters (n; k; d; r; δ) was given in [N. Prakash et al., “Optimal Linear Codes with a Local-Error-Correction Property”, IEEE Int. Symp. Inf. Theory]. In this paper, a LRC achieving this bound is called perfect. Results on the existence and nonexistence of linear perfect (n; k; d; r; δ)-LRCs were given in [W. Song et al., “Optimal locally repairable codes”, IEEE J. Sel. Areas Comm.]. Using matroid theory, these existence and nonexistence results were later strengthened in [T. Westerbäck et al., “On the Combinatorics of Locally Repairable Codes”, Arxiv: 1501.00153], which also provided a general lower bound on the maximal achievable minimum distance d max (n; k; r; δ) that a linear LRC with parameters (n; k; r; δ) can have. This article expands the class of parameters (n; k; d; r; δ) for which there exist perfect linear LRCs and improves the lower bound for d max (n; k; r; δ). Further, this bound is proved to be optimal for the class of matroids that is used to derive the existence bounds of linear LRCs.

We present a construction of demi-matroids, a generalization of matroids, from linear codes over finite Frobenius rings, as well as a Greene-type identity for rank generating functions of demi-matroids. We also prove a MacWilliams-type identity for Hamming support enumerators of linear codes over finite Frobenius rings. As a special case, these results give a combinatorial proof of the MacWilliams identity for Hamming weight enumerators of linear codes over finite Frobenius rings.

In this paper, a link between polymatroid theory and locally repairable codes (LRCs) is established. The codes considered here are completely general in that they are subsets of $A^n$, where $A$ is an arbitrary finite set. Three classes of LRCs are considered, both with and without availability, and for both information-symbol and all-symbol locality. The parameters and classes of LRCs are generalized to polymatroids, and a general- ized Singelton bound on the parameters for these three classes of polymatroids and LRCs is given. This result generalizes the earlier Singleton-type bounds given for LRCs. Codes achieving these bounds are coined perfect, as opposed to the more common term optimal used earlier, since they might not always exist. Finally, new constructions of perfect linear LRCs are derived from gammoids, which are a special class of matroids. Matroids, for their part, form a subclass of polymatroids and have proven useful in analyzing and constructing linear LRCs.