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The order and chain polytopes are two 0/1-polytopes constructed from a finite poset. In this paper, we study the $f$-vectors of these polytopes. We investigate how the order and chain polytopes behave under disjoint unions and ordinal sums of posets, and how the $f$-vectors of these polytopes are expressed in terms of $f$-vectors of smaller polytopes. Our focus is on comparing the $f$-vectors of the order and chain polytope built from the same poset. In our main theorem we prove that for a family of posets built inductively by taking disjoint unions and ordinal sums of posets, for any poset $\mathcal{P}$ in this family the $f$-vector of the order polytope of $\mathcal{P}$ is component-wise at most the $f$-vector of the chain polytope of $\mathcal{P}$., Comment: Fixed typos, slight change to terminology, added one reference

Let $M$ be an arbitrary matroid with circuits $\mathcal{C}(M)$. We propose a definition of a derived matroid $\delta M$ that has as its ground set $\mathcal{C}(M)$. Unlike previous attempts of such a definition, our definition applies to arbitrary matroids, and is completely combinatorial. We prove that the rank of $\delta M$ is bounded from above by $|M|-r(M)$, that it is connected if and only if $M$ is connected. We compute examples including the derived matroids of uniform matroids, the V\'amos matroid and the graphical matroid $M(K_4)$. We formulate conjectures relating our construction to previous definitions of derived matroids., Comment: Errors in Section 6 of a previous version corrected

In this work, a flexible and robust private information retrieval (PIR) scheme based on binary non-maximum distance separable (non-MDS) codes is considered. This combines previous works on PIR schemes based on transitive non-MDS codes on one hand, and PIR from MDS-coded Byzantine and non-responsive servers on the other hand. More specifically, a PIR scheme employing binary Reed-Muller (RM) codes tolerant to colluding, Byzantine, and non-responsive servers is constructed, and bounds for the achievable rates are derived under certain conditions. The construction of such schemes turns out to be much more involved than for MDS codes. Namely, the binary query vectors have to be selected with great care to hit the desired information sets, which is technically challenging as will be shown., Comment: conference submission, 6 pages

Inspired by problems in Private Information Retrieval, we consider the setting where two users need to establish a communication protocol to transmit a secret without revealing it to external observers. This is a question of how large a linear code can be, when it is required to agree with a prescribed code on a collection of coordinate sets. We show how the efficiency of such a protocol is determined by the derived matroid of the underlying linear communication code. Furthermore, we provide several sufficient combinatorial conditions for when no secret transmission is possible.

We define and analyze low-rank parity-check (LRPC) codes over extension rings of the finite chain ring $\mathbb{Z}_{p^r}$, where $p$ is a prime and $r$ is a positive integer. LRPC codes have originally been proposed by Gaborit et al.(2013) over finite fields for cryptographic applications. The adaption to finite rings is inspired by a recent paper by Kamche et al. (2019), which constructed Gabidulin codes over finite principle ideal rings with applications to space-time codes and network coding. We give a decoding algorithm based on simple linear-algebraic operations. Further, we derive an upper bound on the failure probability of the decoder. The upper bound is valid for errors whose rank is equal to the free rank.

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.

In this work, two practical concepts related to private information retrieval (PIR) are introduced and coined full support-rank PIR and strongly linear PIR. Being of full support-rank is a technical, yet natural condition required to prove a converse result for a capacity expression and satisfied by almost all currently known capacity-achieving schemes, while strong linearity is a practical requirement enabling implementation over small finite fields with low subpacketization degree. Then, the capacity of MDS-coded, linear, full support-rank PIR in the presence of colluding servers is derived, as well as the capacity of symmetric, linear PIR with colluding, adversarial, and nonresponsive servers for the recently introduced concept of matched randomness. This positively settles the capacity conjectures stated by Freij-Hollanti et al. and Tajeddine et al. in the presented cases. It is also shown that, further restricting to strongly-linear PIR schemes with deterministic linear interference cancellation, the so-called star product scheme proposed by Freij-Hollanti et al. is essentially optimal and induces no capacity loss.

In this work codes with availability are constructed based on the cyclic \emph{locally repairable code} (LRC) construction by Tamo et al. and their extension to $(r,\rho)$-locality by Chen et al. The minimum distance of these codes is increased by carefully extending their defining set. We give a bound on the dimension of LRCs with availability and orthogonal repair sets and show that the given construction is optimal for a range of parameters.

A private information retrieval (PIR) scheme allows a user to retrieve a file from a database without revealing any information on the file being requested. As of now, PIR schemes have been proposed for several kinds of storage systems, including replicated and MDS-coded data. In this paper, the problem of constructing a PIR scheme on regenerating codes is considered. A regenerating code is a storage code whose codewords are distributed among $n$ nodes, enabling efficient storage of files, as well as low-bandwidth retrieval of files and repair of nodes. In this work, a PIR scheme on regenerating codes is constructed, using the product-matrix (PM) framework of Rashmi, Shah and Kumar. Both the minimum-bandwidth (MBR) and minimum-storage (MSR) settings are considered, and the structure given by the PM framework is used in order to reduce the download communication complexity of our schemes.

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 $(r, {\delta})$, 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.

The problem of Private Information Retrieval (PIR) from coded storage systems with colluding, byzantine, and unresponsive servers is considered. An explicit scheme using an $[n,k]$ Reed-Solomon storage code is designed, protecting against $t$-collusion and handling up to $b$ byzantine and $r$ unresponsive servers, when $n>k+t+2b+r-1$. This scheme achieves a PIR rate of $\frac{n-r-(k+2b+t-1)}{n-r}$. In the case where the capacity is known, namely when $k=1$, it is asymptotically capacity-achieving as the number of files grows. Lastly, the scheme is adapted to symmetric PIR., Comment: This is an extended journal version of the ISIT paper arXiv:1802.03731

Recently, information-theoretic private information retrieval (PIR) from coded storage systems has gained a lot of attention, and a general star product PIR scheme was proposed. In this paper, the star product scheme is adopted, with appropriate modifications, to the case of private (e.g., video) streaming. It is assumed that the files to be streamed are stored on~$n$ servers in a coded form, and the streaming is carried out via a convolutional code. The star product scheme is defined for this special case, and various properties are analyzed for two channel models related to straggling and Byzantine servers, both in the baseline case as well as with colluding servers. The achieved PIR rates for the given models are derived and, for the cases where the capacity is known, the first model is shown to be asymptotically optimal, when the number of stripes in a file is large. The second scheme introduced in this work is shown to be the equivalent of block convolutional codes in the PIR setting. For the Byzantine server model, it is shown to outperform the trivial scheme of downloading stripes of the desired file separately without memory.

A private information retrieval (PIR) scheme on coded storage systems with colluding, byzantine, and non-responsive servers is presented. Furthermore, the scheme can also be used for symmetric PIR in the same setting. An explicit scheme using an $[n,k]$ generalized Reed-Solomon storage code is designed, protecting against $t$-collusion and handling up to $b$ byzantine and $r$ non-responsive servers, when $n\geq n'= (\nu +1) k+t+2b+r-1$, for some integer $\nu \geq 1$. This scheme achieves a PIR rate of $1-\frac{k+2b+t+r-1}{n'}$. In the case where the capacity is known, namely when $k=1$, it is asymptotically capacity achieving as the number of files grows., Comment: This is a preliminary conference version of arxiv:1806.08006

This paper presents private information retrieval (PIR) schemes for coded storage with colluding servers, which are not restricted to maximum distance separable (MDS) codes. PIR schemes for general linear codes are constructed and the resulting PIR rate is calculated explicitly. It is shown that codes with transitive automorphism groups yield the highest possible rates obtainable with the proposed scheme. This rate coincides with the known asymptotic PIR capacity for MDS-coded storage systems without collusion. While many PIR schemes in the literature require field sizes that grow with the number of servers and files in the system, we focus especially on the case of a binary base field, for which Reed- Muller codes serve as an important and explicit class of examples.

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., Comment: 9 pages, 1 figure. Parts of this paper were presented at IZS 2018. This extended arxiv version includes corrected versions of Theorem 1.4 and Proposition 6 that appeared in the IZS 2018 proceedings

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 $\mathbb{F}_{2}$-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., Comment: 14 pages, 2 figures

Given a statistical model for the request frequencies and sizes of data objects in a caching system, we derive the probability density of the size of the file that accounts for the largest amount of data traffic. This is equivalent to finding the required size of the cache for a caching placement that maximizes the expected byte hit ratio for given file size and popularity distributions. The file that maximizes the expected byte hit ratio is the file for which the product of its size and popularity is the highest -- thus, it is the file that incurs the greatest load on the network. We generalize this theoretical problem to cover factors and addends of arbitrary order statistics for given parent distributions. Further, we study the asymptotic behavior of these distributions. We give several factor and addend densities of widely-used distributions, and verify our results by extensive computer simulations., Comment: 24 pages, 6 figures, submitted to IEEE Transactions on Communications

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 distribute 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., Comment: Invited book chapter in Network Coding and Subspace Designs, Springer, to appear

In Private Information Retrieval (PIR), one wants to download a file from a database without revealing to the database which file is being downloaded. Much attention has been paid to the case of the database being encoded across several servers, subsets of which can collude to attempt to deduce the requested file. With the goal of studying the achievable PIR rates in realistic scenarios, we generalize results for coded data from the case of all subsets of servers of size $t$ colluding, to arbitrary subsets of the servers. We investigate the effectiveness of previous strategies in this new scenario, and present new results in the case where the servers are partitioned into disjoint colluding groups., Comment: Updated with a corrected statement of Theorem 2

We present a general framework for Private Information Retrieval (PIR) from arbitrary coded databases, that allows one to adjust the rate of the scheme according to the suspected number of colluding servers. If the storage code is a generalized Reed-Solomon code of length n and dimension k, we design PIR schemes which simultaneously protect against t colluding servers and provide PIR rate 1-(k+t-1)/n, for all t between 1 and n-k. This interpolates between the previously studied cases of t=1 and k=1 and asymptotically achieves the known capacity bounds in both of these cases, as the size of the database grows.

We propose a generic strategic network resource sharing game between a set of players representing operators. The players negotiate which sets of players share given resources, serving users with varying sensitivity to interference. We prove that the proposed game has a Nash equilibrium, to which a greedily played game converges. Furthermore, simulation results show that, when applied to inter-operator spectrum sharing in small-cell indoor office environment, the convergence is fast and there is a significant performance improvement for the operators when compared to the default resource usage configuration.

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., Comment: Conference, submitted

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 $(n,k,d,r,\delta)$ 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 $(n,k,d,r,\delta)$-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. Westerb\"ack 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 $d_{\rm{max}}(n,k,r,\delta)$ that a linear LRC with parameters $(n,k,r,\delta)$ can have. This article expands the class of parameters $(n,k,d,r,\delta)$ for which there exist perfect linear LRCs and improves the lower bound for $d_{\rm{max}}(n,k,r,\delta)$. Further, this bound is proved to be optimal for the class of matroids that is used to derive the existence bounds of linear LRCs., Comment: 5 pages, final version, submitted to the 2016 IEEE International Symposium on Information Theory, presents the results of arXiv:1512.05325 in an abridged form

Computing the theta series of an arbitrary lattice, and more specifically a related quantity known as the flatness factor, has been recently shown to be important for lattice code design in various wireless communication setups. However, the theta series is in general not known in closed form, excluding a small set of very special lattices. In this article, motivated by the practical applications as well as the mathematical problem itself, a simple approximation of the theta series of a lattice is derived. A rigorous analysis of its accuracy is provided. In relation to this, maximum-likelihood decoding in the context of compute-and-forward relaying is studied. Following previous work, it is shown that the related metric can exhibit a flat behavior, which can be characterized by the flatness factor of the decoding function. Contrary to common belief, we note that the decoding metric can be rewritten as a sum over a random lattice only when at most two sources are considered. Using a particular matrix decomposition, a link between the random lattice and the code lattice employed at the transmitter is established, which leads to an explicit criterion for code design, in contrast to implicit criteria derived previously. Finally, candidate lattices are examined with respect to the proposed criterion using the derived theta series approximation., Comment: Submitted

In this paper we study a pair of numerical parameters associated to a graph $G$. One the one hand, one can construct $\text{Hom}(K_2, G)$, a space of homomorphisms from a edge $K_2$ into $G$ and study its (topological) connectivity. This approach dates back to the neighborhood complexes introduced by Lov\'asz in his proof of the Kneser conjecture. In another direction Brightwell and Winkler introduced a graph parameter called the warmth $\zeta(G)$ of a graph $G$, based on asymptotic behavior of $d$-branching walks in $G$ and inspired by constructions in statistical physics. Both the warmth of $G$ and the connectivity of $\text{Hom}(K_2,G)$ provide lower bounds on the chromatic number of $G$. Here we seek to relate these two constructions, and in particular we provide evidence for the conjecture that the warmth of a graph $G$ is always less than three plus the connectivity of $\text{Hom}(K_2, G)$. We succeed in establishing a first nontrivial case of the conjecture, by showing that $\zeta(G) \leq 3$ if $\text{Hom}(K_2,G)$ has an infinite first homology group. We also calculate warmth for a family of `twisted toroidal' graphs that are important extremal examples in the context of $\text{Hom}$ complexes. Finally we show that $\zeta(G) \leq n-1$ if a graph $G$ does not have the complete bipartite graph $K_{a,b}$ for $a+b=n$. This provides an analogue for a similar result in the context of $\text{Hom}$ complexes., Comment: 17 pages, 4 figures; V2: corrections and revisions, incorporating feedback from referees

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., Comment: To appear in the proceedings of the 53rd Annual Allerton Conference on Communication, Control, and Computing