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We study the Random Access Problem in DNA storage, which addresses the challenge of retrieving a specific information strand from a DNA-based storage system. Given that $k$ information strands, representing the data, are encoded into $n$ strands using a code. The goal under this paradigm is to identify and analyze codes that minimize the expected number of reads required to retrieve any of the $k$ information strand, while in each read one of the $n$ encoded strands is read uniformly at random. We fully solve the case when $k=2$, showing that the best possible code attains a random access expectation of $0.914 \cdot 2$. Moreover, we generalize a construction from \cite{GMZ24}, specific to $k=3$, for any value of $k$. Our construction uses $B_{k-1}$ sequences over $\mathbb{Z}_{q-1}$, that always exist over large finite fields. For $k=4$, we show that this generalized construction outperforms all previous constructions in terms of reducing the random access expectation .

The performance of Reed-Solomon codes (RS codes, for short) in the presence of insertion and deletion errors has been studied recently in several papers. In this work, we further study this intriguing mathematical problem, focusing on two regimes. First, we study the question of how well full-length RS codes perform against insertions and deletions. For 2-dimensional RS codes, we fully characterize which codes cannot correct even a single insertion or deletion and show that (almost) all 2-dimensional RS codes correct at least $1$ insertion or deletion error. Moreover, for large enough field size $q$, and for any $k \ge 2$, we show that there exists a full-length $k$-dimensional RS code that corrects $q/10k$ insertions and deletions. Second, we focus on rate $1/2$ RS codes that can correct a single insertion or deletion error. We present a polynomial time algorithm that constructs such codes for $q = O(k^4)$. This result matches the existential bound given in \cite{con2023reed}.

Effective and reliable data retrieval is critical for the feasibility of DNA storage, and the development of random access efficiency plays a key role in its practicality and reliability. In this paper, we study the Random Access Problem, which asks to compute the expected number of samples one needs in order to recover an information strand. Unlike previous work, we took a geometric approach to the problem, aiming to understand which geometric structures lead to codes that perform well in terms of reducing the random access expectation (Balanced Quasi-Arcs). As a consequence, two main results are obtained. The first is a construction for $k=3$ that outperforms previous constructions aiming to reduce the random access expectation. The second, exploiting a result from [1], is the proof of a conjecture from [2] for rate $1/2$ codes in any dimension.

DNA labeling is a tool in molecular biology and biotechnology to visualize, detect, and study DNA at the molecular level. In this process, a DNA molecule is labeled by a set of specific patterns, referred to as labels, and is then imaged. The resulting image is modeled as an $(\ell+1)$-ary sequence, where $\ell$ is the number of labels, in which any non-zero symbol indicates the appearance of the corresponding label in the DNA molecule. The labeling capacity refers to the maximum information rate that can be achieved by the labeling process for any given set of labels. The main goal of this paper is to study the minimum number of labels of the same length required to achieve the maximum labeling capacity of 2 for DNA sequences or $\log_2q$ for an arbitrary alphabet of size $q$. The solution to this problem requires the study of path unique subgraphs of the de Bruijn graph with the largest number of edges and we provide upper and lower bounds on this value.

We investigate the fundamental limits of the recently proposed random access coverage depth problem for DNA data storage. Under this paradigm, it is assumed that the user information consists of $k$ information strands, which are encoded into $n$ strands via a generator matrix $G$. During the sequencing process, the strands are read uniformly at random, as each strand is available in a large number of copies. In this context, the random access coverage depth problem refers to the expected number of reads (i.e., sequenced strands) required to decode a specific information strand requested by the user. This problem heavily depends on the generator matrix $G$, and besides computing the expectation for different choices of $G$, the goal is to construct matrices that minimize the maximum expectation over all possible requested information strands, denoted by $T_{\max}(G)$. In this paper, we introduce new techniques to investigate the random access coverage depth problem, capturing its combinatorial nature and identifying the structural properties of generator matrices that are advantageous. We establish two general formulas to determine $T_{\max}(G)$ for arbitrary generator matrices. The first formula depends on the linear dependencies between columns of $G$, whereas the second formula takes into account recovery sets and their intersection structure. We also introduce the concept of recovery balanced codes and provide three sufficient conditions for a code to be recovery balanced. These conditions can be used to compute $T_{\max}(G)$ for various families of codes, such as MDS, simplex, Hamming, and binary Reed-Muller codes. Additionally, we study the performance of modified systematic MDS and simplex matrices, showing that the best results for $T_{\max}(G)$ are achieved with a specific combination of encoded strands and replication of the information strands.

We present the theory of linear rank-metric codes from the point of view of their fundamental parameters. These are: the minimum rank distance, the rank distribution, the maximum rank, the covering radius, and the field size. The focus of this chapter is on the interplay among these parameters and on their significance for the code's (combinatorial) structure. The results covered in this chapter span from the theory of optimal codes and anticodes to very recent developments on the asymptotic density of MRD codes., Comment: Invited book chapter (to appear)

We propose and develop a theory of Ferrers diagrams and their $q$-rook polynomials solely based on their diagonals. We show that the cardinalities of the diagonals of a Ferrers diagram are equivalent information to their rook numbers, $q$-rook polynomials, and the rank distribution of matrices supported on the diagram. Our approach is based on the concept of \textit{canonical form} of a Ferrers diagrams, and on two simple diagram operations as the main proof tools. In the second part of the paper we develop the same theory for symmetric Ferrers diagrams, considering symmetric and alternating matrices supported on them. As an application of our results, we establish some combinatorial identities linking symmetric and alternating matrices, which do not appear to have an obvious bijective proof, and which generalize some curious results in enumerative combinatorics.

We develop a duality theory of locally recoverable codes (LRCs) and apply it to establish a series of new bounds on their parameters. We introduce and study a refined notion of weight distribution that captures the code's locality. Using a duality result analogous to a MacWilliams identity, we then derive an LP-type bound that improves on the best known bounds in several instances. Using a dual distance bound and the theory of generalized weights, we obtain non-existence results for optimal LRCs over small fields. In particular, we show that an optimal LRC must have both minimum distance and block length relatively small compared to the field size.

We initiate the study of the duality theory of locally recoverable codes, with a focus on the applications. We characterize the locality of a code in terms of the dual code, and introduce a class of invariants that refine the classical weight distribution. In this context, we establish a duality theorem analogous to (but very different from) a MacWilliams identity. As an application of our results, we obtain two new bounds for the parameters of a locally recoverable code, including an LP bound that improves on the best available bounds in several instances.

In 2009, Etzion and Siberstein proposed a conjecture on the largest dimension of a linear space of matrices over a finite field in which all nonzero matrices are supported on a Ferrers diagram and have rank bounded below by a given integer. Although several cases of the conjecture have been established in the past decade, proving or disproving it remains to date a wide open problem. In this paper, we take a new look at the Etzion-Siberstein Conjecture, investigating its connection with rook theory. Our results show that the combinatorics behind this open problem is closely linked to the theory of $q$-rook polynomials associated with Ferrers diagrams, as defined by Garsia and Remmel. In passing, we give a closed formula for the trailing degree of the $q$-rook polynomial associated with a Ferrers diagram in terms of the cardinalities of its diagonals. The combinatorial approach taken in this paper allows us to establish some new instances of the Etzion-Silberstein Conjecture using a non-constructive argument. We also solve the asymptotic version of the conjecture over large finite fields, answering a current open question.

We introduce and explore a new concept of evasive subspace with respect to a collection of subspaces sharing a common dimension, most notably partial spreads. We show that this concept generalises known notions of subspace scatteredness and evasiveness. We establish various upper bounds for the dimension of an evasive subspace with respect to arbitrary partial spreads, obtaining improvements for the Desarguesian ones. We also establish existence results for evasive spaces in a non-constructive way, using a graph theory approach. The upper and lower bounds we derive have a precise interpretation as bounds for the critical exponent of certain combinatorial geometries. Finally, we investigate connections between the notion of evasive space we introduce and the theory of rank-metric codes, obtaining new results on the covering radius and on the existence of minimal vector rank-metric codes.

We investigate two fundamental questions intersecting coding theory and combinatorial geometry, with emphasis on their connections. These are the problem of computing the asymptotic density of MRD codes in the rank metric, and the Critical Problem for combinatorial geometries by Crapo and Rota. Using methods from semifield theory, we derive two lower bounds for the density function of full-rank, square MRD codes. The first bound is sharp when the matrix size is a prime number and the underlying field is sufficiently large, while the second bound applies to the binary field. We then take a new look at the Critical Problem for combinatorial geometries, approaching it from a qualitative, often asymptotic, viewpoint. We illustrate the connection between this very classical problem and that of computing the asymptotic density of MRD codes. Finally, we study the asymptotic density of some special families of codes in the rank metric, including the symmetric, alternating and Hermitian ones. In particular, we show that the optimal codes in these three contexts are sparse.

We consider the problem of describing the typical (possibly) non-linear code of minimum distance bounded from below over a large alphabet. We concentrate on block codes with the Hamming metric and on subspace codes with the injection metric. In sharp contrast with the behavior of linear block codes, we show that the typical non-linear code in the Hamming metric of cardinality $q^{n-d+1}$ is far from having minimum distance $d$, i.e., from being MDS. We also give more precise results about the asymptotic proportion of block codes with good distance properties within the set of codes having a certain cardinality. We then establish the analogous results for subspace codes with the injection metric, showing also an application to the theory of partial spreads in finite geometry.

In this paper, we develop the theory of convolutional codes over finite commutative chain rings. In particular, we focus on maximum distance profile (MDP) convolutional codes and we provide a characterization of these codes, generalizing the one known for fields. Moreover, we relate (reverse) MDP convolutional codes over a finite chain ring with (reverse) MDP convolutional codes over its residue field. Finally, we provide a construction of (reverse) MDP convolutional codes over finite chain rings generalizing the notion of (reverse) superregular matrices., Comment: 19 pages

Motivated by applications to the theory of rank-metric codes, we study the problem of estimating the number of common complements of a family of subspaces over a finite field in terms of the cardinality of the family and its intersection structure. We derive upper and lower bounds for this number, along with their asymptotic versions as the field size tends to infinity. We then use these bounds to describe the general behaviour of common complements with respect to sparseness and density, showing that the decisive property is whether or not the number of spaces to be complemented is negligible with respect to the field size. By specializing our results to matrix spaces, we obtain upper and lower bounds for the number of MRD codes in the rank metric. In particular, we answer an open question in coding theory, proving that MRD codes are sparse for all parameter sets as the field size grows, with only very few exceptions. We also investigate the density of MRD codes as their number of columns tends to infinity, obtaining a new asymptotic bound. Using properties of the Euler function from number theory, we then show that our bound improves on known results for most parameter sets. We conclude the paper by establishing general structural properties of the density function of rank-metric codes.