arXiv subjects

Benjamin Jany

Publications and source records attributed to Benjamin Jany.

17 recordsLinked to original sources

arXiv2026

Linear Code Conversion in the Merge Regime: General Bounds and Reed-Muller Constructions

Erasure codes are a core component of most existing large-scale distributed storage systems, ensuring reliability against node failures. Recent work has shown that adapting code parameters to changing node failure rates can lead to significant storage savings. The default approach is to re-encode the data under a new code, which consumes substantial system resources. Code conversion was introduced to reduce this cost. However, existing work has mainly focused on conversions within specific classes of codes. In this paper, we study scalar linear code conversion in the merge regime for arbitrary linear codes. We derive universal lower bounds on the write and read costs in terms of unchanged and read symbols. The bounds are refined using generalized Hamming weights, which capture support-growth properties of subcodes and can give sharper estimates than minimum-distance-only arguments. We show that the framework recovers known bounds for important special cases and can be strictly stronger when the final code has nontrivial jumps in its generalized Hamming weight hierarchy. We then apply the framework to Reed-Muller codes and construct explicit Reed-Muller convertible codes using the Plotkin decomposition. For a natural Reed-Muller parameter regime, the construction attains the derived write-cost lower bound. For the read cost, the generalized-Hamming-weight analysis is sharp for one initial block, while a gap remains for the other block.

cs.IT↗

arXiv2026

Counting q-Matroids

$q$-Matroids, a $q$-analogue of classical matroids have attracted a lot of attention over the last decade, yet their enumeration remains largely unexplored. In this paper, we study the number of $q$-matroids, paving and sparse-paving $q$-matroids defined on a fixed ground space and with prescribed rank. We derive new lower bounds using constructions from constant-dimension codes and improve existing estimates. On the upper bound side, we develop two approaches: a combinatorial method based on controlling the number of dependent hyperplanes for paving $q$-matroids, and an entropy-based counting argument applicable to classes of $q$-matroids closed under contraction. These techniques yield explicit upper bounds on the logarithmic number of $q$-matroids with fixed rank and ground space. Finally, we analyze the asymptotic behavior of these bounds, and identify gaps between lower and upper estimates, leading to conjectures on the true asymptotic growth.

math.CO↗

arXiv2026

Intersecting Codes and the Connectivity of $q$-Matroids

We investigate the structure of intersecting error-correcting codes, with a particular focus on their connection to matroid theory. We establish properties and bounds for intersecting codes with the Hamming metric and illustrate how these distinguish the subfamily of minimal codes within the family of intersecting codes. We prove that the property of a code being intersecting is characterized by the matroid-theoretic notion of vertical connectivity, showing that intersecting codes are precisely those achieving the highest possible value of this parameter. We then introduce the concept of vertical connectivity for $q$-matroids and link it to the theory of intersecting codes endowed with the rank metric.

math.CO↗

arXiv2026

Convertible Codes for Data and Device Heterogeneity

Distributed storage systems must handle both data heterogeneity, arising from non-uniform access demands, and device heterogeneity, caused by time-varying node reliability. In this paper, we study convertible codes, which enable the transformation of one code into another with minimum cost in the merge regime, addressing the latter. We derive general lower bounds on the read and write costs of linear code conversion, applicable to arbitrary linear codes. We then focus on Reed-Muller codes, which efficiently handle data heterogeneity, addressing the former issue, and construct explicit conversion procedures that, for the first time, combine both forms of heterogeneity for distributed data storage.

cs.IT↗

arXiv2025

The Star Product of Uniformly Random Codes

We consider the problem of determining the expected dimension of the star product of two uniformly random linear codes that are not necessarily of the same dimension. We use a correspondence between the star product and the evaluation of bilinear forms to provide an explicit lower bound on the expected star product dimension. We prove that the expected dimension asymptotically reaches its maximum possible value as the field size increases. Furthermore, we show that the same maximal dimension is achieved asymptotically as the code dimensions increase, subject to a condition bounding their relative growth rates. We also analyze the variance of the star product dimension, providing explicit asymptotic upper bounds. Finally, we discuss the implications of these results for private information retrieval, secure distributed matrix multiplication, quantum error correction, and cryptanalysis.

cs.IT↗

arXiv2025

Polynomial Invariants of q-Matroids and Rank-Metric Codes

It is shown that the Whitney function of a representable q-matroid and the collection of all higher weight enumerators of any representing rank-metric code determine each other via a monomial substitution. Moreover, the q-matroid itself and the collection of all higher support enumerators of the code determine each other. Next, it is proven that the Whitney function of a q-matroid and the Whitney function of its projectivization determine each other via a monomial substitution. Finally, q-matroids with isomorphic projectivizations are studied. It is shown that the projectivizations are isomorphic iff the q-matroids admit a dimension-preserving lattice isomorphism between their lattices of flats. Such q-matroids are called weakly isomorphic.

math.CO↗

arXiv2025

Eigenvalue bounds for the quantum chromatic number of graph powers

The quantum chromatic number, a generalization of the chromatic number, was first defined in relation to the non-local quantum coloring game. We generalize the former by defining the quantum $k$-distance chromatic number $\chi_{kq}(G)$ of a graph $G$, which can be seen as the quantum chromatic number of the $k$-th power graph, $G^k$, and as generalization of the classical $k$-distance chromatic number $\chi_k(G)$ of a graph. It can easily be shown that $\chi_{kq}(G) \leq \chi_k(G)$. In this paper, we strengthen three classical eigenvalue bounds for the $k$-distance chromatic number by showing they also hold for the quantum counterpart of this parameter. This shows that several bounds by Elphick et al. [J. Combinatorial Theory Ser. A 168, 2019, Electron. J. Comb. 27(4), 2020] hold in the more general setting of distance-$k$ colorings. As a consequence we obtain several graph classes for which $\chi_{kq}(G)=\chi_{k}(G)$, thus increasing the number of graphs for which the quantum parameter is known.

math.CO↗

arXiv2025

The Cloud and Flock Polynomials of q-Matroids

We show that the Whitney function of a q-matroid can be determined from the cloud and flock polynomials associated to the cyclic flats. These polynomials capture information about the corank (resp., nullity) of certain spaces whose cyclic core (resp., closure) is the given cyclic flat. Going one step further, we prove that the Whitney function, and in fact the cloud-flock lattice, are determined by the configuration of the q-matroid, which is the abstract lattice of cyclic flats together with the corank-nullity data. Furthermore, we show that the configuration and cloud-flock lattice behave well under duality and direct sums, whereas the Whitney function does not contain enough information to behave well under taking direct sums. As an aside we show that every configuration of a matroid arises as a configuration of a q-matroid, whereas the converse is not true.

math.CO↗

arXiv2024

$t$-Balanced Codes with the Kendall-$\tau$ Metric

We investigate the maximum cardinality and the mathematical structure of error-correcting codes endowed with the Kendall-$\tau$ metric. We establish an averaging bound for the cardinality of a code with prescribed minimum distance, discuss its sharpness, and characterize codes attaining it. This leads to introducing the family of $t$-balanced codes in the Kendall-$\tau$ metric. The results are based on novel arguments that shed new light on the structure of the Kendall-$\tau$ metric space.

math.CO↗

arXiv2023

LRCs: Duality, LP Bounds, and Field Size

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.

cs.IT↗

arXiv2023

Decompositions of q-Matroids Using Cyclic Flats

We study the direct sum of q-matroids by way of their cyclic flats. Using that the rank function of a q-matroid is fully determined by the cyclic flats and their ranks, we show that the cyclic flats of the direct sum of two q-matroids are exactly all the direct sums of the cyclic flats of the two summands. This simplifies the rank function of the direct sum significantly. A q-matroid is called irreducible if it cannot be written as a (non-trivial) direct sum. We provide a characterization of irreducibility in terms of the cyclic flats and show that every q-matroid can be decomposed into a direct sum of irreducible q-matroids, which are unique up to equivalence.

math.CO↗

arXiv2022

Duality and LP Bounds for Codes with Locality

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.

cs.IT↗

arXiv2022

Representability of the Direct Sum of $q$-Matroids

While there are many parallels between matroid theory and $q$-matroid theory, most notably on the level of cryptomorphisms, there are substantial differences when it comes to the direct sum. The direct sum of $q$-matroids has been introduced in the literature only recently. In this short note we show that the direct sum of representable $q$-matroids may not be representable. It remains an open question whether representability of the direct sum can be characterized by the given $q$-matroids.

math.CO↗

arXiv2022

The Projectivization Matroid of a $q$-Matroid

In this paper, we investigate the relation between a $q$-matroid and its associated matroid called the projectivization matroid. The latter arises by projectivizing the groundspace of the $q$-matroid and considering the projective space as the groundset of the associated matroid on which is defined a rank function compatible with that of the $q$-matroid. We show that the projectivization map is a functor from categories of $q$-matroids to categories of matroids, which allows to prove new results about maps of $q$-matroids. We furthermore show the characteristic polynomial of a $q$-matroid is equal to that of the projectivization matroid. We use this relation to establish a recursive formula for the characteristic polynomial of a $q$-matroid in terms of the characteristic polynomial of its minors. Finally we use the projectivization matroid to prove a $q$-analogue of the critical theorem in terms of $\mathbb{F}_{q^m}$-linear rank metric codes and $q$-matroids.

math.CO↗

arXiv2021

Coproducts in Categories of q-Matroids

q-Matroids form the q-analogue of classical matroids. In this paper we introduce various types of maps between q-matroids. These maps are not necessarily linear, but they map subspaces to subspaces and respect the q-matroid structure in certain ways. The various types of maps give rise to different categories of q-matroids. We show that only one of these categories possesses a coproduct. This is the category where the morphisms are linear weak maps, that is, the rank of the image of any subspace is not larger than the rank of the subspace itself. The coproduct in this category is the very recently introduced direct sum of q-matroids.

math.CO↗

arXiv2021

Independent Spaces of q-Polymatroids

This paper is devoted to the study of independent spaces of q-polymatroids. With the aid of an auxiliary q-matroid it is shown that the collection of independent spaces satisfies the same properties as for q-matroids. However, in contrast to q-matroids, the rank value of an independent space does not agree with its dimension. Nonetheless, the rank values of the independent spaces fully determine the q-polymatroid, and this fact can be exploited to derive a cryptomorphism of q-polymatroids. Finally, the notions of minimal spanning spaces, maximally strongly independent spaces, and bases will be elaborated on.

math.CO↗

arXiv2021

q-Polymatroids and Their Relation to Rank-Metric Codes

It is well known that linear rank-metric codes give rise to q-polymatroids. Analogously to matroid theory one may ask whether a given q-polymatroid is representable by a rank-metric code. We provide an answer by presenting an example of a q-matroid that is not representable by any linear rank-metric code and, via a relation to paving matroids, provide examples of various q-matroids that are not representable by F_{q^m}-linear rank-metric codes. We then go on and introduce deletion and contraction for q-polymatroids and show that they are mutually dual and correspond to puncturing and shortening of rank-metric codes. Finally, we introduce a closure operator along with the notion of flats and show that the generalized rank weights of a rank-metric code are fully determined by the flats of the associated q-polymatroid.

cs.IT↗