相关论文: Bounds and Genericity of Sum-Rank-Metric Codes

We develop a probabilistic framework for large-scale dimension bounds in metric geometry, based on padded decompositions, randomized ball carving on net graphs, and the Lov\'asz Local Lemma. For metric measure spaces with volume doubling…

We couple projective limits of probability measures to direct limits of their symmetry groups. We show that the direct limit group is the group of symmetries of the projective limit probability measure. If projective systems of probability…

We construct asymptotically good nested Calderbank-Shor-Steane (CSS) code pairs from Hsu-Anastasopoulos codes and MacKay-Neal codes. In the fixed-degree regime, we prove that the coding rate stays bounded away from zero and that the…

We establish a general formula for the maximum size of finite length block codes with minimum pairwise distance no less than $d$. The achievability argument involves an iterative construction of a set of radius-$d$ balls, each centered at a…

We construct Gray codes over permutations for the rank-modulation scheme, which are also capable of correcting errors under the infinity-metric. These errors model limited-magnitude or spike errors, for which only single-error-detecting…

We derive a general lower bound for the generalized Hamming weights of nested matrix-product codes, with a particular emphasis on the cases with two and three constituent codes. We also provide an upper bound which is reminiscent of the…

The asymptotic dimension of metric spaces is an important notion in geometric group theory introduced by Gromov. The metric spaces considered in this paper are the ones whose underlying spaces are the vertex-sets of graphs and whose metrics…

In the context of error control in random linear network coding, it is useful to construct codes that comprise well-separated collections of subspaces of a vector space over a finite field. In this paper, the metric used is the so-called…

The set of all error-correcting codes C over a fixed finite alphabet F of cardinality q determines the set of code points in the unit square with coordinates (R(C), delta (C)):= (relative transmission rate, relative minimal distance). The…

Rank-metric codes, defined as sets of matrices over a finite field with the rank distance, have gained significant attention due to their applications in network coding and connections to diverse mathematical areas. Initially studied by…

For any positive integer $q\geq 2$ and any real number $\delta\in(0,1)$, let $\alpha_q(n,\delta n)$ denote the maximum size of a subset of $\mathbb{Z}_q^n$ with minimum Hamming distance at least $\delta n$, where…

We investigate the asymptotic rates of length-$n$ binary codes with VC-dimension at most $dn$ and minimum distance at least $\delta n$. Two upper bounds are obtained, one as a simple corollary of a result by Haussler and the other via a…

A locally recoverable code (LRC code) is a code over a finite alphabet such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. In this paper we derive new finite-length and…

This text contains some notes on the Griesmer bound. In particular, we give a geometric proof of the Griesmer bound for the generalized weights and show that a Solomon--Stiffler type construction attains it if the minimum distance is…

Codes on hypergraphs are an extension of the well-studied family of codes on bipartite graphs. Bilu and Hoory (2004) constructed an explicit family of codes on regular t-partite hypergraphs whose minimum distance improves earlier estimates…

We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized…

This paper extends the study of rank-metric codes in extension fields $\mathbb{L}$ equipped with an arbitrary Galois group $G = \mathrm{Gal}(\mathbb{L}/\mathbb{K})$. We propose a framework for studying these codes as subspaces of the group…

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…

In this paper, several infinite families of codes over the extension of non-unital non-commutative rings are constructed utilizing general simplicial complexes. Thanks to the special structure of the defining sets, the principal parameters…

In this paper, we propose and study $r$-minimal codes, a natural extension of minimal codes which have been extensively studied with respect to Hamming metric, rank metric and sum-rank metric. We first propose $r$-minimal codes in a general…