arXiv subjects

Grégory Berhuy

Publications and source records attributed to Grégory Berhuy.

6 recordsLinked to original sources

arXiv2025

On the decomposability of bilinear spaces of dimension four

In this paper, we study the problem of decomposability of bilinear spaces of dimension four without symmetry, as well as the problem of decomposability of split central simple algebras of degree four with an anti-automorphism. In particular, we show that, contrary to the case of symmetric or skew-symmetric bilinear spaces, these two problems are not equivalent. We will also prove that cohomological invariants do not detect decomposability of bilinear spaces of dimension four in general, whereas the determinant does for split central simple algebras of degree four with an anti-automorphism.

math.RA↗

arXiv2024

On the existence of MRD self-dual codes

In this paper, we investigate the existence of self-dual MRD codes $C\subset L^n$, where $L/F$ is an arbitrary field extension of degree $m\geq n$. We then apply our results to the case of finite fields, and prove that if $m=n$ and $F=\mathbb{F}_q$, a self-dual MRD code exists if and only if $q\equiv n\equiv 3 \ [4].$

cs.IT↗

arXiv2021

Minimal and characteristic polynomials of symmetric matrices in characteristic two

Let $k$ be a field of characteristic two. We prove that a non constant monic polynomial $f\in k[X]$ of degree $n$ is the minimal/characteristic polynomial of a symmetric matrix with entries in $k$ if and only if it is not the product of pairwise distinct inseparable irreducible polynomials. In this case, we prove that $f$ is the minimal polynomial of a symmetric matrix of size $n$. We also prove that any element $α\in k_{alg}$ of degree $n\geq 1$ is the eigenvalue of a symmetrix matrix of size $n$ or $n+1$, the first case happening if and only if the minimal polynomial of $α$ is not the product of pairwise distinct inseparable irreducible polynomials.

math.NT↗

arXiv2020

A Quadratic Form Approach to Construction A of Lattices over Cyclic Algebras

We propose a construction of lattices from (skew-) polynomial codes, by endowing quotients of some ideals in both number fields and cyclic algebras with a suitable trace form. We give criteria for unimodularity. This yields integral and unimodular lattices with a multiplicative structure. Examples are provided.

cs.IT↗

arXiv2019

Rank weights for arbitrary finite field extensions

In this paper, we study several definitions of generalized rank weights for arbitrary finite extensions of fields. We prove that all these definitions coincide, generalizing known results for extensions of finite fields.

cs.IT↗

arXiv2014

Bounds of fast decodability of space time block codes, skew-Hermitian matrices, and Azumaya algebras

We study fast lattice decodability of space-time block codes for $n$ transmit and receive antennas, written very generally as a linear combination $\sum_{i=1}^{2l} s_i A_i$, where the $s_i$ are real information symbols and the $A_i$ are $n\times n$ $\mathbb R$-linearly independent complex valued matrices. We show that the mutual orthogonality condition $A_iA_j^* + A_jA_i^*=0$ for distinct basis matrices is not only sufficient but also necessary for fast decodability. We build on this to show that for full-rate ($l = n^2$) transmission, the decoding complexity can be no better than $|S|^{n^2+1}$, where $|S|$ is the size of the effective real signal constellation. We also show that for full-rate transmission, $g$-group decodability, as defined in [1], is impossible for any $g \ge 2$. We then use the theory of Azumaya algebras to derive bounds on the maximum number of groups into which the basis matrices can be partitioned so that the matrices in different groups are mutually orthogonal---a key measure of fast decodability. We show that in general, this maximum number is of the order of only the $2$-adic value of $n$. In the case where the matrices $A_i$ arise from a division algebra, which is most desirable for diversity, we show that the maximum number of groups is only $4$. As a result, the decoding complexity for this case is no better than $|S|^{\lceil l/2 \rceil}$ for any rate $l$.

cs.IT↗