# Patrick SoléAix-Marseille Université | AMU · I2M UMR 7373

Patrick Solé

Docteur-Ingenieur de l' ENST

## About

476

Publications

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Introduction

Discrete Math and their applications: coding theory, algebraic graph theory, with some excursions in analytic number theory

## Publications

Publications (476)

Blockchain is a method of recording information that makes it not feasible for the system to be replaced, attacked, or manipulated. A blockchain is equipped with a notebook that copies and processes the various procedures across the network of computers participating in the blockchain. Digital signature algorithm is one of the cryptographic protoco...

We determine the weight spectrum of $RM(m-3,m)$ for $m\ge 6,$ of $RM(m-4,m)$ for $m\ge 8,$ and of $RM(m-5,m)$ for $m\ge 9.$ The technique used is induction on $m$ based on Corollary 2 of (Shi et al. 2019).

The main motivation of this work is to study and obtain some reversible and DNA codes of length n with better parameters. Here, we first investigate the structure of cyclic and skew cyclic codes over the chain ring R := F 4 [v]/ v 3. We show an association between the codons and the elements of R using a Gray map. Under this Gray map, we study reve...

A necessary condition for a Z 4-code to be a three-weight code for the Lee weight is given. Two special constructions of three-weight codes over Z 4 are derived. The coset graphs of their duals are shown to be strongly 3-walk-regular, a generalization of strongly regular graphs.

A new notion of bent sequence related to Hadamard matrices was introduced recently, motivated by a security application (Solé et al. 2021). In this paper we introduce the analogous notion for complex Hadamard matrices, and we study the self-dual class in length at most 90. We use three competing methods of generation: Brute force, Linear Algebra an...

Strongly walk regular graphs (SWRGs or s-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length 2 are replaced by paths of length s. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible paramete...

We show that no more new distance-regular graphs in the tables of the book of (Brouwer, Cohen, Neumaier, 1989) can be produced by using the coset graph of additive completely regular codes over finite fields.

A new notion of bent sequence related to Hadamard matrices was introduced recently, motivated by a security application (Solé et al, 2021). The subclass of self-dual bent sequences was studied from the standpoint of generation and symmetry (Shi et al., 2022). An analogous study was led for complex Hadamard matrices (Shi et al., submitted). We surve...

An additive cyclic code of length n over F4 can be defined equivalently as an F2[x]/〈xn+1〉-submodule of F4[x]/〈xn+1〉. In this paper we study additive cyclic and complementary dual codes of odd length over F4 with respect to the trace Hermitian inner product and the trace Euclidean inner product. We characterize subfield subcodes and trace codes of...

A linear code is linear complementary dual (LCD) if it meets its dual trivially. LCD codes have been a hot topic recently due to Boolean masking application in the security of embarked electronics (Carlet and Guilley in Pinto et al (eds) Coding theory and applications, Springer, CIMSMS, Berlin, 2015). Additive codes over \(\mathbb {F}_4\) are \(\ma...

A notion of t-designs in the symmetric group on n letters was introduced by Godsil in 1988. In particular, t-transitive sets of permutations form a t-design. We derive upper bounds on the covering radius of these designs, as a function of n and t and in terms of the largest zeros of Charlier polynomials.

Let \({\mathbb {F}}_q\) be a finite field of order \(q=p^m\), where p is an odd prime. This paper presents the study of self-dual and LCD double circulant codes over a class of finite commutative non-chain rings \(R_q={\mathbb {F}}_q+u{\mathbb {F}}_q+u^2{\mathbb {F}}_q+\cdots +u^{q-1}{\mathbb {F}}_q\) where \(u^q=u\). Here, the whole contribution i...

There is a local ring I of order 4, without identity for the multiplication, defined by generators and relations as I=⟨a,b∣2a=2b=0,a2=b,ab=0⟩.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69...

A linear code is linear complementary dual (LCD) if it meets its dual trivially. LCD codes have been a hot topic recently due to Boolean masking application in the security of embarked electronics (Carlet and Guilley, 2014). Additive codes over $\F_4$ are $\F_4$-codes that are stable by codeword addition but not necessarily by scalar multiplication...

We prove that the covering radius of the Melas code
$M(m,q)$
of length
$n=q^{m}-1$
over
$\mathbb {F}_{q}$
is 2 if
$q > 3$
. We also prove that the covering radius of
$M(m,3)$
is 3 is
$m \ge 3$
, the covering radius of
$M(2,3)$
is 4, and the covering radii of
$M(1,2)$
and
$M(1,3)$
are 1.

We construct strongly walk-regular graphs as coset graphs of the duals of codes with three non-zero homogeneous weights over Zpm,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{do...

We show that no more new distance-regular graphs in the tables of the book of (Brouwer, Cohen, Neumaier, 1989) can be produced by using the coset graph of additive completely regular codes over finite fields.

The invention of smart low-power devices and ubiquitous Internet connectivity have facilitated the shift of many labour-intensive jobs into the digital domain. The shortage of skilled workforce and the growing food demand have led the agriculture sector to adapt to the digital transformation. Smart sensors and systems are used to monitor crops, pla...

We study LCD (linear complementary dual) and ACD (additive complementary dual) codes over a noncommutative non-unital ring E with four elements. This is the first attempt to construct LCD codes over a noncommutative non-unital ring. We show that free LCD codes over E are directly related to binary LCD codes. We introduce ACD codes over E. They incl...

A new notion of bent sequence related to Hadamard matrices was introduced recently, motivated by a security application ( Sol\'e et al, 2021). We study the self dual class in length at most $196.$ We use three competing methods of generation: Exhaustion, Linear Algebra and Groebner bases. Regular Hadamard matrices and Bush-type Hadamard matrices pr...

Let $\mathbb{Z}_4$ be the ring of integers modulo $4$. This paper studies mixed alphabets $\mathbb{Z}_4\mathbb{Z}_4[u^3]$-additive cyclic and $\lambda$-constacyclic codes for units $\lambda=1+2u^2,3+2u^2$. First, we obtain the generator polynomials and minimal generating set of additive cyclic codes. Then we extend our study to determine the struct...

Let p be a prime number. Reducible cyclic codes of rank 2 over Zpm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_{p^m}$$\end{document} are shown to have ex...

We present two new constructions of entanglement-assisted quantum error-correcting codes using some fundamental properties of (classical) linear codes in an effective way. The main ideas include linear complementary dual codes and related concatenation constructions. Numerical examples in modest lengths show that our constructions perform better th...

In this paper we study the uncertainty principle (UP) connecting a function over a finite field and its Mattson-Solomon polynomial, which is a kind of Fourier transform in positive characteristic. Three versions of the UP over finite fields are studied, in connection with the asymptotic theory of cyclic codes. We first show that no finite field sat...

Let \begin{document}$ \mathbb{Z}_4 $\end{document} be the ring of integers modulo \begin{document}$ 4 $\end{document}. This paper studies mixed alphabets \begin{document}$ \mathbb{Z}_4\mathbb{Z}_4[u^3] $\end{document}-additive cyclic and \begin{document}$ \lambda $\end{document}-constacyclic codes for units \begin{document}$ \lambda = 1+2u^2,3+2u^2...

Quasi-polycyclic (QP for short) codes over a finite chain ring R are a generalization of quasi-cyclic codes, and these codes can be viewed as an R[x]-submodule of \({\mathcal {R}}_m^{\ell }\), where \({\mathcal {R}}_m:= R[x]/\langle f\rangle \), and f is a monic polynomial of degree m over R. If f factors uniquely into monic and coprime basic irred...

Quasi-polycyclic (QP for short) codes over a finite chain ring $R$ are a generalization of quasi-cyclic codes, and these codes can be viewed as an $R[x]$-submodule of $\mathcal{R}_m^{\ell}$, where $\mathcal{R}_m:= R[x]/\langle f\rangle$, and $f$ is a monic polynomial of degree $m$ over $R$. If $f$ factors uniquely into monic and coprime basic irred...

There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as \(E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle .\) We study a special construction of self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graph...

The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull (LCD codes) and one-dimensional hull by employing the positive characteristic analogues of Gauss sums. These codes are quasi-abelian, and so...

Double Toeplitz (DT) codes are codes with a generator matrix of the form (I,T) with T a Toeplitz matrix, that is to say constant on the diagonals parallel to the main. When T is tridiagonal and symmetric we determine its spectrum explicitly by using Dickson polynomials, and deduce from there conditions for the code to be LCD. Using a special concat...

We investigate fat projective linear codes over
${\mathbb Z}_{p^{m}}$
,
$m\geqslant 2$
, with two nonzero homogeneous weights (“two-weight codes”), building on the graph theory approach developed by Delsarte for codes over fields. Our main result is the classification of such codes under the additional assumption that the columns of a generator...

The decomposition of a quasi-abelian code into shorter linear codes over larger alphabets was given in Jitman and Ling (Des Codes Cryptogr 74:511–531, 2015), extending the analogous Chinese remainder decomposition of quasi-cyclic codes (Ling and Solé in IEEE Trans Inf Theory 47:2751–2760, 2001). We give a concatenated decomposition of quasi-abelian...

We study the codes of the title by the CRT method, that decomposes such codes into constituent codes, which are shorter codes over larger alphabets. Criteria on these constituent codes for self-duality and linear complementary duality of the decomposed codes are derived. The special class of the one-generator codes is given a polynomial representat...

A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in Q-polynomial distance-regular graphs. An approximation of their cumulative distribution function, based on the notion of Christoffel function...

In this paper, for q = pm (p is prime) such that q ≡ 1 (mod e), we study skew constacyclic codes over a class of non-chain rings Re,q=Fq[u]/〈ue−1〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}...

The automorphism group of the Zetterberg code Z of length 17 (also a quadratic residue code) is a rank three group whose orbits on the coordinate pairs determine two strongly regular graphs equivalent to the Paley graph attached to the prime 17. As a consequence, codewords of a given weight of Z are the characteristic vectors of the blocks of a PBI...

A notion of $t$-designs in the symmetric group on $n$ letters was introduced by Godsil in 1988. In particular $t$-transitive sets of permutations form a $t$-design. We derive upper bounds on the covering radius of these designs, as a function of $n$ and $t$ and in terms of the largest zeros of Charlier polynomials.

For a prime p, let \({\mathbb {F}}_{q^2}\) be the finite field of \(q^2\) elements where \(q=p^m\) and \(m\ge 1\) is an integer. In this paper, we study constacyclic and skew constacyclic codes of length n over a class of finite commutative non-chain rings \({\mathcal {A}}_{q,r}={\mathbb {F}}_{q^2}[u_1,u_2,\dots ,u_r]/\langle u^2_i-\gamma _i u_i,u_...

There is a local ring $E$ of order $4,$ without identity for the multiplication, defined by generators and relations as $E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle.$ We study a special construction of self-orthogonal codes over $E,$ based on combinatorial matrices related to two-class association schemes, Strongly Regular Gr...

There is a local ring I of order 4, without identity for the multiplication, defined by generators and relations as $$\begin{aligned} I=\langle a,b \mid 2a=2b=0,\, a^{2}=b,\, \,ab=0 \rangle . \end{aligned}$$We give a natural map between linear codes over I and additive codes over \({\mathbb{F}}_{4},\) that allows for efficient computations. We stud...

A notion of $t$-designs in the symmetric group on $n$ letters was introduced by Godsil in 1988. In particular $t$-transitive sets of permutations form a $t$-design. We derive lower bounds for $t=1$ and $t=2$ by a power moment method. For general $n,$ and $t\le n/2,$ we give a linear programming lower bound involving Charlier polynomials. For specif...

The codewords of weight 10 of the [42, 21, 10] extended binary quadratic residue code are shown to hold a design of parameters 3 − ( 42 , 10 , 18 ). Its automorphism group is isomorphic to P S L ( 2 , 41 ). Its existence can be explained neither by a transitivity argument, nor by the Assmus–Mattson theorem.

There is a local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by generators and relations as [Formula: see text] We study a recursive construction of self-orthogonal codes over [Formula: see text] We classify, up to permutation equivalence, self-orthogonal codes of length [Formula: see text]...

There is a special local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by [Formula: see text] We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over [Formula: s...

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2, and they gave constructions and decoding algorithms. Cyclic codes were consi...

The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull ( LCD codes) and one-dimensional hull by employing the positive characteristic analogues of Gauss sums. These codes are quasi-abelian, and s...

Double Toeplitz (DT) codes are codes with a generator matrix of the form $(I,T)$ with $T$ a Toeplitz matrix, that is to say constant on the diagonals parallel to the main. When $T$ is tridiagonal and symmetric we determine its spectrum explicitly by using Dickson polynomials, and deduce from there conditions for the code to be LCD. Using a special...

Double Toeplitz (shortly DT) codes are introduced here as a generalization of double circulant codes. We show that such a code is isodual, hence formally self-dual. Self-dual DT codes are characterized as double circulant or double negacirculant. Likewise, even DT binary codes are characterized as double circulants. Numerical examples obtained by e...

A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in $Q$-polynomial distance-regular graphs. An approximation of their cumulative distribution function, based on the notion of Christoffel functio...

Let p be a prime of the form \(p=mt+1\), where integers \(t\ge 1, m\ge 2\) and \(R_m=\mathbb {F}_p[u]/\langle u^m-1\rangle .\) Thus, \(R_m\) is a finite commutative non-chain ring. For a given unit \(\lambda \in R_m\), we study \(\lambda \)-constacyclic codes of length n over \(R_m\). The necessary and sufficient conditions for these codes to conta...

This preprint has been withdrawn because result is not new, see [J. Li and D. Wan, Counting subset sums of finite abelian groups, J. Combin. Theory, Ser. A, 119(1) (2012), 170-182]

Let q be an odd prime power, and denote by Fq the finite field with q elements. In this paper, we consider the ring R=Fq+uFq+vFq, where u2=u,v2=v,uv=vu=0 and study double circulant and double negacirculant codes over this ring. We first obtain the necessary and sufficient conditions for a double circulant code to be self-dual (resp. LCD). Then we e...

The codewords of weight $10$ of the $[42,21,10]$ extended binary quadratic residue code are shown to hold a design of parameters $3-(42,10,18).$ Its automorphism group is isomorphic to $PSL(2,41)$. Its existence can be explained neither by a transitivity argument, nor by the Assmus-Mattson theorem.

In this paper, for each of six families of three-valued \begin{document}$ m $\end{document}-sequence correlation, we construct an infinite family of five-weight codes from trace codes over the ring \begin{document}$ R = \mathbb{F}_2+u\mathbb{F}_2 $\end{document}, where \begin{document}$ u^2 = 0. $\end{document} The trace codes have the algebraic st...

This paper mainly study \begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document}-additive codes. A Gray map from \begin{document}$ \mathbb{Z}_{2}^{\alpha}\times\mathbb{Z}_{4}^{\beta}[u] $\end{document} to \begin{document}$ \mathbb{Z}_{4}^{\alpha+2\beta} $\end{document} is defined, and we prove that is a weight preserving and distance prese...

Strongly walk-regular graphs can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters in the binary and ternary case for medium size code lengths. Additionally some theoretical insights on the properties of the feasible parameters...

Two infinite families of Z4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_4$$\end{document}-codes with two nonzero Lee weights are constructed by their gen...

Double polycirculant codes are introduced here as a generalization of double circulant codes. When the matrix of the polyshift is a companion matrix of a trinomial, we show that such a code is isodual, hence formally self-dual. Numerical examples show that the codes constructed have optimal or quasi-optimal parameters amongst formally self-dual cod...

Polycyclic codes are a powerful generalization of cyclic and constacyclic codes. Their algebraic structure is studied here by the theory of invariant subspaces from linear algebra. As an application, a bound on the minimum distance of these codes is derived which outperforms, in some cases, the natural analogue of the BCH bound.

A (t,n)-secret sharing scheme is a method of distribution of information among n participants such that any t>1 of them can reconstruct the secret but any t−1 cannot. A ramp secret sharing scheme is a relaxation of that protocol that allows that some (t−1)-coalitions could reconstruct the secret. In this work, we explore some ramp secret sharing sc...

Linear codes over finite rings are described here as trace codes for a suitable generalization of the trace called a GF-trace. Cyclic codes over Galois rings are given a trace description as well. The main tools are the notion of trace dual bases, in the case of linear codes, and of normal bases of an extension ring over a ring, in the case of cycl...

There exist two semi-local rings of order 6 without identity for the multiplication. We classify up to coordinate permutation self-orthogonal codes of length n and size 6 n/2 over these rings (called here quasi self-dual codes or QSD) till the length n = 8. To any such code is attached canonically a ℤ 6 -code, which, when self-dual, produces an uni...

There is a local ring H of order 4, without identity for the multiplication, defined by generators and relations as H =〈a, b | 2a = 2b = 0, a 2 = 0, b 2 = b, ab = ba = 0〉. We classify self orthogonal codes of length n and size 2 n (called here quasi self-dual codes or QSD) up to the length n = 6. In particular, we classify quasi Type IV codes (a su...

In this paper, we examine a secret sharing scheme based on polynomials over finite fields. In the presented scheme, the shares can be used for the reconstruction of the secret using polynomial multiplication. This scheme is both ideal and perfect.

In this paper we study the uncertainty principle (UP) connecting a function over a finite field and its Mattson-Solomon polynomial, which is a kind of Fourier transform in positive characteristic. Three versions of the UP over finite fields are studied, in connection with the asymptotic theory of cyclic codes. The naive version is the direct analog...

We study mixed alphabet cyclic and constacyclic codes over the two alphabets Z4, the ring of integers modulo 4, and its quadratic extension Z4[u]=Z4+uZ4,u2=0. Their generator polynomials and minimal spanning sets are obtained. Further, under new Gray maps, we find cyclic, quasi-cyclic codes over Z4 as the Gray images of both λ-constacyclic and skew...

A finite group is said to be metacyclic if it has a cyclic normal subgroup N such that G∕N is cyclic. A code over a finite field F is called metacyclic if it is a left ideal in the group algebra FG, with G a metacyclic group. Metacyclic codes are generalizations of dihedral codes, and can be constructed as quasi-cyclic codes with an extra automorph...

We investigate the largest number of nonzero weights of quasi-cyclic codes. In particular, we focus on the function ΓQ(n; ‘; k; q); that is defined to be the largest number of nonzero weights a quasi-cyclic code of index gcd(‘; n), length n and dimension k over 픽q can have, and connect it to similar functions related to linear and cyclic codes. We...

The dual of the Kasami code of length $q^2–1$, with $q$ a power of $2$, is constructed by concatenating a cyclic MDS code of length $q+1$ over $F_q$ with a Simplex code of length $q–1$. This yields a new derivation of the weight distribution of the Kasami code, a new description of its coset graph, and a new proof that the Kasami code is completely...

A secret sharing scheme is a method of assigning shares for a secret to some participants such that only some distinguished subsets of these subsets can recover the secret while other subsets cannot. Such schemes can be used for sharing a private key, for digital signatures or sharing the key that can be used to decrypt the content of a file. There...

In this correspondence, we investigate the covering radius of various types of repetition codes over Z p k ( k ≥ 2 ) with respect to the Lee distance. We determine the exact covering radius of the various repetition codes, which have been constructed using the zero divisors and units in Z p k . We also derive the lower and upper bounds on the cover...

Secret sharing is one of the most important cryptographic protocols. Secret sharing schemes (SSS) have been created to that end. This protocol requires a dealer and several participants. The dealer divides the secret into several pieces ( the shares), and one share is given to each participant. The secret can be recovered once a subset of the parti...

Double circulant codes of length 2n over the non-local ring R=Fq+uFq,u2=u,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R=\mathbb {F}_{q}+u\mathbb {F}_{q}, u^{2}=u,$\e...

We construct strongly walk-regular graphs as coset graphs of the duals of codes with three non-zero homogeneous weights over $\mathbb{Z}_{p^m},$ for $p$ a prime, and more generally over chain rings of depth $m$, and with a residue field of size $q$, a prime power. Infinite families of examples are built from Kerdock and generalized Teichm\"uller co...

Let $p$ be a prime number. Irreducible cyclic codes of length $p^2-1$ and dimension $2$ over the integers modulo $p^h$ are shown to have exactly two nonzero Hamming weights. The construction uses the Galois ring of characteristic $p^h$ and order $p^{2h}.$ When the check polynomial is primitive, the code meets the Griesmer bound of (Shiromoto, Storm...