Patrick Solé

Aix-Marseille University | AMU · I2M UMR 7373

We show that the large Cartesian powers of any graph have log-concave valencies with respect to a ffxed vertex. We show that the series of valencies of distance regular graphs is log-concave, thus improving on a result of (Taylor, Levingston, 1978). Consequences for strongly regular graphs, two-weight codes, and completely regular codes are derived...

Ternary isodual codes and their duals are shown to support 3-designs under mild symmetry conditions. These designs are held invariant by a double cover of the permutation part of the automorphism group of the code. Examples of interest include extended quadratic residues (QR) codes of lengths 14 and 38 whose automorphism groups are PSL(2, 13) and P...

Weighing matrices with entries in the complex cubic and sextic roots of unity are employed to construct Hermitian self-dual codes and Hermitian linear complementary dual codes over the finite field $\mathrm {GF}(4).$ The parameters of these codes are explored for small matrix orders and weights.

Consider the ring Be=Fq+uFq+⋯+ue−1Fq, ue=u (e≥2), where Fq denotes the finite field having q=pm elements (for m≥1 and a prime p), and q≡1(mode−1). Skew constacyclic codes over Be are studied in this paper. We present their generator polynomial and describe the criteria for their complementary duality. Moreover, we derive criteria for these codes to...

The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring R=Fp+w1Fp+w2Fp+w22Fp+w1w2Fp+w1w22Fp, where w12−α2=0, w1w2=w2w1, w23−β2w2=0, and α,β∈Fp∖{0}, for a prime p. We begin by introducing a Gray map defined over R, which is associated with an invertible matrix. We demonstrate its advantages over the can...

Double Toeplitz (shortly DT) codes are introduced here as a generalization of double circulant codes. The authors show that such a code is isodual, hence formally self-dual (FSD). FSD codes form a far-reaching generalization of self-dual codes, the most important class of codes of rate one-half. Self-dual DT codes are characterized as double circul...

Triorthogonal matrices were introduced in Quantum Information Theory in connection with distillation of magic states (Bravyi and Haah (2012)). We give an algorithm to construct binary triorthogonal matrices from binary self-dual codes. Further, we generalize to this setting the classical coding techniques of shortening and extending. We also give s...

We construct a ternary [49,25,7] code from the row span of a Jacobsthal matrix. It is equivalent to a Generalized Quadratic Residue (GQR) code in the sense of van Lint and MacWilliams (1978). These codes are the abelian generalizations of the quadratic residue (QR) codes which are cyclic. The union of the [50,25,8] extension of the said code and it...

Triorthogonal matrices were introduced in quantum information theory in connection with distillation of magic states (Bravyi and Haah in Phys Rev A 86:052329, 2012). We give an algorithm to construct binary triorthogonal matrices from binary self-dual codes. Further, we generalize to this setting the classical coding techniques of shortening and ex...

Irreducible cyclic codes of length p2-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ p^2 - 1 $$\end{document} are constructed as two-weight codes over a chain ring w...

The problem of determining the largest possible number of distinct Hamming weights in several classes of codes over finite fields was studied recently in several papers (Shi et al. in Des Codes Cryptogr 87(1):87–95, 2019, in IEEE Trans Inf Theory 66(11):6855–6862, 2020; Chen et al. in IEEE Trans Inf Theory 69(2):995–1004, 2022). A further problem i...

In this paper, we investigate cyclic codes over the ring E of order 4 and characteristic 2 defined by generators and relations as E=⟨a,b∣2a=2b=0,a2=a,b2=b,ab=a,ba=b⟩. This is the first time that cyclic codes over the ring E are studied. Each cyclic code of length n over E is identified uniquely by the data of an ordered pair of binary cyclic codes...

Let p be a prime number and \(q=p^m\) for some positive integer m. In this paper, we find the possible Hermitian hull dimensions of \(\lambda \)-constacyclic codes over \(R_e={\mathbb {F}}_{q^2}+u{\mathbb {F}}_{q^2} +u^2{\mathbb {F}}_{q^2}+\cdots +u^{e-1}{\mathbb {F}}_{q^2}\), \(u^e=1\) where \({\mathbb {F}}_{q^2}\) is the finite field of \(q^2\) e...

In this note, we study skew cyclic and skew constacyclic codes over the mixed alphabet R=FqR1R2, where q=pm, p is an odd prime with m odd and R1=Fq+uFq with u2=u, and R2=Fq+uFq+vFq with u2=u,v2=v,uv=vu=0. Such codes consist of the juxtaposition of three codes of the same size over Fq,R1, and R2, respectively. We investigate the generator polynomial...

In this paper, we derive a mass formula for the self-orthogonal codes and self-dual codes over a non-commutative non-unitary ring, namely, Ep=a,b|pa=pb=0,a2=a,b2=b,ab=a,ba=b, where a≠b and p is any odd prime. We also give a classification of self-orthogonal codes and self-dual codes over Ep, where p=3,5, and 7, in short lengths.

The build-up method is a powerful class of propagation rules that generate self-dual codes over finite fields and unitary rings. Recently, it was extended to non-unitary rings of order 4, to generate quasi self-dual codes. In the present paper, we introduce three such propagation rules to generate self-orthogonal, self-dual and quasi self-dual code...

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as H=⟨a,b∣2a=2b=0,a2=0,b2=b,ab=ba=0⟩. This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length n. W...

The build-up method is a powerful class of propagation rules that generate self-dual codes over finite fields and unitary rings. Recently, it was extended to non-unitary rings of order four to generate quasi self-dual codes. In the present paper we introduce three such propagation rules to generate self-orthogonal, one-sided self-dual, and self-dua...

We study the structure of self-orthogonal and self-dual codes over two non-unital rings of order four, namely, the commutative ring I=a,b|2a=2b=0,a2=b,ab=0 and the noncommutative ring E=a,b|2a=2b=0,a2=a,b2=b,ab=a,ba=b. We use these structures to give mass formulas for self-orthogonal and self-dual codes over these two rings, that is, we give the fo...

There is a non-unital ring [Formula: see text] of order 4 defined by generators and relations as [Formula: see text]. In this paper, we present special constructions of linear codes over [Formula: see text] from the adjacency matrices of two class association schemes. These consist of either Strongly Regular Graphs (SRGs) or Doubly Regular Tourname...

Modular strongly regular graphs have been introduced by Greaves et al. (Linear Algebra Appl 639:50–80, 2022). We show that a related class of isodual codes is asymptotically good. Equiangular tight frames over finite fields also introduced by the same authors in 2022 are shown here to connect with self-dual codes. We give several examples of equian...

We study the probability of an undetected error for general q-ary codes. We give upper and lower bounds on this quantity, by the Linear Programming and the Polynomial methods, as a function of the length, size, and minimum distance. Sharper bounds are obtained in the important special case of binary Hamming codes. Finally, several examples are give...

The authors study the binary codes spanned by the adjacency matrices of the strongly regular graphs (SRGs) on at most two hundred vertices whose existence is unknown. The authors show that in length less than one hundred they cannot be cyclic, except for the exceptions of the SRGs of parameters (85, 42, 20, 21) and (96, 60, 38, 36). In particular,...

In this paper, we establish a mass formula for self-orthogonal codes, quasi self-dual codes, and self-dual codes over commutative non-unital rings $ {{\mathit {I}_p}} = \left < a, b | pa = pb = 0, a^2 = b, ab = 0 \right > $, where $ p $ is an odd prime. We also give a classification of the three said classes of codes over $ {{\mathit {I}_p}} $ wher...

Let $ E $ and $ I $ denote the two non-unital rings of order 4 in the notation of (Fine, 93) 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 $ and $ I = \langle a, b \mid 2a = 2b = 0, a^2 = b, ab = 0\rangle $. Recently, Alahmadi et al classified quasi self-dual (QSD) codes over the...

Linear codes with complementary duals, or LCD codes, have recently been applied to side-channel and fault injection attack-resistant cryptographic countermeasures. We explain that over characteristic two fields, they exist whenever the permanent of any generator matrix is non-zero. Alternatively, in the binary case, the matroid represented by the c...

A ramp secret sharing scheme is a cryptographic method to encode a secret s into multiple sharess1, s2, … , sn that only from specified subsets of the shares one can recover s. In this paper, we construct a strong rampsecret sharing scheme based on finite fields.

We recall a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields, analogous to the well-known BCH bound for cyclic codes. This BCH-like bound serves as a foundation for proposing some minimum-distance lower bounds for single-generator quasi-twisted (QT) codes. Associating each QT code with a constacyclic code...

In this study, we consider the Euclidean and Galois hulls of multi-twisted (MT) codes over a finite field $\mathbb{F}_{p^e}$ of characteristic $p$. Let $\mathbf{G}$ be a generator polynomial matrix (GPM) of a MT code $\mathcal{C}$. For any $0\le \kappa<e$, the $\kappa$-Galois hull of $\mathcal{C}$, denoted by $h_\kappa\left(\mathcal{C}\right)$, is...

Quantum codes are crucial building blocks of quantum computers. With a self-dual quantum code is attached, canonically, a unique stabilised quantum state. Improving on a previous publication, we show how to determine the coefficients on the basis of kets in these states. Two important ingredients of the proof are algebraic graph theory and quadrati...

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.

In this paper, we present a basic theory of the duality of linear codes over three of the non-unital rings of order four; namely I, E , and H as denoted in (Fine, 1993). A new notion of duality is introduced in the case of the non-commutative ring E . The notion of self-dual codes with respect to this duality coincides with that of quasi self...

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...

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...

A new notion of bent sequence related to Hadamard matrices was introduced recently, motivated by a security application (Solé, et al., 2021). The authors study the self-dual class in length at most 196. The authors use three competing methods of generation: Exhaustion, Linear Algebra and Gröbner bases. Regular Hadamard matrices and Bush-type Hadama...

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...

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...

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...