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Patrick Sole

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PhD Télécom Paris Tech 1987, HDR Nice Sophia Antipolis 1993.
Directeur de Recherche au CNRS.
Institut de Mathématiques de Marseille, Marseilles, France.
Supervisors: Gérard D. Cohen

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Papers by Patrick Sole

Additive complementary dual codes over F 4

HAL (Le Centre pour la Communication Scientifique Directe), Jul 5, 2022

A linear code is linear complementary dual (LCD) if it meets its dual trivially. LCD codes have b... more 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. An additive code over F 4 is additive complementary dual (ACD) if it meets its dual trivially. The aim of this research is to study such codes which meet their dual trivially. All the techniques and problems used to study LCD codes are potentially relevant to ACD codes. Interesting constructions of ACD codes from binary codes are given with respect to the trace Hermitian and trace Euclidean inner product. The former product is relevant to quantum codes.

On the structure of 1-generator quasi-polycyclic codes over finite chain rings

Journal of Applied Mathematics and Computing

Quasi-polycyclic (QP for short) codes over a finite chain ring R are a generalization of quasi-cy... more 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 R ℓ m , where R m := R[x]/ f , and f is a monic polynomial of degree m over R. If f factors uniquely into monic and coprime basic irreducibles, then their algebraic structure allow us to characterize the generator polynomials and the minimal generating sets of 1-generator QP codes as R-modules. In addition, we also determine the parity check polynomials for these codes by using the strong Gröbner bases. In particular, via Magma system, some quaternary codes with new parameters are derived from these 1-generator QP codes.

Some ternary cubic two-weight codes

arXiv (Cornell University), Dec 3, 2016

We study trace codes with defining set L, a subgroup of the multiplicative group of an extension ... more We study trace codes with defining set L, a subgroup of the multiplicative group of an extension of degree m of the alphabet ring F 3 + uF 3 + u 2 F 3 , with u 3 = 1. These codes are abelian, and their ternary images are quasi-cyclic of co-index three (a.k.a. cubic codes). Their Lee weight distributions are computed by using Gauss sums. These codes have three nonzero weights when m is singly-even and |L| = 3 3m −3 2m 2. When m is odd, and |L| = 3 3m −3 2m 2 , or |L| = 3 3m − 3 2m and m is a positive integer, we obtain two new infinite families of two-weight codes which are optimal. Applications of the image codes to secret sharing schemes are also given.

Construction of MDS self-dual codes from orthogonal matrices

arXiv (Cornell University), Oct 25, 2016

In this paper 1 , we give algorithms and methods of construction of self-dual codes over finite f... more In this paper 1 , we give algorithms and methods of construction of self-dual codes over finite fields using orthogonal matrices. Randomization in the orthogonal group, and code extension are the main tools. Some optimal, almost MDS, and MDS self-dual codes over both small and large prime fields are constructed.

Covering Radius of Melas Codes

IEEE Transactions on Information Theory, 2022

We prove that the covering radius of the Melas code M (m, q) of length n = q m − 1 over Fq is 2 i... more

On isodual double Toeplitz codes

ArXiv, 2021

Double Toeplitz (shortly DT) codes are introduced here as a generalization of double circulant co... more 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 exhaustive search show that the codes constructed have best-known minimum distance, up to one unit, amongst formally self-dual codes, and sometimes improve on the known values. Over F4 an explicit construction of DT codes, based on quadratic residues in a prime field, performs equally well. We show that DT codes are asymptotically good over Fq. Specifically, we construct DT codes arbitrarily close to the asymptotic varshamov-Gilbert bound for codes of rate one half.

Three-weight codes and the quintic construction

ArXiv, 2016

We construct a class of three-Lee-weight and two infinite families of five-Lee-weight codes over ... more We construct a class of three-Lee-weight and two infinite families of five-Lee-weight codes over the ring $R=\mathbb{F}_2 +v\mathbb{F}_2 +v^2\mathbb{F}_2 +v^3\mathbb{F}_2 +v^4\mathbb{F}_2,$ where $v^5=1.$ The same ring occurs in the quintic construction of binary quasi-cyclic codes. %The length of these codes depends on the degree $m$ of ring extension. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Given a linear Gray map, we obtain three families of binary abelian codes with few weights. In particular, we obtain a class of three-weight codes which are optimal. Finally, an application to secret sharing schemes is given.

Optimal three-weight cubic codes

ArXiv, 2016

In this paper, we construct an infinite family of three-weight binary codes from linear codes ove... more In this paper, we construct an infinite family of three-weight binary codes from linear codes over the ring $R=\mathbb{F}_2+v\mathbb{F}_2+v^2\mathbb{F}_2$, where $v^3=1.$ These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distributions are computed by employing character sums. The three-weight binary linear codes which we construct are shown to be optimal when $m$ is odd and $m>1$. They are cubic, that is to say quasi-cyclic of co-index three. An application to secret sharing schemes is given.

Trace Codes with Few Weights over Fp+uFp

ArXiv, 2016

We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the chain ring ... more We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the chain ring Fp+uFp. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. Then by using a linear Gray map, we obtain an infinite family of abelian codes with few weights over Fp. In particular, we obtain an infinite family of two-weight codes which meets the Griesmer bound with equality. Finally, an application to secret sharing schemes is given.

Self-orthogonal codes over a non-unital ring and combinatorial matrices

Designs, Codes and Cryptography, 2021

There is a local ring E of order 4, without identity for the multiplication, defined by generator... more There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as E = a, b | 2a = 2b = 0, a 2 = a, b 2 = b, ab = a, ba = b. We study a special construction of self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), and Doubly Regular Tournaments (DRT). We construct quasi self-dual codes over E, and Type IV codes, that is, quasi self-dual codes whose all codewords have even Hamming weight. All these codes can be represented as formally self-dual additive codes over F 4. The classical invariant theory bound for the weight enumerators of this class of codes improves the known bound on the minimum distance of Type IV codes over E.

Good Stabilizer Codes from Quasi-Cyclic Codes over F4 and F9

2019 IEEE International Symposium on Information Theory (ISIT), 2019

We apply quantum Construction X on quasi-cyclic codes with large Hermitian hulls over F4 and F9 t... more We apply quantum Construction X on quasi-cyclic codes with large Hermitian hulls over F4 and F9 to derive good qubit and qutrit stabilizer codes, respectively. In several occasions we obtain quantum codes with stricly improved parameters than the current record. In numerous other occasions we obtain quantum codes with best-known performance. For the qutrit ones we supply a systematic construction to fill some gaps in the literature.

Construction of isodual codes from polycirculant matrices

Designs, Codes and Cryptography, 2020

Double polycirculant codes are introduced here as a generalization of double circulant codes. Whe... more 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 codes. Self-duality, the trivial case of isoduality, can only occur over F 2 in the double circulant case. Building on an explicit infinite sequence of irreducible trinomials over F 2 , we show that binary double polycirculant codes are asymptotically good.

How Many Weights Can a Cyclic Code Have?

IEEE Transactions on Information Theory, 2019

Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimensi... more Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), whose nonzero codewords have period equal to the length of the code. Asymptotics are derived on the function Γ(k, q), that is defined as the largest number of nonzero weights a cyclic code of dimension k over F q can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller codes, either binary or q-ary, as well as in the q-ary Hamming code are determined, two difficult results of independent interest.

On self-dual and LCD quasi-twisted codes of index two over a special chain ring

Cryptography and Communications, 2018

Let q be a prime power, and let F q denote the finite field of order q. Consider the chain ring R... more Let q be a prime power, and let F q denote the finite field of order q. Consider the chain ring R k = F q [u]/ u k with k ≥ 1 an integer. We study self-dual and LCD quasi-twisted codes of index two and twisting constant λ over R k for the metric induced by the standard Gray map. Some special factorizations of x m − λ over R k are studied. By random coding, we obtain four classes of asymptotically good self-dual λ-circulant codes and four classes of asymptotically good LCD λ-circulant codes over R k. Keywords Double circulant codes • Gray map • Self-dual codes • LCD codes • Quasi-twisted codes Mathematics Subject Classification (2010) 94 B15 • 94 B25 • 05 E30 This research is supported by National Natural Science Foundation of China (61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133) and Key projects of support program for outstanding young talents in Colleges and Universities (gxyqZD2016008).

Constructions of optimal LCD codes over large finite fields

Finite Fields and Their Applications, 2018

In this paper, we prove existence of optimal complementary dual codes (LCD codes) over large fini... more In this paper, we prove existence of optimal complementary dual codes (LCD codes) over large finite fields. We also give methods to generate orthogonal matrices over finite fields and then apply them to construct LCD codes. Construction methods include random sampling in the orthogonal group, code extension, matrix product codes and projection over a self-dual basis.

Two-weight and three-weight codes from trace codes over Fp+uFp+vFp+uvFp

Discrete Mathematics, 2018

We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the non-chain r... more We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the non-chain ring F p + uF p + vF p + uvF p , where u 2 = 0, v 2 = 0, uv = vu. These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. With a linear Gray map, we obtain a class of abelian three-weight codes and two-weight codes over F p. In particular, the two-weight codes we describe are shown to be optimal by application of the Griesmer bound. We also discuss their dual Lee distance. Finally, an application to secret sharing schemes is given.

On self-dual negacirculant codes of index two and four

Designs, Codes and Cryptography, 2018

We study the asymptotic performance of quasi-twisted codes viewed as modules in the ring R = F q ... more We study the asymptotic performance of quasi-twisted codes viewed as modules in the ring R = F q [x]/ x n +1 , when they are self-dual and of length 2n or 4n. In particular, in order for the decomposition to be amenable to analysis, we study factorizations of x n + 1 over F q , with n twice an odd prime, containing only three irreducible factors, all self-reciprocal. We give arithmetic conditions bearing on n and q for this to happen. Given a fixed q, we show these conditions are met for infinitely many n's, provided a refinement of Artin primitive root conjecture holds. This number theory conjecture is known to hold under GRH (Generalized Riemann Hypothesis). We derive a modified Varshamov-Gilbert bound on the relative distance of the codes considered, building on exact enumeration results for given n and q.

$Research paper thumbnail of On self-dual and LCD double circulant and double negacirculant codes over Fq+uFq$\mathbb {F}_{q}+u\mathbb {F}_{q}$$

On self-dual and LCD double circulant and double negacirculant codes over Fq+uFq$\mathbb {F}_{q}+u\mathbb {F}_{q}$

Cryptography and Communications, 2019

Double circulant codes of length 2n over the semilocal ring R = F q + uF q , u 2 = u, are studied... more Double circulant codes of length 2n over the semilocal ring R = F q + uF q , u 2 = u, are studied when q is an odd prime power, and −1 is a square in F q. Double negacirculant codes of length 2n are studied over R when n is even and q is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length 2n is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length 4n over F q are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below. The parameters of examples of the modest length are computed. Several such codes are optimal.

On Linear Complementary Pairs of Codes

IEEE Transactions on Information Theory, 2018

We study linear complementary pairs (LCP) of codes (C, D), where both codes belong to the same al... more We study linear complementary pairs (LCP) of codes (C, D), where both codes belong to the same algebraic code family. We especially investigate constacyclic and quasicyclic LCP of codes. We obtain characterizations for LCP of constacyclic codes and LCP of quasi-cyclic codes. Our result for the constacyclic complementary pairs extends the characterization of linear complementary dual (LCD) cyclic codes given by Yang and Massey. We observe that when C and D are complementary and constacyclic, the codes C and D ⊥ are equivalent to each other. Hence, the security parameter min(d(C), d(D ⊥)) for LCP of codes is simply determined by one of the codes in this case. The same holds for a special class of quasi-cyclic codes, namely 2D cyclic codes, but not in general for all quasi-cyclic codes, since we have examples of LCP of double circulant codes not satisfying this conclusion for the security parameter. We present examples of binary LCP of quasi-cyclic codes and obtain several codes with better parameters than known binary LCD codes. Finally, a linear programming bound is obtained for binary LCP of codes and a table of values from this bound is presented in the case d(C) = d(D ⊥). This extends the linear programming bound for LCD codes. Index Terms-Constacyclic code, quasi-cyclic code, LCP of codes, linear programming bound.

How many weights can a linear code have?

Designs, Codes and Cryptography, 2018

We study the combinatorial function L(k, q), the maximum number of nonzero weights a linear code ... more We study the combinatorial function L(k, q), the maximum number of nonzero weights a linear code of dimension k over F q can have. We determine it completely for q = 2, and for k = 2, and provide upper and lower bounds in the general case when both k and q are ≥ 3. A refinement L(n, k, q), as well as nonlinear analogues N (M, q) and N (n, M, q), are also introduced and studied.

Additive complementary dual codes over F 4

HAL (Le Centre pour la Communication Scientifique Directe), Jul 5, 2022

A linear code is linear complementary dual (LCD) if it meets its dual trivially. LCD codes have b... more 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. An additive code over F 4 is additive complementary dual (ACD) if it meets its dual trivially. The aim of this research is to study such codes which meet their dual trivially. All the techniques and problems used to study LCD codes are potentially relevant to ACD codes. Interesting constructions of ACD codes from binary codes are given with respect to the trace Hermitian and trace Euclidean inner product. The former product is relevant to quantum codes.

On the structure of 1-generator quasi-polycyclic codes over finite chain rings

Journal of Applied Mathematics and Computing

Quasi-polycyclic (QP for short) codes over a finite chain ring R are a generalization of quasi-cy... more 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 R ℓ m , where R m := R[x]/ f , and f is a monic polynomial of degree m over R. If f factors uniquely into monic and coprime basic irreducibles, then their algebraic structure allow us to characterize the generator polynomials and the minimal generating sets of 1-generator QP codes as R-modules. In addition, we also determine the parity check polynomials for these codes by using the strong Gröbner bases. In particular, via Magma system, some quaternary codes with new parameters are derived from these 1-generator QP codes.

Some ternary cubic two-weight codes

arXiv (Cornell University), Dec 3, 2016

We study trace codes with defining set L, a subgroup of the multiplicative group of an extension ... more We study trace codes with defining set L, a subgroup of the multiplicative group of an extension of degree m of the alphabet ring F 3 + uF 3 + u 2 F 3 , with u 3 = 1. These codes are abelian, and their ternary images are quasi-cyclic of co-index three (a.k.a. cubic codes). Their Lee weight distributions are computed by using Gauss sums. These codes have three nonzero weights when m is singly-even and |L| = 3 3m −3 2m 2. When m is odd, and |L| = 3 3m −3 2m 2 , or |L| = 3 3m − 3 2m and m is a positive integer, we obtain two new infinite families of two-weight codes which are optimal. Applications of the image codes to secret sharing schemes are also given.

Construction of MDS self-dual codes from orthogonal matrices

arXiv (Cornell University), Oct 25, 2016

In this paper 1 , we give algorithms and methods of construction of self-dual codes over finite f... more In this paper 1 , we give algorithms and methods of construction of self-dual codes over finite fields using orthogonal matrices. Randomization in the orthogonal group, and code extension are the main tools. Some optimal, almost MDS, and MDS self-dual codes over both small and large prime fields are constructed.

Covering Radius of Melas Codes

IEEE Transactions on Information Theory, 2022

We prove that the covering radius of the Melas code M (m, q) of length n = q m − 1 over Fq is 2 i... more

On isodual double Toeplitz codes

ArXiv, 2021

Double Toeplitz (shortly DT) codes are introduced here as a generalization of double circulant co... more 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 exhaustive search show that the codes constructed have best-known minimum distance, up to one unit, amongst formally self-dual codes, and sometimes improve on the known values. Over F4 an explicit construction of DT codes, based on quadratic residues in a prime field, performs equally well. We show that DT codes are asymptotically good over Fq. Specifically, we construct DT codes arbitrarily close to the asymptotic varshamov-Gilbert bound for codes of rate one half.

Three-weight codes and the quintic construction

ArXiv, 2016

We construct a class of three-Lee-weight and two infinite families of five-Lee-weight codes over ... more We construct a class of three-Lee-weight and two infinite families of five-Lee-weight codes over the ring $R=\mathbb{F}_2 +v\mathbb{F}_2 +v^2\mathbb{F}_2 +v^3\mathbb{F}_2 +v^4\mathbb{F}_2,$ where $v^5=1.$ The same ring occurs in the quintic construction of binary quasi-cyclic codes. %The length of these codes depends on the degree $m$ of ring extension. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Given a linear Gray map, we obtain three families of binary abelian codes with few weights. In particular, we obtain a class of three-weight codes which are optimal. Finally, an application to secret sharing schemes is given.

Optimal three-weight cubic codes

ArXiv, 2016

In this paper, we construct an infinite family of three-weight binary codes from linear codes ove... more In this paper, we construct an infinite family of three-weight binary codes from linear codes over the ring $R=\mathbb{F}_2+v\mathbb{F}_2+v^2\mathbb{F}_2$, where $v^3=1.$ These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distributions are computed by employing character sums. The three-weight binary linear codes which we construct are shown to be optimal when $m$ is odd and $m>1$. They are cubic, that is to say quasi-cyclic of co-index three. An application to secret sharing schemes is given.

Trace Codes with Few Weights over Fp+uFp

ArXiv, 2016

We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the chain ring ... more We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the chain ring Fp+uFp. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. Then by using a linear Gray map, we obtain an infinite family of abelian codes with few weights over Fp. In particular, we obtain an infinite family of two-weight codes which meets the Griesmer bound with equality. Finally, an application to secret sharing schemes is given.

Self-orthogonal codes over a non-unital ring and combinatorial matrices

Designs, Codes and Cryptography, 2021

There is a local ring E of order 4, without identity for the multiplication, defined by generator... more There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as E = a, b | 2a = 2b = 0, a 2 = a, b 2 = b, ab = a, ba = b. We study a special construction of self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), and Doubly Regular Tournaments (DRT). We construct quasi self-dual codes over E, and Type IV codes, that is, quasi self-dual codes whose all codewords have even Hamming weight. All these codes can be represented as formally self-dual additive codes over F 4. The classical invariant theory bound for the weight enumerators of this class of codes improves the known bound on the minimum distance of Type IV codes over E.

Good Stabilizer Codes from Quasi-Cyclic Codes over F4 and F9

2019 IEEE International Symposium on Information Theory (ISIT), 2019

We apply quantum Construction X on quasi-cyclic codes with large Hermitian hulls over F4 and F9 t... more We apply quantum Construction X on quasi-cyclic codes with large Hermitian hulls over F4 and F9 to derive good qubit and qutrit stabilizer codes, respectively. In several occasions we obtain quantum codes with stricly improved parameters than the current record. In numerous other occasions we obtain quantum codes with best-known performance. For the qutrit ones we supply a systematic construction to fill some gaps in the literature.

Construction of isodual codes from polycirculant matrices

Designs, Codes and Cryptography, 2020

Double polycirculant codes are introduced here as a generalization of double circulant codes. Whe... more 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 codes. Self-duality, the trivial case of isoduality, can only occur over F 2 in the double circulant case. Building on an explicit infinite sequence of irreducible trinomials over F 2 , we show that binary double polycirculant codes are asymptotically good.

How Many Weights Can a Cyclic Code Have?

IEEE Transactions on Information Theory, 2019

Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimensi... more Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), whose nonzero codewords have period equal to the length of the code. Asymptotics are derived on the function Γ(k, q), that is defined as the largest number of nonzero weights a cyclic code of dimension k over F q can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller codes, either binary or q-ary, as well as in the q-ary Hamming code are determined, two difficult results of independent interest.

On self-dual and LCD quasi-twisted codes of index two over a special chain ring

Cryptography and Communications, 2018

Let q be a prime power, and let F q denote the finite field of order q. Consider the chain ring R... more Let q be a prime power, and let F q denote the finite field of order q. Consider the chain ring R k = F q [u]/ u k with k ≥ 1 an integer. We study self-dual and LCD quasi-twisted codes of index two and twisting constant λ over R k for the metric induced by the standard Gray map. Some special factorizations of x m − λ over R k are studied. By random coding, we obtain four classes of asymptotically good self-dual λ-circulant codes and four classes of asymptotically good LCD λ-circulant codes over R k. Keywords Double circulant codes • Gray map • Self-dual codes • LCD codes • Quasi-twisted codes Mathematics Subject Classification (2010) 94 B15 • 94 B25 • 05 E30 This research is supported by National Natural Science Foundation of China (61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133) and Key projects of support program for outstanding young talents in Colleges and Universities (gxyqZD2016008).

Constructions of optimal LCD codes over large finite fields

Finite Fields and Their Applications, 2018

In this paper, we prove existence of optimal complementary dual codes (LCD codes) over large fini... more In this paper, we prove existence of optimal complementary dual codes (LCD codes) over large finite fields. We also give methods to generate orthogonal matrices over finite fields and then apply them to construct LCD codes. Construction methods include random sampling in the orthogonal group, code extension, matrix product codes and projection over a self-dual basis.

Two-weight and three-weight codes from trace codes over Fp+uFp+vFp+uvFp

Discrete Mathematics, 2018

We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the non-chain r... more We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the non-chain ring F p + uF p + vF p + uvF p , where u 2 = 0, v 2 = 0, uv = vu. These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. With a linear Gray map, we obtain a class of abelian three-weight codes and two-weight codes over F p. In particular, the two-weight codes we describe are shown to be optimal by application of the Griesmer bound. We also discuss their dual Lee distance. Finally, an application to secret sharing schemes is given.

On self-dual negacirculant codes of index two and four

Designs, Codes and Cryptography, 2018

We study the asymptotic performance of quasi-twisted codes viewed as modules in the ring R = F q ... more We study the asymptotic performance of quasi-twisted codes viewed as modules in the ring R = F q [x]/ x n +1 , when they are self-dual and of length 2n or 4n. In particular, in order for the decomposition to be amenable to analysis, we study factorizations of x n + 1 over F q , with n twice an odd prime, containing only three irreducible factors, all self-reciprocal. We give arithmetic conditions bearing on n and q for this to happen. Given a fixed q, we show these conditions are met for infinitely many n's, provided a refinement of Artin primitive root conjecture holds. This number theory conjecture is known to hold under GRH (Generalized Riemann Hypothesis). We derive a modified Varshamov-Gilbert bound on the relative distance of the codes considered, building on exact enumeration results for given n and q.

$Research paper thumbnail of On self-dual and LCD double circulant and double negacirculant codes over Fq+uFq$\mathbb {F}_{q}+u\mathbb {F}_{q}$$

On self-dual and LCD double circulant and double negacirculant codes over Fq+uFq$\mathbb {F}_{q}+u\mathbb {F}_{q}$

Cryptography and Communications, 2019

Double circulant codes of length 2n over the semilocal ring R = F q + uF q , u 2 = u, are studied... more Double circulant codes of length 2n over the semilocal ring R = F q + uF q , u 2 = u, are studied when q is an odd prime power, and −1 is a square in F q. Double negacirculant codes of length 2n are studied over R when n is even and q is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length 2n is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length 4n over F q are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below. The parameters of examples of the modest length are computed. Several such codes are optimal.

On Linear Complementary Pairs of Codes

IEEE Transactions on Information Theory, 2018

We study linear complementary pairs (LCP) of codes (C, D), where both codes belong to the same al... more We study linear complementary pairs (LCP) of codes (C, D), where both codes belong to the same algebraic code family. We especially investigate constacyclic and quasicyclic LCP of codes. We obtain characterizations for LCP of constacyclic codes and LCP of quasi-cyclic codes. Our result for the constacyclic complementary pairs extends the characterization of linear complementary dual (LCD) cyclic codes given by Yang and Massey. We observe that when C and D are complementary and constacyclic, the codes C and D ⊥ are equivalent to each other. Hence, the security parameter min(d(C), d(D ⊥)) for LCP of codes is simply determined by one of the codes in this case. The same holds for a special class of quasi-cyclic codes, namely 2D cyclic codes, but not in general for all quasi-cyclic codes, since we have examples of LCP of double circulant codes not satisfying this conclusion for the security parameter. We present examples of binary LCP of quasi-cyclic codes and obtain several codes with better parameters than known binary LCD codes. Finally, a linear programming bound is obtained for binary LCP of codes and a table of values from this bound is presented in the case d(C) = d(D ⊥). This extends the linear programming bound for LCD codes. Index Terms-Constacyclic code, quasi-cyclic code, LCP of codes, linear programming bound.

How many weights can a linear code have?

Designs, Codes and Cryptography, 2018

We study the combinatorial function L(k, q), the maximum number of nonzero weights a linear code ... more We study the combinatorial function L(k, q), the maximum number of nonzero weights a linear code of dimension k over F q can have. We determine it completely for q = 2, and for k = 2, and provide upper and lower bounds in the general case when both k and q are ≥ 3. A refinement L(n, k, q), as well as nonlinear analogues N (M, q) and N (n, M, q), are also introduced and studied.