Linear codes with complementary duals

Linear codes with complementary duals are linear codes whose intersection with their duals are trivial, shortly named LCD codes. In this paper we outline a construction for LCD codes over finite fields of order q using weighing matrices and their orbit matrices. The LCD codes constructed can be of any length dimension according to the choice of matrices used in their construction. As a special case, LCD codes of length 2n and dimension n are constructed which also have the property of being formally self-dual. Alternatively, under a condition depending on q that the codes are not LCD, this method constructs self-dual codes. To illustrate the method we construct LCD codes from weighing matrices, including the Paley conference matrices and Hadamard matrices. We also extend the construction to Hermitian LCD codes over the finite field of order 4. In addition, we propose a decoding algorithm that can be feasible for the LCD codes obtained from some of the given methods.

[Formula: see text]-Direct codes are an extension to the class of linear codes having complementary duals (LCD codes). Defined over a finite field [Formula: see text], it is comprised of [Formula: see text] linear codes [Formula: see text], [Formula: see text] with [Formula: see text], where [Formula: see text] is the dual with respect to [Formula: see text]. A 2-Direct code [Formula: see text] with respect to [Formula: see text] is comprised of only LCD codes: [Formula: see text]. On the contrary, two LCD codes do not set up a [Formula: see text]-Direct code in general. This paper presents some important and generalized results on [Formula: see text]-Direct codes, in that it attempts to construct [Formula: see text]-Direct codes from LCD codes. The class of [Formula: see text]-cyclic maximum rank distance (MRD) codes as having complementary duals over [Formula: see text] are generalized. Dual bases such as self-dual basis and self-dual normal basis play a crucial role in constructions. Further, construction implausibility of [Formula: see text]-Direct codes from almost self-dual bases is also dealt. Results obtained are demonstrated through examples.

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 columns of the matrix has an odd number of bases. We explain how Grassmannian varieties as well as linear and quadratic complexes are connected with LCD codes. Accessing the classification of polarities, we relate the binary LCD codes of dimension k to the two kinds of symmetric non-singular binary matrices, to certain truncated Reed–Muller codes, and to the geometric codes of planes in finite projective space via the self-orthogonal codes of dimension k.

Binary and ternary leading-systematic LCD codes from special functions

A linear code $C$ is called a linear complementary dual code (LCD code) if $C \cap C^\perp = {0}$ holds. LCD codes have many applications in cryptography, communication systems, data storage, and quantum coding theory. In this dissertation we show that a necessary and sufficient condition for a cyclic code $C$ over $\Z_4$ of odd length to be an LCD code is that $C=\big( f(x) \big)$ where $f$ is a self-reciprocal polynomial in $\Z_{4}[X]$ which is also in our paper \cite{GK1}. We then extend this result and provide a necessary and sufficient condition for a cyclic code $C$ of length $N$ over a finite chain ring $R=\big(R,\m=(\gamma),\kappa=R/\m \big)$ with $\nu(\gamma)=2$ to be an LCD code. In \cite{DKOSS} a linear programming bound for LCD codes and the definition for $\text{LD}_{2}(n, k)$ for binary LCD $[n, k]$-codes are provided. Thus, in a different direction, we find the formula for $\text{LD}_{2}(n, 2)$ which appears in \cite{GK2}. In 2020, Pang et al. defined binary $\text{LCD}\; [n,k]$ codes with biggest minimal distance, which meets the Griesmer bound \cite{Pang}. We give a correction to and provide a different proof for \cite[Theorem 4.2]{Pang}, provide a different proof for \cite[Theorem 4.3]{Pang}, examine properties of LCD ternary codes, and extend some results found in \cite{Harada} for any $q$ which is a power of an odd prime.

We recall why linear codes with complementary duals (LCD codes) play a role in counter-measures to passive and active side-channel analyses on embedded cryptosystems. The rate and the minimum distance of such LCD codes must be as large as possible. We investigate constructions.

We recall why linear codes with complementary duals (LCD codes) play a role in counter-measures to passive and active side-channel analyses on embedded cryptosystems. The rate and the minimum distance of such LCD codes must be as large as possible. We investigate constructions.

A multisecret-sharing scheme is a method for distributing n randomly associated secrets s 1, s 2,..., s n among a set of attendants. Linear codes with complementary duals, LCD codes, form an important class of linear codes, while weighing orthogonal matrices play a significant role in cryptography. In this paper, we design novel multisecret-sharing schemes based on LCD codes derived from weighing orthogonal matrices. We analyze the access structures, coalition statistics, security aspects, and information-theoretic efficiency compared to existing multisecret-sharing methods.

Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. LCD codes have been used in certain communication systems. It is recently found that LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for given length and code size. Constructing LCD MDS codes is thus of significance in theory and practice. Recently, Jin constructed several classes of LCD MDS codes through generalized Reed-Solomon codes. In this paper, a different approach is proposed to obtain new LCD MDS codes from generalized Reed-Solomon codes. Consequently, new code constructions are provided and certain previously known results by Jin are extended.

Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. These codes were first introduced by Massey in 1964. Nowadays, LCD codes are extensively studied in the literature and widely applied in data storage, cryptography, etc. In this paper, we prove some properties of binary LCD codes using their shortened and punctured codes. We also present some inequalities for the largest minimum weight $d_{LCD}(n,k)$ of binary LCD [n,k] codes for given length n and dimension k. Furthermore, we give two tables with the values of $d_{LCD}(n,k)$ for $k\le 32$ and $n\le 40$, and two tables with classification results.

A linear code with a complementary dual (or An LCD code) is defined to be a linear code C whose dual code C ⊥ satisfies C ∩ C ⊥= $\left \{ \mathbf {0}\right \} $ . Let L D (n, k) denote the maximum of possible values of d among [n, k, d] binary LCD codes. We give the exact values of L D (n, k) for k = 2 for all n and some bounds on L D (n, k) for other cases. From our results and some direct search we obtain a complete table for the exact values of L D (n, k) for 1 ≤ k ≤ n ≤ 12. As a consequence, we also derive bounds on the dimensions of LCD codes with fixed lengths and minimum distances.

In this work, we first generalize the $$\sigma $$-LCD codes over finite fields to $$\sigma $$-LCD codes over finite chain rings. Under suitable conditions, linear codes over finite chain rings that are $$\sigma $$-LCD codes are characterized. Then we provide a necessary and sufficient condition for free constacyclic codes over finite chain rings to be $$\sigma $$-LCD. We also get some new binary LCD codes of different lengths which come from Gray images of constacyclic $$\sigma $$-LCD codes over $$\mathbb {F}_{2}+\gamma \mathbb {F}_{2}+\gamma ^2\mathbb {F}_{2}$$. Finally, for special finite chain rings $$\mathbb {F}_{q}+\gamma \mathbb {F}_{q}$$, we define a new Gray map $$\Phi $$ from $$(\mathbb {F}_{q}+\gamma \mathbb {F}_{q})^n$$ to $$\mathbb {F}_{q}^{2n}$$, and by using $$\sigma $$-LCD codes over finite chain rings $$\mathbb {F}_{q}+\gamma \mathbb {F}_{q}$$, we construct new entanglement-assisted quantum error-correcting (abbreviated to EAQEC) codes with maximal entanglement and parts of them are MDS EAQEC codes.

Hermitian linear complementary dual codes are linear codes whose intersections with their Hermitian dual codes are trivial. The largest minimum weight among quaternary Hermitian linear complementary dual codes of dimension $2$ is known for each length. We give the complete classification of optimal quaternary Hermitian linear complementary dual codes of dimension $2$. Hermitian linear complementary dual codes are linear codes whose intersections with their Hermitian dual codes are trivial. The largest minimum weight among quaternary Hermitian linear complementary dual codes of dimension $2$ is known for each length. We give the complete classification of optimal quaternary Hermitian linear complementary dual codes of dimension $2$.

The design of codes for distributed storage systems that protects from node failures has been studied for years, and locally repairable code (LRC) is such a method that gives a solution for fast recovery of node failures. Linear complementary dual code (LCD code) is useful for preventing malicious attacks, which helps to secure the system. In this paper, we combine LRC and LCD code by integration of enhancing security and repair efficiency, and propose some techniques for constructing LCD codes with their localities determined. On the basis of these methods and inheriting previous achievements of optimal LCD codes, we give optimal or near-optimal [n, k, d, r] LCD codes for k ≤ 6 and n ≥ k+1 with relatively small locality, mostly r ≤ 3. Since all of our obtained codes are distance-optimal, in addition, we show that the majority of them are r-optimal and the other 63 codes are all near r-optimal, according to CM bound.

LCD codes are linear codes with important cryptographic applications. Recently, a method has been presented to transform any linear code into an LCD code with the same parameters when it is supported on a finite field with cardinality larger than 3. Hence, the study of LCD codes is mainly open for binary and ternary fields. Subfield-subcodes of $J$-affine variety codes are a generalization of BCH codes which have been successfully used for constructing good quantum codes. We describe binary and ternary LCD codes constructed as subfield-subcodes of $J$-affine variety codes and provide some new and good LCD codes coming from this construction.