LCD Codes Research Articles

On the minimum weights of quaternary Hermitian LCD codes

On the minimum weights of quaternary Hermitian LCD codes

Two classes of LCD BCH codes over finite fields

BCH codes form a special subclass of cyclic codes and have been extensively studied in the past decades. Determining the parameters of BCH codes, however, has been an important but difficult problem. Recently, in order to further investigate the dual codes of BCH codes, the concept of dually-BCH codes was proposed. In this paper, we study BCH codes of lengths qm+1q+1 and qm+1 over the finite field Fq, both of which are LCD codes. The dimensions of narrow-sense BCH codes of length qm+1q+1 with designed distance δ=ℓqm−12+1 are determined, where q>2 and 2≤ℓ≤q−1. Lower bounds on the minimum distances of the dual codes of narrow-sense BCH codes of length qm+1 are developed for odd q, which are good in some cases. Moreover, sufficient and necessary conditions for the even-like subcodes of narrow-sense BCH codes of length qm+1 being dually-BCH codes are presented, where q is odd and m≢0(mod4).

Several Families of Self-Orthogonal Codes and Their Applications in Optimal Quantum Codes and LCD Codes

Several Families of Self-Orthogonal Codes and Their Applications in Optimal Quantum Codes and LCD Codes

Explicit Non-special Divisors of Small Degree, Algebraic Geometric Hulls, and LCD Codes from Kummer Extensions

Explicit Non-special Divisors of Small Degree, Algebraic Geometric Hulls, and LCD Codes from Kummer Extensions

Binary and ternary leading-systematic LCD codes from special functions

Linear complementary dual codes (LCD codes for short) are an important subclass of linear codes which have nice applications in communication systems, cryptography, consumer electronics and information protection. In the literature, it has been proved that an [n,k,d] Euclidean LCD code over Fq with q>3 exists if there is an [n,k,d] linear code over Fq, where q is a prime power. However, the existence of binary and ternary Euclidean LCD codes has not been totally characterized. Hence it is interesting to construct binary and ternary Euclidean LCD codes with new parameters. In this paper, we construct new families of binary and ternary leading-systematic Euclidean LCD codes from some special functions including semibent functions, quadratic functions, almost bent functions, and planar functions. These LCD codes are not constructed directly from such functions, but come from some self-orthogonal codes constructed with such functions. Compared with known binary and ternary LCD codes, the LCD codes in this paper have new parameters.

New constructions of optimal binary LCD codes

Linear complementary dual (LCD) codes provide an optimum linear coding solution for the two-user binary adder channel. LCD codes also can be used against side-channel attacks and fault non-invasive attacks. In this paper, we obtain a lower bound on the distance of binary LCD codes through expanded codes. We give necessary and sufficient conditions to extend binary [n,k] LCD codes to binary [n+1,k] and [n+1,k+1] LCD codes. A sufficient condition for a binary [n,k,d−1] LCD code constructed from a binary [n+1,k,d]LCDo,e code is also given. Finally, we construct some new binary LCD codes by using our LCD bounds and some methods such as puncturing, shortening, expanding and extension. In particular, we improve some previously known range of dLCD(n,k) of lengths 38≤n≤40 and dimensions 9≤k≤15. We also obtain some values or range of dLCD(n,k) with 41≤n≤50 and 6≤k≤n−6.

Classification of Certain Cyclic LCD Codes

We show that a necessary and sufficient condition for a cyclic code C of length N over a finite chain ring R (whose maximal ideal has nilpotence 2) to be an LCD code is that C = (f(x)), where f(X) is a self-reciprocal monic divisor of XN − 1 in R[X] and x = X + (XN − 1) in R[X]/(XN − 1). A similar, but slightly different, theorem was proved in 2019 by Z. Liu and J. Wang for general finite chain rings (Theorem 25 in [5]). We provide two proofs, both completely different than the proof of Liu and Wang.

Characterizations of the minimum weights of LCD codes of large dimensions

Characterizations of the minimum weights of LCD codes of large dimensions

On Locality of Some Binary LCD Codes

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.

Ternary self-orthogonal codes from weakly regular bent functions and their application in LCD Codes

Ternary self-orthogonal codes from weakly regular bent functions and their application in LCD Codes

Connections between Linear Complementary Dual Codes, Permanents and Geometry

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.

On the Construction of Quantum and LCD Codes from Cyclic Codes over the Finite Commutative Rings

Let Fq be a field of order q, where q is a power of an odd prime p, and α and β are two non-zero elements of Fq. The primary goal of this article is to study the structural properties of cyclic codes over a finite ring R=Fq[u1,u2]/⟨u12−α2,u22−β2,u1u2−u2u1⟩. We decompose the ring R by using orthogonal idempotents Δ1,Δ2,Δ3, and Δ4 as R=Δ1R⊕Δ2R⊕Δ3R⊕Δ4R, and to construct quantum-error-correcting (QEC) codes over R. As an application, we construct some optimal LCD codes.

Construction for both self-dual codes and LCD codes

<p style='text-indent:20px;'>From a given [<i>n</i>, <i>k</i>] code <i>C</i>, we give a method for constructing many [<i>n</i>, <i>k</i>] codes <i>C</i><sup>'</sup> such that the hull dimensions of <i>C</i> and <i>C</i><sup>'</sup> are identical. This method can be applied to constructions of both self-dual codes and linear complementary dual codes (LCD codes for short). Using the method, we construct 661 new inequivalent extremal doubly even [56, 28, 12] codes. Furthermore, constructing LCD codes by the method, we improve some of the previously known lower bounds on the largest minimum weights of binary LCD codes of length 26 ≤ <i>n</i> ≤ 40.</p>

Construction of quaternary Hermitian LCD codes

We introduce a general construction of many Hermitian LCD [n, k] codes from a given Hermitian LCD [n, k] code. Furthermore, we present some results on punctured codes and shortened codes of quaternary Hermitian LCD codes. As an application, we improve some of the previously known lower bounds on the largest minimum weights of quaternary Hermitian LCD codes of length $$26 \le n \le 30$$ .

On the Structure of Binary LCD Codes Having an Automorphism of Odd Prime Order

The aim of this work is to study the structure and properties of the binary LCD codes having an automorphism of odd prime order and to present a method for their construction.

On quantum and LCD Codes Over The Ring $$F_q+vF_q+v^2F_q$$

On quantum and LCD Codes Over The Ring $$F_q+vF_q+v^2F_q$$

Group LCD and group reversible LCD codes

In this paper, we give a new method for constructing LCD codes. We employ group rings and a well known map that sends group ring elements to a subring of the n×n matrices to obtain LCD codes. Our construction method guarantees that our LCD codes are also group codes, namely, the codes are ideals in a group ring. We show that with a certain condition on the group ring element v, one can construct non-trivial group LCD codes. Moreover, we also show that by adding more constraints on the group ring element v, one can construct group LCD codes that are reversible. We present many examples of binary group LCD codes of which some are optimal and group reversible LCD codes with different parameters.

Factorization of some polynomials over finite local commutative rings and applications to certain self-dual and LCD codes

Factorization of some polynomials over finite local commutative rings and applications to certain self-dual and LCD codes

On the classification of quaternary optimal Hermitian LCD codes

We propose a method for classifying quaternary Hermitian LCD codes having large minimum weights. For example, we classify quaternary optimal Hermitian LCD codes of dimension 3.

A new method for constructing linear codes with small hulls

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 sometimes doubly circulant. Some sufficient conditions for a linear code to be an LCD code (resp. a linear code with one-dimensional hull) are presented. It is worth mentioning that we present a lower bound on the minimum distances of the constructed linear codes. As an application, using these conditions, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) with respect to the online Database of Grassl.