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.
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 $n \times n$ 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 exam…
Linear complementary dual codes (LCD codes) are codes whose intersections with their dual codes are trivial. These codes were introduced by Massey in 1992. LCD codes have wide applications in data storage, communication systems, and cryptography. Niederreiter-Rosenbloom-Tsfasman LCD codes (NRT-LCD codes) were introduced by Heqian, Guangku and Wei as a generalization of LCD codes for the NRT metric space $M_{n,s}(\mathbb{F}_{q})$. In this paper, we study LCD$[n\times s,k]$, the maximum minimum NRT distance among all binary $[n\times s,k]$ NRT-LCD codes. We prove the existence (non-existence) of…
Linear complementary dual (LCD) codes are linear codes which intersect their dual codes trivially, which have been of interest and extensively studied due to their practical applications in computational complexity and information protection. In this paper, we give some methods for constructing LCD codes over small finite fields by modifying some typical methods for constructing linear codes. We show that all odd-like binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes can be constructed using the modified methods. Our results improve the known lower bounds on the largest mi…
Both linear complementary dual (LCD) codes and maximum distance separable (MDS) codes have good algebraic structures, and they have interesting practical applications such as communication systems, data storage, quantum codes, and so on. So far, most of LCD MDS codes have been constructed by employing generalized Reed-Solomon codes. In this paper we construct some classes of new Euclidean LCD MDS codes and Hermitian LCD MDS codes which are not monomially equivalent to Reed-Solomon codes, called LCD MDS codes of non-Reed- Solomon type. Our method is based on the constructions of Beelen et al. (…
We give two methods for constructing many linear complementary dual (LCD for short) codes from a given LCD code, by modifying some known methods for constructing self-dual codes. Using the methods, we construct binary LCD codes and quaternary Hermitian LCD codes, which improve the previously known lower bound on the largest minimum weights.
Locally recoverable (LRC) codes provide a solution to single node failure in distributed storage systems, where it is a very common problem. On the other hand, linear complementary dual (LCD) codes are useful in fault injections attacks on storage systems. In this paper, we establish a connection between LRC codes and LCD codes. We derive some conditions on the construction of cyclic LRC codes so that they are also LCD codes. A lower bound on the minimum distance of such codes is determined. Some examples have been given to explain the construction.
Linear complementary dual (LCD) codes, which is a class of linear codes introduced by Massey, have been extensively studied in literature recently. It has been shown that LCD codes can help to improve the security of the information processed by sensitive devices, especially against so-called side-channel attacks (SCA) and fault invasive attacks. In this paper, Tables are presented of good quaternary Hermitian LCD codes and there are used in the construction of puncturing, extending, shortening and combination codes. Results including tables 3 of the best-known quaternary Hermitian LCD codes o…
Maximum distance separable (MDS) codes are optimal where the minimum distance cannot be improved for a given length and code size. Twisted Reed-Solomon codes over finite fields were introduced in 2017, which are generalization of Reed-Solomon codes. Twisted Reed-Solomon codes can be applied in cryptography which prefer the codes with large minimum distance. MDS codes can be constructed from twisted Reed-Solomon codes, and most of them are not equivalent to Reed-Solomon codes. In this paper, we first generalize twisted Reed-Solomon codes to generalized twisted Reed-Solomon codes, then we give s…
Let $\mathbb{F}_q$ be a finite field of order $q$, a prime power integer such that $q=et+1$ where $t\geq 1,e\geq 2$ are integers. In this paper, we study cyclic codes of length $n$ over a non-chain ring $R_{e,q}=\mathbb{F}_q[u]/\langle u^e-1\rangle$. We define a Gray map $φ$ and obtain many { maximum-distance-separable} (MDS) and optimal $\mathbb{F}_q$-linear codes from the Gray images of cyclic codes. Under certain conditions we determine { linear complementary dual} (LCD) codes of length $n$ when $\gcd(n,q)\neq 1$ and $\gcd(n,q)= 1$, respectively. It is proved that { a} cyclic code $\mathcal…
Absolute coset leaders were first proposed by the authors which have advantages in constructing binary LCD BCH codes. As a continue work, in this paper we focus on ternary linear codes. Firstly, we find the largest, second largest, and third largest absolute coset leaders of ternary primitive BCH codes. Secondly, we present three classes of ternary primitive BCH codes and determine their weight distributions. Finally, we obtain some LCD BCH codes and calculate some weight distributions. However, the calculation of weight distributions of two of these codes is equivalent to that of Kloosterman …
In \cite{anote}, Wu and Shi studied $ l $-Galois LCD codes over finite chain ring $\mathcal{R}=\mathbb{F}_q+u\mathbb{F}_q$, where $u^2=0$ and $ q=p^e$ for some prime $p$ and positive integer $e$. In this work, we extend the results to the finite non chain ring $ \mathcal{R} =\mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q+uv\mathbb{F}_q$, where $u^2=u,v^2=v $ and $ uv=vu $. We define a correspondence between $ l $-Galois dual of linear codes over $ \mathcal{R} $ and $ l $-Galois dual of its component codes over $ \mathbb{F}_q .$ Further, we construct Euclidean LCD and $ l $-Galois LCD codes from line…
Linear complementary dual (LCD) codes are linear codes that intersect with their dual codes trivially. We study the largest minimum weight $d_2(n,k)$ among all binary LCD $[n,k]$ codes and the largest minimum weight $d_3(n,k)$ among all ternary LCD $[n,k]$ codes. The largest minimum weights $d_2(n,5)$ and $d_3(n,4)$ are partially determined. We also determine the largest minimum weights $d_2(n,n-5)$, $d_3(n,n-i)$ for $i \in \{2,3,4\}$, and $d_3(n,k)$ for $n \in \{11,12,\ldots,19\}$.
We propose a method for a classification of quaternary Hermitian LCD codes having large minimum weights. As an example, we give a classification of quaternary optimal Hermitian LCD codes of dimension 3.
Non-GRS type Euclidean and Hermitian LCD codes and Their Applications for EAQECCs
College of Mathematical Sciences, Sichuan Normal University（四川师范大学数学科学学院）；School of Mathematics, Southwest Jiaotong University（西南交通大学数学系）；School of Information Science and Technology, Southwest Jiaotong University（西南交通大学信息科学与技术学院）
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 $…
