Ankan Shaw, Sanjit Bhowmick, Satya Bagchi
We establish a complete classification of binary group codes with complementary duals for a finite group and explicitly determine the number of linear complementary dual (LCD) cyclic group codes by using cyclotomic cosets. The dimension and the minimum distance for LCD group codes are explored. Finally, we find a connection between LCD MDS group codes and maximal ideals.
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Linear complementary dual codes (LCD) are linear codes satisfying $C\cap C^{\perp}=\{0\}$. Under suitable conditions, matrix-product codes that are complementary dual codes are characterized. We construct LCD codes using quasi-orthogonal matrices. Some asymptotic results are derived.
Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study binary linear complementary dual $[n,k]$ codes with the largest minimum weight among all binary linear complementary dual $[n,k]$ codes. We characterize binary linear complementary dual codes with the largest minimum weight for small dimensions. A complete classification of binary linear complementary dual $[n,k]$ codes with the largest minimum weight is also given for $1 \le k \le n \le 16$.
Two families of complementary codes over finite fields $\mathbb{F}_q$ are studied, where $q=r^2$ is square: i) Hermitian complementary dual linear codes, and ii) trace Hermitian complementary dual subfield linear codes. Necessary and sufficient conditions for a linear code (resp., a subfield linear code) to be Hermitian complementary dual (resp., trace Hermitian complementary dual) are determined. Constructions of such codes are given together their parameters. Some illustrative examples are provided as well.
Abelian codes and complementary dual codes form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, a family of abelian codes with complementary dual in a group algebra $\mathbb{F}_{p^ν}[G]$ has been studied under both the Euclidean and Hermitian inner products, where $p$ is a prime, $ν$ is a positive integer, and $G$ is an arbitrary finite abelian group. Based on the discrete Fourier transform decomposition for semi-simple group algebras and properties of ideas in local group algebras, the characteri…
We show that any ternary Euclidean (resp.\ quaternary Hermitian) linear complementary dual $[n,k]$ code contains a Euclidean (resp.\ Hermitian) linear complementary dual $[n,k-1]$ subcode for $2 \le k \le n$. As a consequence, we derive a bound on the largest minimum weights among ternary Euclidean linear complementary dual codes and quaternary Hermitian linear complementary dual codes.
We give a complete classification of binary linear complementary dual codes of lengths up to $13$ and ternary linear complementary dual codes of lengths up to $10$.
Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study the largest minimum weight $d(n,k)$ among all binary linear complementary dual $[n,k]$ codes. We determine $d(n,4)$ for $n \equiv 2,3,4,5,6,9,10,13 \pmod{15}$, and $d(n,5)$ for $n \equiv 3,4,5,7,11,19,20,22,26 \pmod{31}$. Combined with known results, the values $d(n,k)$ are also determined for $n \le 24$.
Linear Complementary Equi-Dual Codes
本文研究线性互补等对偶码，提出其一个必要条件，并猜想该条件也是充分的，同时给出若干支持性结论。
LCD codes are linear codes that intersect with their dual trivially. Quasi cyclic codes that are LCD are characterized and studied by using their concatenated structure. Some asymptotic results are derived. Hermitian LCD codes are introduced to that end and their cyclic subclass is characterized. Constructions of QCCD codes from codes over larger alphabets are given.
The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension $2$. In this paper, we give some conditions for the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As an application, we completely determine the largest minimum weights for dimension $3$, by using a classification of some quaternary codes. In addition, for a positive integer $s$, a maximal entanglement entanglement-assisted quantum $[[21s+5,3,16s+3;21s+2]]$ codes is constructed for the first time from a quaternary Hermitian linear com…
Trace duality and additive complementary pairs of additive cyclic codes over finite chain rings
本文研究有限交换环上加性循环码互补对的代数结构，证明其组成码均为自由模，给出构成互补对的充要条件，还构造了有限链环上的互补对并证明两码满足迹对偶的置换等价关系。
$\mathbb{F}_{2}\mathbb{F}_{4}$-Additive Complementary Dual Codes
本文研究F2F4上加法互补对偶码的结构与性质，给出其为ACD码的充分条件，并证明其二元像是LCD码时原码必为ACD码，且可构造距离最优二元LCD码。
On Euclidean, Hermitian and symplectic quasi-cyclic complementary dual codes
本文从码字层面给出LCD码的新刻画，建立单生成元拟循环码在三种内积下为LCD码的充要条件，并构造出多个参数优良的LCD码，其中辛LCD码对应结果优于已知最优四元埃尔米特LCD码。
Extremal Self-Dual Codes and Linear Complementary Dual Codes from Double Circulant Codes
本文研究二元双循环码的自对偶、极值及 LCD 性质，给出自对偶充要条件、长度至44的极值判定，并建立加边 DC 码成为 LCD 码的充分条件。
In 1964, Massey introduced a class of codes with complementary duals which are called Linear Complimentary Dual (LCD for short) codes. He showed that LCD codes have applications in communication system, side-channel attack (SCA) and so on. LCD codes have been extensively studied in literature. On the other hand, MDS codes form an optimal family of classical codes which have wide applications in both theory and practice. The main purpose of this paper is to give an explicit construction of several classes of LCD MDS codes, using tools from algebraic function fields. We exemplify this constructi…
Linear complementary dual quasi-cyclic codes of index 2
本文提出了一种多项式方法，研究了有限域上指数为2的LCD准循环码的构造与条件，并给出了相关结果。
We establish the existence of optimal maximal entanglement entanglement-assisted quantum $[[n,k,d;n-k]]_2$ codes for $(n,k,d)=(14,6,7)$, $(15,7,7)$, $(17,6,9)$, $(17,7,8)$, $(19,7,9)$ and $(20,7,10)$. These codes are obtained from quaternary Hermitian linear complementary dual codes. We also give some observation on the largest minimum weights.
It was shown by Massey that linear complementary dual (LCD for short) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert-Varshamov (GV for short) bound. Until now, the GV bound still remains to be the best asymptotical lower bound for LCD codes. In this paper, we show that an algebraic geometry code over a finite field of even characteristic is equivalent to an LCD code and consequently there exists a family of LCD codes that are equivalent to algebraic geometry codes and exceed the asymptotical GV bound.
Given an LCD group code $C$ in a group algebra $KG$, we inspect kinship between $C$ and $G$, more precisely between the subgroup structures of $G$ and $C$. Under some special circumstances our inspection provides an estimation for various parameters of a group code $C$. When $C$ is MDS, the inter relation between $K$ and $G$ becomes more impressive. Application of Sylow theorem facilitated us to explore the inter relation between $G$ and $K$ (when $C$ is LCD and MDS) in more general way and finally we get the result of Cruz and Willems (Lemma $4.4$) as a simple sequel.
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 potential…
Linear codes with complementary-duals (LCD) are linear codes that intersect with their dual trivially. Multinegacirculant codes of index $2$ that are LCD are characterized algebraically and some good codes are found in this family. Exact enumeration is performed for indices 2 and 3, and for all indices $t$ for a special case of the co-index by using their concatenated structure. Asymptotic existence results are derived for the special class of such codes that are one-generator and have co-index a power of two by means of Dickson polynomials. This shows that there are infinite families of LCD m…
Linear Complementary dual codes and Linear Complementary pairs of AG codes in function fields
本文研究函数域上代数几何码的线性互补对（LCP）与线性互补对偶（LCD）码，通过构造亏格 $g-1$ 的非特殊除子对，在 Kummer 扩张、超椭圆函数域和椭圆曲线中显式构造了相应的 LCP 码与 LCD AG 码。
Linear complementary dual codes were defined by Massey in 1992, and were used to give an optimum linear coding solution for the two user binary adder channel. In this paper, we define the analog of LCD codes over fields in the ambient space with mixed binary and quaternary alphabets. These codes are additive, in the sense that they are additive subgroups, rather than linear as they are not vector spaces over some finite field. We study the structure of these codes and we use the canonical Gray map from this space to the Hamming space to construct binary LCD codes in certain cases.
Linear complementary dual (LCD) codes introduced by Massey are the codes whose intersections with their dual codes are trivial. It can help to improve the security of the information processed by sensitive devices, especially against side-channel attacks (SCA) and fault invasive attacks. In this paper, By construction of puncturing, extending, shortening and combination codes, many good ternary LCD codes are presented. We give a Table 1 with the values of $d_{LCD}(n,k)$ for length $ n \leq 20$. In addition, Many of these ternary LCD codes given in this paper are optimal which are saturating th…
Linear complementary dual (LCD) maximum distance separable (MDS) codes are constructed to given specifications. For given $n$ and $r<n$, with $n$ or $r$ (or both) odd, MDS LCD $(n,r)$ codes are constructed over finite fields whose characteristic does not divide $n$. Series of LCD MDS codes are constructed to required rate and required error-correcting capability. Given the field $GF(q)$ and $n/(q-1)$, LCD MDS codes of length $n$ and dimension $r$ are explicitly constructed over $GF(q)$ for all $r<n$ when $n$ is odd and for all odd $r<n$ when $n$ is even. For given dimension and given error-cor…
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. 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 (\cite{Jin}, IEEE Trans. Inf. Theory, 2016) con…
Hermitian linear complementary dual codes are linear codes whose intersection with their Hermitian dual code is 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 $2$.
For $k \ge 2$ and a positive integer $d_0$, we show that if there exists no quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$ and Hermitian dual distance greater than or equal to $2$, then there exists no quaternary Hermitian linear complementary dual $[n,k,d]$ code with $d \ge d_0$ and Hermitian dual distance $1$. As a consequence, we generalize a result by Araya, Harada and Saito on the nonexistence of some quaternary Hermitian linear complementary dual codes.
In this paper, we investigate the rank-metric codes which are proposed by Delsarte and Gabidulin to be complementary dual codes. We point out the relationship between Delsarte complementary dual codes and Gabidulin complementary dual codes. In finite field $\mathbb{F}_{q}^{m}$, we construct two classes of Gabidulin LCD MRD codes by self-dual basis (or almost self-dual basis) of $\mathbb{F}_{q}^{m}$ over $\mathbb{F}_{q}$. Under a suitable condition, we determine a sufficient condition for Delsarte optimal anticodes to be LCD codes over $\mathbb{F}_{q}$.
Affine Cartesian codes with complementary duals
本文研究广义仿射笛卡尔码的线性互补对偶（LCD）性质，通过缩放坐标构造求值码，并给出LCD广义里德-所罗门码及更一般情形的刻画。
Linear Complementary Dual Codes Constructed from Reinforcement Learning
本研究针对编码理论中的LCD码构造问题，采用好奇心驱动的强化学习方法，通过设计奖励与状态动作映射，结合随机网络蒸馏优化探索，构造出纠错性能更优、最小距离界更好的二元和三元LCD码。
The Neighbor Graph of Linear Complementary Dual (LCD) Codes
本文研究有限域上线性互补对偶（LCD）码的邻接关系及其诱导图，确定LCD码的LCD邻居数量，证明邻接图在任意有限域上的正则性及二元和奇特征情形下结构子图的正则性，为LCD码提供图论框架。
Linear complementary dual (LCD) codes is a class of linear codes introduced by Massey in 1964. LCD codes have been extensively studied in literature recently. In addition to their applications in data storage, communications systems, and consumer electronics, LCD codes have been employed in cryptography. More specifically, it has been shown that LCD codes can also help improve the security of the information processed by sensitive devices, especially against so-called side-channel attacks (SCA) and fault non-invasive attacks. In this paper, we are interested in the construction of particular a…
We consider (symmetric, non-degenerate) bilinear spaces over a finite field and investigate the properties of their $\ell$-complementary subspaces, i.e., the subspaces that intersect their dual in dimension $\ell$. This concept generalizes that of a totally isotropic subspace and, in the context of coding theory, specializes to the notions of self-orthogonal, self-dual and linear-complementary-dual (LCD) codes. In this paper, we focus on the enumerative and asymptotic combinatorics of all these objects, giving formulas for their numbers and describing their typical behavior (rather than the be…
A note on the hull and linear complementary pair of cyclic codes
本文利用基本对偶零点与迹表示刻画有限域上长度 $q^m-1$ 的循环码的 LCD 性质和 LCP，并给出任意长度循环码包络维数及两码交集维数公式。
Construction of Additive Complementary Dual Codes Over Finite Fields
本文研究有限域 $\mathbb{F}_{q^2}$ 上关于迹内积的 ACD 码，利用生成矩阵刻画并由线性码及已有 ACD 码构造新码，得到参数优于已知最佳线性码的多类 ACD 码。
This article provides a completion to theories of information based on entropy, resolving a longstanding question in its axiomatization as proposed by Shannon and pursued by Jaynes. We show that Shannon's entropy function has a complementary dual function which we call "extropy." The entropy and the extropy of a binary distribution are identical. However, the measure bifurcates into a pair of distinct measures for any quantity that is not merely an event indicator. As with entropy, the maximum extropy distribution is also the uniform distribution, and both measures are invariant with respect t…
Galois LCD codes over mixed alphabets
该研究针对有限链环生成的混合字母表上的伽罗瓦线性互补对偶码，刻画了其类型与伽罗瓦不变性条件，并通过研究特定参数下的$\mathbb{F}_p\mathbb{F}_p[θ]$-线性码格雷像，得到了$\mathbb{F}_p$-线性互补对偶码。
Linear complementary pairs of skew constacyclic codes
研究斜常循环码的线性互补对性质，利用其代数结构给出刻画并构造具有设计安全参数的码对，扩展 skew BCH codes 并利用保重自同构群实现 LCP 构造。
We give a short and elementary proof of the fact that for a linear complementary pair $(C,D)$, where $C$ and $D$ are $2$-sided ideals in a group algebra, $D$ is uniquely determined by $C$ and the dual code $D^\perp$ is permutation equivalent to $C$. This includes earlier results of Carlet et al. and Güneri et al. on nD cyclic codes which have been proved by subtle and lengthy calculations in the space of polynomials.
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 $…
Linear complementary dual (LCD) codes and linear complementary pair (LCP) of codes over finite fields have been intensively studied recently due to their applications in cryptography, in the context of side-channel and fault injection attacks. The security parameter for an LCP of codes $(C,D)$ is defined as the minimum of the minimum distances $d(C)$ and $d(D^\bot)$. It has been recently shown that if $C$ and $D$ are both 2-sided group codes over a finite field, then $C$ and $D^\bot$ are permutation equivalent. Hence the security parameter for an LCP of 2-sided group codes $(C,D)$ is simply $d…
On codes induced from Hadamard matrices
本文利用Hadamard矩阵构造和分析线性分组码和卷积码，推导出多种类型的编码结构。
On reversible and reversible-complementary DNA codes over $\mathbb{F}_{4}$
本文提出构造和计数 $\mathbb{F}_{4}$ 上满足逆序及逆序互补约束的DNA线性码的方法，通过模论刻画生成矩阵，优于现有方法，并给出最小汉明距离的界。
Linear complementary pairs (LCP) of codes play an important role in armoring implementations against side-channel attacks and fault injection attacks. One of the most common ways to construct LCP of codes is to use Euclidean linear complementary dual (LCD) codes. In this paper, we first introduce the concept of linear codes with $σ$ complementary dual ($σ$-LCD), which includes known Euclidean LCD codes, Hermitian LCD codes, and Galois LCD codes. As Euclidean LCD codes, $σ$-LCD codes can also be used to construct LCP of codes. We show that, for $q > 2$, all q-ary linear codes are $σ$-LCD and th…
On Linear Complementary Pairs of Algebraic Geometry Codes over Finite Fields
本文研究有限域上代数几何码的线性互补对，给出椭圆曲线显式构造、安全参数与对偶等价条件，并探讨MDS情形。
From a given $[n, k]$ code $C$, we give a method for constructing many $[n, k]$ codes $C'$ such that the hull dimensions of $C$ and $C'$ 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 $n=26,28 \le n \le 40$.
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