Makoto Araya, Masaaki Harada
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.
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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.
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 $2$.
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 $12 \le n \le 30$.
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.
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…
A classification of quaternary Hermitian self-dual codes of length 20 is given. Using this classification, a classification of extremal quaternary Hermitian self-dual codes of length 22 is also given.
Shortest LCD embeddings of binary, ternary and quaternary linear codes
本文研究将线性码嵌入到最优LCD码中的最短LCD嵌入方法，通过确定需添加的列数并刻画所有可能形式，从短汉明码出发得到最小距离4的最优LCD码，并发现若干参数更优的新三元和四元LCD码。
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…
A method for constructing quaternary Hermitian self-dual codes and an application to quantum codes
本文提出四元修正四 μ-循环 Hermitian 自对偶码的构造方法，首次得到两个 [56,28,16] 码，改进已知下界并导出量子 [[56,0,16]] 码。
We give restrictions on the weight enumerators of ternary near-extremal self-dual codes of length divisible by $12$ and quaternary near-extremal Hermitian self-dual codes of length divisible by $6$. We consider the weight enumerators for which there is a ternary near-extremal self-dual code of length $12m$ for $m =3,4,5,6$. Also we consider the weight enumerators for which there is a quaternary near-extremal Hermitian self-dual code of length $6m$ for $m =4,5,6$.
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.
In this work, we give three new techniques for constructing Hermitian self-dual codes over commutative Frobenius rings with a non-trivial involutory automorphism using $λ$-circulant matrices. The new constructions are derived as modifications of various well-known circulant constructions of self-dual codes. Applying these constructions together with the building-up construction, we construct many new best known quaternary Hermitian self-dual codes of lengths 26, 32, 36, 38 and 40.
Linear codes with small hulls over finite fields have been extensively studied due to their practical applications in computational complexity and information protection. In this paper, we develop a general method to determine the exact value of $D_4^H(n,k,1)$ for $n\leq 12$ or $k\in \{1,2,3,n-1,n-2,n-3\}$, where $D_4^H(n,k,1)$ denotes the largest minimum distance among all quaternary linear $[n,k]$ codes with one-dimensional Hermitian hull. As a consequence, we solve a conjecture proposed by Mankean and Jitman on the largest minimum distance of a quaternary linear code with one-dimensional He…
Symplectic self-orthogonal and LCD codes from the Plotkin sum construction
本文给出 Plotkin 和构造所得线性码成为辛自正交码与 LCD 码的判据，并利用 ℓ-交对和广义 Reed-Muller 码构造优良码类，证明其渐近良好且部分四元码性能超越已知最佳结果。
Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. It was proved that asymptotically good Hermitian LCD codes exist. The objective of this paper is to construct some cyclic Hermitian LCD codes over finite fields and analyse their parameters. The dimensions of these codes are settled and the lower bounds on their minimum distances are presented. Most Hermitian LCD codes presented in this paper are not BCH codes. In addition, we employ Hermitian LCD codes to propose a Hermit…
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.
Hermitian LCD $2$-Quasi Abelian Codes over Finite Chain Rings
本文研究有限域与有限链环上的埃尔米特LCD 2-拟阿贝尔码，完成低相对最小重量码的计数，证明两类环上均存在渐近优的对应码类。
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…
Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) have been of much interest from many researchers due to their theoretical significant and practical implications. However, little work has been done on LCD M…
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。
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.
In this paper, we study the complementary dual codes in more general setting (which are called Galois LCD codes) by a uniform method. A necessary and sufficient condition for linear codes to be Galois LCD codes is determined, and constacyclic codes to be Galois LCD codes are characterized. Some illustrative examples which constacyclic codes are Galois LCD MDS codes are provided as well. In particular, we study Hermitian LCD constacyclic codes. Finally, we present a construction of a class of Hermitian LCD codes which are also MDS codes.
New distance bounds for one-generator quasi-cyclic codes with applications to Hermitian LCD codes and entanglement-assisted quantum codes
本文提出单生成元准循环码最小距离的下界，通过穿孔和缩短构造，应用于搜索获得改进的Hermitian LCD码及最大纠缠纠缠辅助量子码。
Non-GRS type Euclidean and Hermitian LCD codes and Their Applications for EAQECCs
本文基于非GRS类型的线性码构造了欧几里得LCD码、赫尔米特LCD码及小维度包络码，并给出了相关上界，同时应用这些码构造了EAQECCs，并证明了GRL码在k>ℓ时非GRS。
Linear codes with complementary duals (LCD) have a great deal of significance amongst linear codes. Maximum distance separable (MDS) codes are also an important class of linear codes since they achieve the greatest error correcting and detecting capabilities for fixed length and dimension. The construction of linear codes that are both LCD and MDS is a hard task in coding theory. In this paper, we study the constructions of LCD codes that are MDS from negacyclic codes over finite fields of odd prime power $q$ elements. We construct four families of MDS negacyclic LCD codes of length $n|\frac{q…
Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual are trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. In this paper, we introduce a general construction of LCD codes from any linear codes. Further, we show that any linear code over $\mathbb F_{q} (q>3)$ is equivalent to an Euclidean LCD code and any linear code over $\mathbb F_{q^2} (q>2)$ is equi…
In this work, we study and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the direct sum construction and the $(\mathbf{u},\mathbf{u+v})$-construction. Some new criteria for the resulting codes derived from these two propagation rules being self-dual, self-orthogonal, or linear complementary dual (LCD) codes are given. As an application, we construct some linear codes with prescribed hull dimensions, many new binary, ternary Euclidean formally self-dual (FSD) LCD codes, and quaternary Hermitian FSD LCD codes. Some new even-like, odd-like, …
Construction Methods for Galois LCD codes over Finite Fields
该研究提出多种Hermitian与Galois LCD码构造方法，得到了参数更优的三元LCD码与F_{2^3}上的最优2-Galois LCD码，并将欧几里得内积的相关结果推广至σ-内积。
The aim of this paper is to give conditions for the equivalency between skew constacyclic codes, skew cyclic codes and skew negacyclic codes defined over semi-local rings. Also, we provide construction and an enumeration of Euclidean and Hermitian skew LCD cyclic codes over $ \mathbb{F}_{p^{t}}+ v \mathbb{F}_{p^{t}} $. Several optimal LCD codes are obtained from Gray images of LCD skew codes over semi-local rings.
A further study on the mass formula for linear codes with prescribed hull dimension
本文研究具有指定壳维数的线性码质量公式，通过酉群和辛群对LCD码的轨道作用，提出一般技术得到闭式质量公式及渐近结果。
A simple construction of quaternary hermitian self-orthogonal codes with parameters $[2n+1,k+1]$ and $[2n+2,k+2]$ from a given pair of self-orthogonal $[n,k]$ codes, and its link to quantum codes is considered. As an application, an optimal quaternary linear $[28,20,6]$ dual containing code is found that yields a new optimal $[[28,12,6]]$ quantum code.
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.
Quaternary Conjucyclic Codes with an Application to EAQEC Codes
本文研究四元域上加法共轭循环码的性质，推导其对偶与迹壳，并利用迹码构造性能良好的纠缠辅助量子纠错码。
In this work, our main objective is to construct quantum codes from quasi-twisted (QT) codes. At first, a necessary and sufficient condition for Hermitian self-orthogonality of QT codes is introduced by virtue of the Chinese Remainder Theorem (CRT). Then we utilize these self-orthogonal QT codes to provide quantum codes via the famous Hermitian Construction. Moreover, we present a new construction method of q-ary quantum codes, which can be viewed as an effective generalization of the Hermitian Construction. General QT codes that are not self-orthogonal are also employed to construct quantum c…
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. (…
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…
Explicit non-special divisors of small degree, algebraic geometric hulls, and LCD codes from Kummer extensions
本文通过Kummer扩张的有理位和Weierstrass半群显式构造非特殊因子，得到指定包络维数的代数几何码及LCD码，并回答了Ballet-Le Brigand的开放问题。
In this paper, we give a method for constructing linear codes with small hulls by generalizing the method in \cite{LCD-T-matric}. As a result, we obtain many optimal Euclidean LCD codes and Hermitian LCD codes, which improve the previously known lower bound on the largest minimum distance. We also obtain many optimal codes with one-dimension hull. Furthermore, we give three tables about formally self-dual LCD codes.
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。
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…
The hull $H(C)$ of a linear code $C$ is defined by $H(C)=C \cap C^\perp$. A linear code with a complementary dual (LCD) is a linear code with $H(C)=\{0\}$. The dimension of the hull of a code is an invariant under permutation equivalence. For binary and ternary codes the dimension of the hull is also invariant under monomial equivalence and we show that this invariant is determined by the extended weight enumerator of the code.\\ The hull of a code is not invariant under monomial equivalence if $q\geq 4$. We show that every ${\mathbb F}_q $-linear code is monomial equivalent with an LCD code i…
On Galois LCD codes and LCPs of codes over mixed alphabets
本文在有限交换链环框架下研究混合字母表的Galois LCD码与线性码的LCP，推导计数公式、完成小长度码的单项等价分类，并给出其在掩码方案安全分析和两用户加法信道编码中的应用。
The intersection ${\bf C}\bigcap {\bf C}^{\perp}$ (${\bf C}\bigcap {\bf C}^{\perp_h}$) of a linear code ${\bf C}$ and its Euclidean dual ${\bf C}^{\perp}$ (Hermitian dual ${\bf C}^{\perp_h}$) is called the Euclidean (Hermitian) hull of this code. It is natural to consider the hull-variation problem when a linear code ${\bf C}$ is transformed to an equivalent code ${\bf v} \cdot {\bf C}$. In this paper we introduce the maximal hull dimension as an invariant of a linear code with respect to the equivalent transformations. Then some basic properties of the maximal hull dimension are studied. We p…
Twisted and Twisted Linearized Reed--Solomon Codes, LCD and ACD MDS constructions
本文研究了在求和-秩度量下扭曲线性化Reed-Solomon码的子族，提出了码为线性互补双码的条件，并构造了无限族的加法互补双码和最大距离分离码。
$W-δ-μ$ dual codes and LCD codes
引入环境空间$F_q^n$上的新内积作为欧几里得、埃尔米特和$\delta$内积的推广，研究对偶码的性质、自正交码、自对偶码、含对偶码和LCD码的定义与存在条件，并计算重复码、二元码和$\lambda$-常循环码的对偶码，最后将内积概念推广到半单环上的码。
Hermitian hull-variation of vector rank-metric codes and self-orthogonal generalized Gabidulin codes
本文研究了矢量秩度量码的Hermite护套变化问题，证明了除了一个参数对外，此类码的Hermite护套维度可以降至其等价类中的任意更小值，并且特别地，每种此类码都等价于Hermite LCD码。接着，本文探讨了具有预设Hermite护套维度的最大秩距离码的存在性。为此，引入了有限域扩展的缩放迹自反基的概念，并利用该概念为每个素数幂构造Hermite自正交广义Gabidulin码，结合护套变化定理，这将产生达到所有可接受Hermite护套维度的最大秩距离码。
Non-special Divisors, LCPs of Codes, and LCD Codes on Kummer Extensions
研究库默尔扩张上非特殊因子的算术刻画，基于此在GK曲线上构造度g-1的非特殊因子，并利用纯间隙构造度g的有效非特殊因子族，进而建立代数几何码的LCP和LCD码的通用框架。
QEC and EAQEC Codes from Hermitian Sums and Hulls of Cyclic Codes over $\mathbb{F}_2 \times (\mathbb{F}_2+v\mathbb{F}_2)$
本文通过确定环 $\mathbb{F}_2 imes (\mathbb{F}_2+v\mathbb{F}_2)$ 上循环码的 Hermitian 核与 Hermitian 和的生成多项式，利用量子构造 X 的 Hermitian 对偶版本构造量子纠错码，并基于该环上的线性互补对偶码通过矩阵乘积码方法导出纠缠辅助量子纠错码族。
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…
We present new constructions of binary quantum codes from quaternary linear Hermitian self-dual codes. Our main ingredients for these constructions are nearly self-orthogonal cyclic or duadic codes over F_4. An infinite family of $0$-dimensional binary quantum codes is provided. We give minimum distance lower bounds for our quantum codes in terms of the minimum distance of their ingredient linear codes. We also present new results on the minimum distance of linear cyclic codes using their fixed subcodes. Finally, we list many new record-breaking quantum codes obtained from our constructions.
Special matrices over finite fields and their applications to quantum error-correcting codes
本文针对定义矩阵满足 $AA^†$ 为 $(D,\tau)$-单项式的矩阵积码，推导 Hermitian 壳维数及 HDC、AHDC、HSO、AHSO、Hermitian LCD 等判定条件，并给出最优最小距离下界构造。
Subfield codes of linear codes over finite fields have recently received much attention. Some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, the $q$-ary subfield codes $C_{f,g}^{(q)}$ of six different families of linear codes $C_{f,g}$ parametrized by two functions $f, g$ over a finite field $F_{q^m}$ are considered and studied, respectively. The parameters and (Hamming) weight distribution of $C_{f,g}^{(q)}$ and their punctured codes $\bar{C}_{f,g}^{(q)}$ are explicitly determined. The parameters of the duals o…
Explicit Factorization of $x^{p+1}-1$ over $\mathbb{Z}_{p^e}$: A Structural Approach via Dickson Polynomials
本文通过建立提升过程与辅助多项式 $V(x)$ 根之间的结构同构，揭示了与 Dickson 多项式的确定性联系，并开发了线性时间算法 Dickson-Engine，用于在整数剩余环上显式分解 $x^{p+1}-1$，从而构造近最优的 Hermitian 对称循环码。
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