Fang-Wei Fu's research while affiliated with Nankai University and other places

Publications (288)

Article
Let Fq2 be the finite field of q2 elements, where q is a power of a prime number, and let D2n=〈x,y|xn=1,y2=1,yxy=xn−1〉 be the dihedral group of 2n elements. Left ideals of the group algebra Fq2[D2n] are known as left dihedral codes over Fq2 of length 2n, and abbreviated as left D2n-codes. Let gcd(n,q)=1. In this paper, we give an explicit represent...
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Two linear codes are said to be a linear ℓ-intersection pair if their intersection has dimension ℓ. Guenda et al. (Des Codes Cryptogr. 88, 133–152, 2020) constructed most of the linear ℓ-intersection pairs of MDS codes and we complement their results by constructing some linear ℓ-intersection pairs of MDS codes over Fq\documentclass[12pt]{minimal}...
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Bent functions \(f: V_{n}\rightarrow \mathbb {F}_{p}\) play an important role in constructing partial difference sets, where \(V_{n}\) denotes an n-dimensional vector space over \(\mathbb {F}_{p}\), p is an odd prime. In [2, 3], the so-called vectorial dual-bent functions are considered to construct partial difference sets. In [2], Çeşmelioğlu et a...
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This paper presents two modifications for Loidreau’s cryptosystem, a rank metric-based cryptosystem constructed by using Gabidulin codes in the McEliece setting. Recently a polynomial-time key recovery attack was proposed to break this cryptosystem in some cases. To prevent this attack, we propose the use of subcodes to disguise the secret codes in...
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Visual cryptography scheme (VCS) is a method of encrypting secret image into n noise-like shares. The secret image can be reconstructed by stacking adequate shares. In the past two decades, many schemes have been proposed to realize the cheating prevention visual cryptography scheme (CPVCS). Significantly, Ren et al. [9] first introduced the idea o...
Preprint
Linear codes with a few weights are an important class of codes in coding theory and have attracted a lot of attention. In this paper, we present several constructions of $q$-ary linear codes with two or three weights from vectorial dual-bent functions, where $q$ is a power of an odd prime $p$. The weight distributions of the constructed $q$-ary li...
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Constructions of optimal locally repairable codes (LRCs) achieving Singleton-type bound have been exhaustively investigated in recent years. In this paper, we consider new bounds and constructions of Singleton-optimal LRCs with minmum distance $d=6$, locality $r=3$ and minimum distance $d=7$ and locality $r=2$, respectively. Firstly, we establish e...
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In this paper, two constructions of frequency-hopping sequence (FHS for short) sets with respect to aperiodic Hamming correlation (AHC for short) are presented. Based on the presented constructions, we obtain several classes of optimal or near optimal FHS sets from cyclic codes and some known FHS sets with periodic partial Hamming correlation (PPHC...
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This paper presents a key recovery attack on a rank metric based cryptosystem proposed by Lau and Tan at ACISP 2018, which uses Gabidulin codes as the underlying decodable code. This attack is shown to cost polynomial time and therefore completely breaks the cryptosystem. Specifically, we convert the problem of recovering the private key into solvi...
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Permutation codes are studied due to their numerous applications in various applications, such as power line communications, block ciphers, and coding for storage. In this paper, we study perfect permutation codes in Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsb...
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A linear code is called an MDS self-dual code if it is both an MDS code and a self-dual code with respect to the Euclidean inner product. The parameters of such codes are completely determined by the code length. In this paper, we consider new constructions of MDS self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. Th...
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Let p be an odd prime and m and s positive integers, with m even. Let further Fpm be the finite field of pm elements and R=Fpm+uFpm (u2=0). Then R is a finite chain ring of p2m elements, and there is a Gray map from RN onto Fpm2N which preserves distance and orthogonality, for any positive integer N. It is an interesting approach to obtain self-dua...
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In this paper, we study the conjecture that there doesn’t exist bent-negabent rotation symmetric Boolean functions. We prove that the conjecture is true for almost all the cases based on the properties of autocorrelation spectra and the enumeration formulas of orbits.
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Based on good algebraic structures and practicabilities, generalized quasi-cyclic (GQC) codes play important role in coding theory. In this paper, we study some results on GQC codes over $\mathbb{Z}_4$ including the normalized generating set, the minimum generating set and the normalized generating set of their dual codes. As an application, new $\...
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Recent research shows that the class of rotation symmetric Boolean functions is potentially rich in functions of cryptographic significance. In this paper, some classes of 2m-variable (m is an odd integer) 1-resilient rotation symmetric Boolean functions are got, whose nonlinearity and algebraic degree are studied. For the first time, we obtain 2m-...
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Secret sharing was firstly proposed in 1979 by Shamir and Blakley respectively. To avoid deficiencies of original schemes, researchers presented improvement schemes, among which the multi-secret sharing scheme (MSS) is significant. There are three categories of MSSs, however, we focus on multi-stage secret sharing scheme (MSSS) recovering secrets w...
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Bent functions $f: V_{n}\rightarrow \mathbb{F}_{p}$ with certain additional properties play an important role in constructing partial difference sets, where $V_{n}$ denotes an $n$-dimensional vector space over $\mathbb{F}_{p}$, $p$ is an odd prime. In \cite{Cesmelioglu1,Cesmelioglu2}, the so-called vectorial dual-bent functions are considered to co...
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Locally repairable codes (LRCs) with $(r,\delta)$ locality were introduced by Prakash \emph{et al.} into distributed storage systems (DSSs) due to their benefit of locally repairing at least $\delta-1$ erasures via other $r$ survival nodes among the same local group. An LRC achieving the $(r,\delta)$ Singleton-type bound is called an optimal $(r,\d...
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Since Noar and Shamir introduced visual cryptography scheme (VCS), the cheating problem of VCS has absorbed much attention of scholars. The current researches on cheating immune have one or more of these serious disadvantages: (1) each share has extra pixel expansion, (2) some methods need extra verification shares to determine the share is genuine...
Article
Secret sharing was proposed primarily in 1979 to solve the problem of key distribution. In recent decades, researchers have proposed many improvement schemes. Among all these schemes, verifiable multi-secret sharing (VMSS) schemes are studied sufficiently, which share multiple secrets simultaneously and perceive malicious dealer as well as particip...
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Hierarchical secret sharing is an important key management technique since it is specially customized for hierarchical organizations with different departments allocated with different privileges, such as the government agencies or companies. Hierarchical access structures have been widely adopted in secret sharing schemes, where efficiency is the...
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For any odd positive integer n, we express cyclic codes over \({\mathbb {Z}}_4\) of length 4n in a new way. Based on the expression of each cyclic code \({\mathcal {C}}\), we provide an efficient encoder and determine the type of \({\mathcal {C}}\). In particular, we give an explicit representation and enumeration for all distinct self-dual cyclic...
Article
Visual cryptography scheme (VCS) is a branch of secret sharing, in which a secret image is encrypted into n noise-like shares. The secret image can be reconstructed by stacking sufficient shares. VCS with multiple decryptions can further expand their application scope. However, the investigations on the existing scheme with multiple decryptions are...
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This paper presents a new technique for disturbing the algebraic structure of linear codes in code-based cryptography. This is a new attempt to exploit Gabidulin codes in the McEliece setting and almost all the previous cryptosystems of this type have been completely or partially broken. To be specific, we introduce the so-called semilinear transfo...
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In the rank modulation scheme for flash memories, permutation codes have been studied. In this paper, we study perfect permutation codes in Sn, the set of all permutations on n elements, under the Kendall τ-metric. We answer one open problem proposed by Buzaglo and Etzion. That is, proving the nonexistence of perfect codes in Sn, under the Kendall...
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This paper presents two public-key cryptosystems based on the so-called expanded Gabidulin codes, which are constructed by expanding Gabidulin codes over the base field. Exploiting the fast decoder of Gabidulin codes, we propose an efficient algorithm to decode these new codes when the noise vector satisfies a certain condition. Additionally, these...
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Using tools developed in a recent work by Shen and the second author, in this paper we carry out an in-depth study on the average decoding error probability of the random matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas f...
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In this paper, we determine the weight distribution of several classes of double cyclic codes over Galois rings by Gauss sums.
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Multiply constant-weight codes (MCWCs) were introduced recently to improve the reliability of certain physically unclonable function response. In this paper, two methods of constructing MCWCs are presented following the concatenation methodology. In other words, MCWCs are constructed by concatenating approximate outer codes and inner codes. Besides...
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Let \(\mathbb {F}_{2^m}\) be a finite field of \(2^m\) elements and denote \(R=\mathbb {F}_{2^m}[u]/\langle u^k\rangle \)\(=\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}+\cdots +u^{k-1}\mathbb {F}_{2^m}\) (\(u^k=0\)), where k is an integer satisfying \(k\ge 2\). For any odd positive integer n, an explicit representation for every self-dual cyclic code over...
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Recently, Galindo et al. introduced the concept of asymmetric entanglement-assisted quantum error-correcting codes (AEAQECCs) from Calderbank-Shor-Steane (CSS) construction. In general, it's difficult to determine the required number of maximally entangled states of an AEAQECC, which is associated with the dimension of the intersection of the two c...
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In this paper, we put forward a brand-new idea of masking the algebraic structure of linear codes used in code-based cryptography. Specially, we introduce the so-called semilinear transformations in coding theory, make a thorough study on their algebraic properties and then creatively apply them to the construction of code-based cryptosystems. Note...
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This paper presents a new family of linear codes, namely the expanded Gabidulin codes. Exploiting the existing fast decoder of Gabidulin codes, we propose an efficient algorithm to decode these new codes when the noise vector satisfies a certain condition. Furthermore, these new codes enjoy an excellent error-correcting capability because of the op...
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In this work, we introduce the FqR-linear skew cyclic codes, wh ere q=ps is a prime power and R=Fq+uFq with u2=0. We provide the algebraic structure of these codes. The dual codes of separable linear skew cyclic codes are also presented. Finally, by using the Gray map from FqR to Fq, we get some optimal linear codes over Fq.
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In this paper, we study the dual of generalized bent functions $f: V_{n}\rightarrow \mathbb{Z}_{p^k}$ where $V_{n}$ is an $n$-dimensional vector space over $\mathbb{F}_{p}$ and $p$ is an odd prime, $k$ is a positive integer. It is known that weakly regular generalized bent functions always appear in pairs since the dual of a weakly regular generali...
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The parameters of a $q$ -ary MDS Euclidean self-dual codes are completely determined by its length and the construction of MDS Euclidean self-dual codes with new length has been widely investigated in recent years. In this paper, we give a further study on the construction of MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes...
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Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in CDMA communication systems. In this paper, we first study the Jacobi sums over Galois rings of arbitrary characteristics and completely determine their absolute values, which extends the work in [34], where the Jacobi sums over Galo...
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Let p be any odd prime number and let m, k be arbitrary positive integers. The construction for self-dual cyclic codes of length \(p^k\) over the Galois ring \(\mathrm{GR}(p^2,m)\) is the key to construct self-dual cyclic codes of length \(p^kn\) over the integer residue class ring \({\mathbb {Z}}_{p^2}\) for any positive integer n satisfying \(\ma...
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In this paper, we study the structure of \({\mathbb {F}}_q\)-linear skew cyclic codes over \({\mathbb {F}}_{q^2}\). Some good \({\mathbb {F}}_q\)-linear skew cyclic codes over \({\mathbb {F}}_{q^2}\) are constructed. Moreover, as an application, some good quantum codes are obtained by \({\mathbb {F}}_q\)-linear skew cyclic codes over \({\mathbb {F}...
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The parameters of MDS self-dual codes are completely determined by the code length. In this paper, we utilize generalized Reed-Solomon (GRS) codes and extended GRS codes to construct MDS self-dual (self-orthogonal) codes and MDS almost self-dual codes over. The main idea of our constructions is to choose suitable evaluation points such that the cor...
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Plateaued functions as an extension of bent functions play a significant role in cryptography, coding theory, sequences and combinatorics. In 2019, Hod\v{z}i\'{c} et al. designed Boolean plateaued functions in spectral domain and provided some efficient construction methods in spectral domain. However, in their constructions, the Walsh support of B...
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Let \(R=\mathbb {Z}_{4}+u\mathbb {Z}_{4}\) be a finite non-chain ring, where u2 = 1. In this paper, we consider (σ, δ)-skew quasi-cyclic codes over the ring R, where σ is an automorphism of R and δ is an inner σ-derivation of R. We determine the structure of 1-generator (σ, δ)-skew quasi-cyclic codes over R and give a sufficient condition for 1-gen...
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Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square \begin{document}$ q $\end{document}, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of \begin{document}$ q $\end{document}-ary MDS Euclidean self-dual codes of lengths in...
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Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in code division multiple access communication systems. In this paper, several classes of codebooks are introduced, whose maximum cross-correlation amplitudes asymptotically achieve the corresponding Welch bound and Levenshtein bound....
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An (r,δ)-locally repairable code ((r, δ)-LRC for short) was introduced by Prakash et al. [14] for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of r-LRCs produced by Gopalan et al. [5]. An (r, δ)-LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. [2]...
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Optimal codebooks meeting the Welch bound with equality are desirable in many areas, such as direct spread code division multiple access communications, compressed sensing and so on. However, it is difficult to construct such optimal codebooks. There have been a number of attempts to construct codebooks nearly meeting the Welch bound. In this paper...
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Secret sharing was proposed primarily in 1979 to solve the problem of key distribution. In recent decades, researchers have proposed many improvement schemes. Among all these schemes, the verifiable multi-secret sharing (VMSS) schemes are studied sufficiently, which share multiple secrets simultaneously and perceive malicious dealer as well as part...
Article
Repair locality has been an important metric in a distributed storage system (DSS). Erasure codes with small locality are more popular in a DSS, which means fewer available nodes participating in the repair process of failed nodes. Locally repairable codes (LRCs) as a new coding scheme have given more rise to the system performance and attracted a...
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A verifiable multi‐secret sharing (VMSS) scheme allows distributors to share multiple secrets simultaneously and can detect fraud by both distributors and participants. After analysing the security of the VMSS schemes proposed by Dehkordi and Mashhadi in 2015, the authors point out that they could not detect the fraudulent behaviour of the dealer....
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Compressed sensing theory provides a new approach to acquire data as a sampling technique and makes sure that sparse signals can be reconstructed from few measurements. The construction of compressed sensing matrices is a central problem in compressed sensing theory. In this paper, the deterministic sparse compressed sensing matrices from constant...
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In this paper, we determine the bounds on covering radius of repetition codes, simplex codes and MacDonald codes over F2R with respect to the Chinese Euclidean distance.
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A locally repairable code (LRC) is a linear code such that every code symbol can be recovered by accessing a small number of other code symbols. In this paper, we study bounds and constructions of LRCs from the viewpoint of paritycheck matrices. Firstly, a simple and unified framework based on parity-check matrix to analyze the bounds of LRCs is pr...
Preprint
An $(r, \delta)$-locally repairable code ($(r, \delta)$-LRC for short) was introduced by Prakash et al. \cite{Prakash2012} for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of $r$-LRCs produced by Gopalan et al. \cite{Gopalan2012}. An $(r, \delta)$-LRC is said to be optimal if it achieves...
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Information-theoretic security is considered in the paradigm of network coding in the presence of wiretappers, who can access one arbitrary edge subset up to a certain size, also referred to as the security level. Secure network coding is applied to prevent the leakage of the source information to the wiretappers. In this paper, we consider the pro...
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Let q be a prime power with \(\mathrm{gcd}(q,6)=1\). Let \(R={\mathbb {F}}_{q^2}+u{\mathbb {F}}_{q^2}+v{\mathbb {F}}_{q^2}+uv{\mathbb {F}}_{q^2}\), where \(u^2=u\), \(v^2=v\) and \(uv=vu\). In this paper, we give the definition of linear skew constacyclic codes over \({\mathbb {F}}_{q^2}R\). By the decomposition method, we study the structural prop...
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Constructing minimal linear codes is an interesting research topic due to their applications in coding theory and cryptography. Ashikhmin and Barg pointed out that wmin∕wmax>(q−1)∕q is a sufficient condition for a linear code over the finite field Fq to be minimal, where wmin and wmax respectively denote the minimum and maximum nonzero weights in a...
Preprint
The parameters of a $q$-ary MDS Euclidean self-dual codes are completely determined by its length and the construction of MDS Euclidean self-dual codes with new length has been widely investigated in recent years. In this paper, we give a further study on the construction of MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and...
Preprint
A linear code is called an MDS self-dual code if it is both an MDS code and a self-dual code with respect to the Euclidean inner product. The parameters of such codes are completely determined by the code length. In this paper, we consider new constructions of MDS self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. Th...
Article
Linear code with locality [Formula: see text] and availability [Formula: see text] is that the value at each coordinate [Formula: see text] can be recovered from [Formula: see text] disjoint repairable sets each containing at most [Formula: see text] other coordinates. This property is particularly useful for codes in distributed storage systems be...
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We correct some mistakes in the paper “A mass formula for negacyclic codes of length 2k and some good negacyclic codes over \(\mathbb {Z}_{4}+u\mathbb {Z}_{4}\)” (Bandi et al. Cryptogr. Commun. 9, 241–272, 2017).
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External difference families (EDFs for short) are a type of block designs that each nonidentity element arises a fixed number of times as a difference between elements in distinct blocks. An EDF with the property that the union of its blocks covers the nonidentity elements exactly once is called a near-complete EDF. In this letter, we obtain some n...
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Locally repairable codes with locality r (r-LRCs for short) were introduced by Gopalan et al. [1] to recover a failed node of the code from at most other r available nodes. And then (r,δ)-locally repairable codes ((r,δ)-LRCs for short) were produced by Prakash et al. [2] for tolerating multiple failed nodes. An r-LRC can be viewed as an (r,2)-LRC....
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In this paper, we investigate all irreducible factors of \(x^{l_{1}^{m_{1}}l_{2}^{m_{2}}} - a\) over \(\mathbb {F}_{q}\) and obtain all primitive idempotents in \(\mathbb {F}_{q}[x]/\langle x^{l_{1}^{m_{1}}l_{2}^{m_{2}}} - a \rangle \), where \(a \in \mathbb {F}_{q}^{*}\), l1, l2 are two distinct odd prime divisors of qt − 1 with \(\gcd (l_{1}l_{2}...
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Codebooks with small inner-product correlation are preferred in many practical applications such as direct spread code division multiple access communications, coding theory, compressed sensing and so on. Heng et al. (IEEE Trans Inf Theory 63(10):6179–6187, 2017), Heng (Discrete Appl Math 250:227–240, 2018) and Luo and Cao (IEEE Trans Inf Theory 64...
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In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take al...
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Let m be an arbitrary positive integer and D8m be the dihedral group of order 8m, i.e., D8m = [x; y | x4m = 1; y2 = 1,yxy = x-1.]. Left ideals of the dihedral group algebra F2[D8m] are called binary left dihedral codes of length 8m, and abbreviated as binary left D8m-codes. In this paper, we give an explicit representation and enumeration for all d...
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Let $\mathbb{F}_{2^m}$ be a finite field of $2^m$ elements, and $R=\mathbb{F}_{2^m}[u]/\langle u^k\rangle=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}+\ldots+u^{k-1}\mathbb{F}_{2^m}$ ($u^k=0$) where $k$ is an integer satisfying $k\geq 2$. For any odd positive integer $n$, an explicit representation for every self-dual cyclic code over $R$ of length $2n$ and...
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Let $\mathbb{F}_{2^m}$ be a finite field of $2^m$ elements, and $R=\mathbb{F}_{2^m}[u]/\langle u^k\rangle=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}+\ldots+u^{k-1}\mathbb{F}_{2^m}$ ($u^k=0$) where $k$ is an integer satisfying $k\geq 2$. For any odd positive integer $n$, an explicit representation for every self-dual cyclic code over $R$ of length $2n$ and...
Article
Let F2m be a finite field of cardinality 2m, R=F2m[u]∕〈u⁴〉=F2m+uF2m+u²F2m+u³F2m(u⁴=0) which is a finite chain ring, and n is an odd positive integer. For any δ,α∈F2m×, an explicit representation for the dual code of any (δ+αu²)-constacyclic code over R of length 2n is given. And some dual codes of (1+u²)-constacyclic codes over R of length 14 are c...
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In this paper, the Sphere-packing bound, Wang-Xing-Safavi-Naini bound, Johnson bound and Gilbert-Varshamov bound on the subspace code of length [Formula: see text], size [Formula: see text], minimum subspace distance [Formula: see text] based on [Formula: see text]-dimensional totally singular subspace in the [Formula: see text]-dimensional orthogo...
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Let F 2 m be a finite field of cardinality 2 m , R = F 2 m + uF 2 m (u 2 = 0) and s, n be positive integers such that n is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring R of length 2 s n and provide a calculation method to obtain all distinct codes. Moreover, we obtain a clear form...
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Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $R=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ $(u^2=0)$ and $s,n$ be positive integers such that $n$ is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring $R$ of length $2^sn$ and provide a calculation method to obtain all distinct...
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Compressed sensing theory is a sampling technique which provides a fundamentally new approach to data acquisition and makes sure that a sparse signal can be reconstructed from few measurements by taking full use of sparsity. In this paper, firstly, the deterministic compressed sensing matrices using a sparse optimal compressed sensing matrix and co...
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In this paper, we study the decoding error probability of linear codes over the erasure channel under the list decoding. The notion of the q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup> -incorrigible sets of linear codes is introduced to characterize its decoding error probability under the...
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A verifiable multi-secret sharing (VMSS) scheme enables the dealer to share multiple secrets, and the deception of both participants and the dealer can be detected. After analyzing the security of VMSS schemes proposed by Mashhadi and Dehkordi in 2015, we illustrate that they cannot detect some deception of the dealer. By using nonhomogeneous linea...
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In this paper, we consider the problem for which lengths an MDS Euclidean self-dual code over Fq exists. This problem is completely solved for the case where q is even. For q is odd, some q-ary MDS Euclidean self-dual codes were obtained in the literature. In the present paper, we construct six new classes of q-ary MDS Euclidean self-dual codes by...
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Locally repairable codes (LRCs) are introduced in distributed storage systems due to their low repair overhead. An LRC is called optimal if its minimum distance attains the Singleton-like upper bound. Chen et al. (2018) recently studied the constructions of optimal (r,δ)-LRCs with length n(q+1) and (r+δ-1)n, where many classes of optimal cyclic con...
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In this letter, as a generalization of Luo et al.'s constructions, a construction of codebook, which meets the Welch bound asymptotically, is proposed. The parameters of codebook presented in this paper are new in some cases.