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In this paper, we propose a class of threshold secret sharing schemes with repairing function between shares without the help of the dealer, that we called repairable threshold secret sharing schemes. Specifically, if a share fails, such as broken or lost, it will be repaired just by some other shares. A construction of such repairable threshold secret sharing schemes is designed by applying linearized polynomials and regenerating codes in distributed storage systems. In addition, a new repairing rate is introduced to characterize the performance and efficiency of the repairing function. Then an achievable upper bound on the repairing rate is derived, which implies the optimality of the repair and describes the security between different shares. Under this optimality of the repair, we further discuss traditional information rate and also indicate its optimality, that can describe the efficiency of secret sharing schemes in the aspect of storage. Finally, by applying the minimum bandwidth regenerating (MBR) codes, our construction designs repairable threshold secret sharing schemes achieving both optimal repairing and information rates simultaneously., Comment: One column and 17 pages, submitted for publication

In this paper, we introduce a new definitions of the Gray weight and the Gray map for linear codes over $\mathbb{Z}_9+u\mathbb{Z}_9$ with $u^2=u$. Some results on self-dual codes over this ring are investigated. Further, the structural properties of quadratic residue codes are also considered. Two self-dual codes with parameters $[22,11,5]$ and $[24,12,9]$ over $\mathbb{Z}_9$ are obtained., Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1402.6771

In network communication, the source often transmits messages at several different information rates within a session. How to deal with information transmission and network error correction simultaneously under different rates is introduced in this paper as a variable-rate network error correction problem. Apparently, linear network error correction MDS codes are expected to be used for these different rates. For this purpose, designing a linear network error correction MDS code based on the existing results for each information rate is an efficient solution. In order to solve the problem more efficiently, we present the concept of variable-rate linear network error correction MDS codes, that is, these linear network error correction MDS codes of different rates have the same local encoding kernel at each internal node. Further, we propose an approach to construct such a family of variable-rate network MDS codes and give an algorithm for efficient implementation. This approach saves the storage space for each internal node, and resources and time for the transmission on networks. Moreover, the performance of our proposed algorithm is analyzed, including the field size, the time complexity, the encoding complexity at the source node, and the decoding methods. Finally, a random method is introduced for constructing variable-rate network MDS codes and we obtain a lower bound on the success probability of this random method, which shows that this probability will approach to one as the base field size goes to infinity., Comment: Single column, 34 pages, submitted for publication. arXiv admin note: text overlap with arXiv:1311.7466, arXiv:1011.1377

Linear codes are considered over the ring $\mathbb{Z}_4+v\mathbb{Z}_4$, where $v^2=v$. Gray weight, Gray maps for linear codes are defined and MacWilliams identity for the Gray weight enumerator is given. Self-dual codes, construction of Euclidean isodual codes, unimodular complex lattices, MDS codes and MGDS codes over $\mathbb{Z}_4+v\mathbb{Z}_4$ are studied. Cyclic codes and quadratic residue codes are also considered. Finally, some examples for illustrating the main work are given., Comment: 25 pages

In this paper, we study distributed storage problems over unidirectional ring networks, whose storage nodes form a directed ring and data is transmitted along the same direction. The original data is distributed to store on these nodes. Each user can connect one and only one storage node to download the total data. A lower bound on the reconstructing bandwidth to recover the original data for each user is proposed, and it is achievable for arbitrary parameters. If a distributed storage scheme can achieve this lower bound with equality for every user, we say it an optimal reconstructing distributed storage scheme (ORDSS). Furthermore, the repair problem for a failed storage node in ORDSSes is under consideration and a tight lower bound on the repair bandwidth is obtained. In particular, we indicate the fact that for any ORDSS, every storage node can be repaired with repair bandwidth achieving the lower bound with equality. In addition, we present two constructions for ORDSSes of arbitrary parameters, called MDS construction and ED construction, respectively. Particularly, ED construction, by using the concept of Euclidean division, is more efficient by our analysis in detail., Comment: 28 pages, one column, 12 figures, submitted for possible publication. arXiv admin note: substantial text overlap with arXiv:1401.5168

In this paper, we study distributed storage problems over unidirectional ring networks. A lower bound on the reconstructing bandwidth to recover total original data for each user is proposed, and it is achievable for arbitrary parameters. If a distributed storage scheme can achieve this lower bound with equality for each user, we say it an optimal reconstructing distributed storage scheme (ORDSS). Furthermore, the repair problem for a failed storage node in ORDSSes is under consideration and a tight lower bound on the repair bandwidth for each storage node is obtained. Particularly, we indicate the fact that for any ORDSS, every storage node can be repaired with repair bandwidth achieving the lower bound with equality. In addition, we present an efficient approach to construct ORDSSes for arbitrary parameters by using the concept of Euclidean division. Finally, we take an example to characterize the above approach., Comment: two columns, 5 pages, 8 figures, and submitted to the ISIT 2014

In the practical network communications, many internal nodes in the network are required to not only transmit messages but decode source messages. For different applications, four important classes of linear network codes in network coding theory, i.e., linear multicast, linear broadcast, linear dispersion, and generic network codes, have been studied extensively. More generally, when channels of communication networks are noisy, information transmission and error correction have to be under consideration simultaneously, and thus these four classes of linear network codes are generalized to linear network error correction (LNEC) coding, and we say them LNEC multicast, broadcast, dispersion, and generic codes, respectively. Furthermore, in order to characterize their efficiency of information transmission and error correction, we propose the (weakly, strongly) extended Singleton bounds for them, and define the corresponding optimal codes, i.e., LNEC multicast/broadcast/dispersion/generic MDS codes, which satisfy the corresponding Singleton bounds with equality. The existences of such MDS codes are discussed in detail by algebraic methods and the constructive algorithms are also proposed., Comment: Single column, 38 pages. Submitted for possible publication

In this paper, we mainly consider quasi-cyclic (QC) codes over finite chain rings. We study module structures and trace representations of QC codes, which lead to some lower bounds on the minimum Hamming distance of QC codes. Moreover, we investigate the structural properties of 1-generator QC codes. Under some conditions, we discuss the enumeration of 1-generator QC codes and describe how to obtain the one and only one generator for each 1-generator QC code., Comment: 26

In this work, we study a class of generalized quasi-cyclic (GQC) codes called skew GQC codes. By the factorization theory of ideals, we give the Chinese Remainder Theorem over the skew polynomial ring, which leads to a canonical decomposition of skew GQC codes. We also focus on some characteristics of skew GQC codes in details. For a 1-generator skew GQC code, we define the parity-check polynomial, determine the dimension and give a lower bound on the minimum Hamming distance. The skew quasi-cyclic (QC) codes are also discussed briefly., Comment: 19

Generalized quasi-cyclic (GQC) codes with arbitrary lengths over the ring $\mathbb{F}_{q}+u\mathbb{F}_{q}$, where $u^2=0$, $q=p^n$, $n$ a positive integer and $p$ a prime number, are investigated. By the Chinese Remainder Theorem, structural properties and the decomposition of GQC codes are given. For 1-generator GQC codes, minimal generating sets and lower bounds on the minimum distance are given. As a special class of GQC codes, quasi-cyclic (QC) codes over $\mathbb{F}_q+u\mathbb{F}_q$ are also discussed briefly in this paper., Comment: 17 pages

In this paper, we construct an authentication scheme for multi-receivers and multiple messages based on a linear code $C$. This construction can be regarded as a generalization of the authentication scheme given by Safavi-Naini and Wang. Actually, we notice that the scheme of Safavi-Naini and Wang is constructed with Reed-Solomon codes. The generalization to linear codes has the similar advantages as generalizing Shamir's secret sharing scheme to linear secret sharing sceme based on linear codes. For a fixed message base field $\f$, our scheme allows arbitrarily many receivers to check the integrity of their own messages, while the scheme of Safavi-Naini and Wang has a constraint on the number of verifying receivers $V\leqslant q$. And we introduce access structure in our scheme. Massey characterized the access structure of linear secret sharing scheme by minimal codewords in the dual code whose first component is 1. We slightly modify the definition of minimal codewords in \cite{Massey93}. Let $C$ be a $[V,k]$ linear code. For any coordinate $i\in \{1,2,\cdots,V\}$, a codeword $\vec{c}$ in $C$ is called minimal respect to $i$ if the codeword $\vec{c}$ has component 1 at the $i$-th coordinate and there is no other codeword whose $i$-th component is 1 with support strictly contained in that of $\vec{c}$. Then the security of receiver $R_i$ in our authentication scheme is characterized by the minimal codewords respect to $i$ in the dual code $C^\bot$., Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1303.0930

Stopping sets and stopping set distribution of a linear code play an important role in the performance analysis of iterative decoding for this linear code. Let $C$ be an $[n,k]$ linear code over $\f$ with parity-check matrix $H$, where the rows of $H$ may be dependent. Let $[n]=\{1,2,...,n\}$ denote the set of column indices of $H$. A \emph{stopping set} $S$ of $C$ with parity-check matrix $H$ is a subset of $[n]$ such that the restriction of $H$ to $S$ does not contain a row of weight 1. The \emph{stopping set distribution} $\{T_{i}(H)\}_{i=0}^{n}$ enumerates the number of stopping sets with size $i$ of $C$ with parity-check matrix $H$. Denote $H^{*}$ the parity-check matrix consisting of all the non-zero codewords in the dual code $C^{\bot}$. In this paper, we study stopping sets and stopping set distributions of some residue algebraic geometry (AG) codes with parity-check matrix $H^*$. First, we give two descriptions of stopping sets of residue AG codes. For the simplest AG codes, i.e., the generalized Reed-Solomon codes, it is easy to determine all the stopping sets. Then we consider AG codes from elliptic curves. We use the group structure of rational points of elliptic curves to present a complete characterization of stopping sets. Then the stopping sets, the stopping set distribution and the stopping distance of the AG code from an elliptic curve are reduced to the search, counting and decision versions of the subset sum problem in the group of rational points of the elliptic curve, respectively. Finally, for some special cases, we determine the stopping set distributions of AG codes from elliptic curves., Comment: 17 pages

Network coding provides the advantage of maximizing the usage of network resources, and has great application prospects in future network communications. However, the properties of network coding also make the pollution attack more serious. In this paper, we give an unconditional secure authentication scheme for network coding based on a linear code $C$. Safavi-Naini and Wang gave an authentication code for multi-receivers and multiple messages. We notice that the scheme of Safavi-Naini and Wang is essentially constructed with Reed-Solomon codes. And we modify their construction slightly to make it serve for authenticating subspace codes over linear network. Also, we generalize the construction with linear codes. The generalization to linear codes has the similar advantages as generalizing Shamir's secret sharing scheme to linear secret sharing sceme based on linear codes. One advantage of this generalization is that for a fixed message space, our scheme allows arbitrarily many receivers to check the integrity of their own messages, while the scheme with Reed-Solomon codes has a constraint on the number of verifying receivers. Another advantage is that we introduce access structure in the generalized scheme. Massey characterized the access structure of linear secret sharing scheme by minimal codewords in the dual code whose first component is 1. We slightly modify the definition of minimal codewords. Let $C$ be a $[V,k]$ linear code. For any coordinate $i\in \{1,2,\cdots,V\}$, a codeword $\vec{c}$ in $C$ is called minimal respect to $i$ if the codeword $\vec{c}$ has component 1 at the $i$-th coordinate and there is no other codeword whose $i$-th component is 1 with support strictly contained in that of $\vec{c}$. Then the security of receiver $R_i$ in our authentication scheme is characterized by the minimal codewords respect to $i$ in the dual code $C^\bot$., Comment: 18 pages

We analyze the security of the authentication code against pollution attacks in network coding given by Oggier and Fathi and show one way to remove one very strong condition they required. Actually, we find a way to attack their authentication scheme. In their scheme, they considered that if some malicious nodes in the network collude to make pollution in the network flow or make substitution attacks to other nodes, they thought these malicious nodes must solve a system of linear equations to recover the secret parameters. Then they concluded that their scheme is an unconditional secure scheme. Actually, note that the authentication tag in the scheme of Oggier and Fathi is nearly linear on the messages, so it is very easy for any malicious node to make pollution attack in the network flow, replacing the vector of any incoming edge by linear combination of his incoming vectors whose coefficients have sum 1. And if the coalition of malicious nodes can carry out decoding of the network coding, they can easily make substitution attack to any other node even if they do not know any information of the private key of the node. Moreover, even if their scheme can work fruitfully, the condition in their scheme $H\leqslant M$ in a network can be removed, where $H$ is the sum of numbers of the incoming edges at adversaries. Under the condition $H\leqslant M$, $H$ may be large, so we need large parameter $M$ which increases the cost of computation a lot. On the other hand, the parameter $M$ can not be very large as it can not exceed the length of original messages., Comment: 9 pages. arXiv admin note: text overlap with arXiv:0909.3146 by other authors

In practice, since many communication networks are huge in scale, or complicated in structure, or even dynamic, the predesigned linear network codes based on the network topology is impossible even if the topological structure is known. Therefore, random linear network coding has been proposed as an acceptable coding technique for the case that the network topology cannot be utilized completely. Motivated by the fact that different network topological information can be obtained for different practical applications, we study the performance analysis of random linear network coding by analyzing some failure probabilities depending on these different topological information of networks. We obtain some tight or asymptotically tight upper bounds on these failure probabilities and indicate the worst cases for these bounds, i.e., the networks meeting the upper bounds with equality. In addition, if the more topological information of the network is utilized, the better upper bounds are obtained. On the other hand, we also discuss the lower bounds on the failure probabilities., Comment: 5 pages, submitted to ISIT 2013. And add the interpretation of the notation $CUT_{t,k}^{out}$ to original manuscript (in column 2, page 2) submitted