168,306 research outputs found

    Euclidean and Hermitian LCD MDS codes

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    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 MDS codes. In particular, determining the existence of qq-ary [n,k][n,k] LCD MDS codes for various lengths nn and dimensions kk is a basic and interesting problem. In this paper, we firstly study the problem of the existence of qq-ary [n,k][n,k] LCD MDS codes and completely solve it for the Euclidean case. More specifically, we show that for q>3q>3 there exists a qq-ary [n,k][n,k] Euclidean LCD MDS code, where 0knq+10\le k \le n\le q+1, or, q=2mq=2^{m}, n=q+2n=q+2 and k=3orq1k= 3 \text{or} q-1. Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes

    Randomly punctured Reed--Solomon codes achieve list-decoding capacity over linear-sized fields

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    Reed--Solomon codes are a classic family of error-correcting codes consisting of evaluations of low-degree polynomials over a finite field on some sequence of distinct field elements. They are widely known for their optimal unique-decoding capabilities, but their list-decoding capabilities are not fully understood. Given the prevalence of Reed-Solomon codes, a fundamental question in coding theory is determining if Reed--Solomon codes can optimally achieve list-decoding capacity. A recent breakthrough by Brakensiek, Gopi, and Makam, established that Reed--Solomon codes are combinatorially list-decodable all the way to capacity. However, their results hold for randomly-punctured Reed--Solomon codes over an exponentially large field size 2O(n)2^{O(n)}, where nn is the block length of the code. A natural question is whether Reed--Solomon codes can still achieve capacity over smaller fields. Recently, Guo and Zhang showed that Reed--Solomon codes are list-decodable to capacity with field size O(n2)O(n^2). We show that Reed--Solomon codes are list-decodable to capacity with linear field size O(n)O(n), which is optimal up to the constant factor. We also give evidence that the ratio between the alphabet size qq and code length nn cannot be bounded by an absolute constant. Our proof is based on the proof of Guo and Zhang, and additionally exploits symmetries of reduced intersection matrices. With our proof, which maintains a hypergraph perspective of the list-decoding problem, we include an alternate presentation of ideas of Brakensiek, Gopi, and Makam that more directly connects the list-decoding problem to the GM-MDS theorem via a hypergraph orientation theorem

    On the maximal dimension of a completely entangled subspace for finite level quantum systems

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    Let Hi\mathcal{H}_i be a finite dimensional complex Hilbert space of dimension did_i associated with a finite level quantum system AiA_i for i=i,1,2,...,ki = i, 1,2, ..., k. A subspace SH=HA1A2...Ak=H1H2...HkS \subset \mathcal{H} = \mathcal{H}_{A_{1} A_{2}... A_{k}} = \mathcal{H}_1 \otimes \mathcal{H}_2 \otimes ... \otimes \mathcal{H}_k is said to be {\it completely entangled} if it has no nonzero product vector of the form u1u2...uku_1 \otimes u_2 \otimes ... \otimes u_k with uiu_i in Hi\mathcal{H}_i for each ii. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that maxSEdimS=d1d2...dk(d1+...+dk)+k1\max_{S \in \mathcal{E}} \dim S = d_1 d_2... d_k - (d_1 + ... + d_k) + k - 1 where E\mathcal{E} is the collection of all completely entangled subspaces. When H1=H2\mathcal{H}_1 = \mathcal{H}_2 and k=2k = 2 an explicit orthonormal basis of a maximal completely entangled subspace of H1H2\mathcal{H}_1 \otimes \mathcal{H}_2 is given. We also introduce a more delicate notion of a {\it perfectly entangled} subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a problem

    On Hull-Variation Problem of Equivalent Linear Codes

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    The intersection CC{\bf C}\bigcap {\bf C}^{\perp} (CCh{\bf C}\bigcap {\bf C}^{\perp_h}) of a linear code C{\bf C} and its Euclidean dual C{\bf C}^{\perp} (Hermitian dual Ch{\bf C}^{\perp_h}) is called the Euclidean (Hermitian) hull of this code. The construction of an entanglement-assisted quantum code from a linear code over Fq{\bf F}_q or Fq2{\bf F}_{q^2} depends essentially on the Euclidean hull or the Hermitian hull of this code. Therefore it is natural to consider the hull-variation problem when a linear code C{\bf C} is transformed to an equivalent code vC{\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. A general method to construct hull-decreasing or hull-increasing equivalent linear codes is proposed. We prove that for a nonnegative integer hh satisfying 0hn10 \leq h \leq n-1, a linear [2n,n]q[2n, n]_q self-dual code is equivalent to a linear hh-dimension hull code. On the opposite direction we prove that a linear LCD code over F2s{\bf F}_{2^s} satisfying d2d\geq 2 and d2d^{\perp} \geq 2 is equivalent to a linear one-dimension hull code under a weak condition. Several new families of negacyclic LCD codes and BCH LCD codes over F3{\bf F}_3 are also constructed. Our method can be applied to the generalized Reed-Solomon codes and the generalized twisted Reed-Solomon codes to construct arbitrary dimension hull MDS codes. Some new EAQEC codes including MDS and almost MDS entanglement-assisted quantum codes are constructed. Many EAQEC codes over small fields are constructed from optimal Hermitian self-dual codes.Comment: 33 pages, minor error correcte

    An information-theoretic view of network management

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    We present an information-theoretic framework for network management for recovery from nonergodic link failures. Building on recent work in the field of network coding, we describe the input-output relations of network nodes in terms of network codes. This very general concept of network behavior as a code provides a way to quantify essential management information as that needed to switch among different codes (behaviors) for different failure scenarios. We compare two types of recovery schemes, receiver-based and network-wide, and consider two formulations for quantifying network management. The first is a centralized formulation where network behavior is described by an overall code determining the behavior of every node, and the management requirement is taken as the logarithm of the number of such codes that the network may switch among. For this formulation, we give bounds, many of which are tight, on management requirements for various network connection problems in terms of basic parameters such as the number of source processes and the number of links in a minimum source-receiver cut. Our results include a lower bound for arbitrary connections and an upper bound for multitransmitter multicast connections, for linear receiver-based and network-wide recovery from all single link failures. The second is a node-based formulation where the management requirement is taken as the sum over all nodes of the logarithm of the number of different behaviors for each node. We show that the minimum node-based requirement for failures of links adjacent to a single receiver is achieved with receiver-based schemes

    Distance-regular graphs

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    This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

    The decoding failure probability of MDPC codes

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    Moderate Density Parity Check (MDPC) codes are defined here as codes which have a parity-check matrix whose row weight is O(n)O(\sqrt{n}) where nn is the length nn of the code. They can be decoded like LDPC codes but they decode much less errors than LDPC codes: the number of errors they can decode in this case is of order Θ(n)\Theta(\sqrt{n}). Despite this fact they have been proved very useful in cryptography for devising key exchange mechanisms. They have also been proposed in McEliece type cryptosystems. However in this case, the parameters that have been proposed in \cite{MTSB13} were broken in \cite{GJS16}. This attack exploits the fact that the decoding failure probability is non-negligible. We show here that this attack can be thwarted by choosing the parameters in a more conservative way. We first show that such codes can decode with a simple bit-flipping decoder any pattern of O(nloglognlogn)O\left(\frac{\sqrt{n} \log \log n}{\log n}\right) errors. This avoids the previous attack at the cost of significantly increasing the key size of the scheme. We then show that under a very reasonable assumption the decoding failure probability decays almost exponentially with the codelength with just two iterations of bit-flipping. With an additional assumption it has even been proved that it decays exponentially with an unbounded number of iterations and we show that in this case the increase of the key size which is required for resisting to the attack of \cite{GJS16} is only moderate
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