Sphere Packings, Lattices and Groups

Nowadays, a large amount of information has to be transmitted or processed. This implies high-power processing, large memory density, and increased energy consumption. In several applications, such as imaging, radar, speech recognition, and data acquisition, the signals involved can be considered sparse or compressive in some domain. The compressive sensing theory could be a proper candidate to deal with these constraints. It can be used to recover sparse or compressive signals with fewer measurements than the traditional methods. Two problems must be addressed by compressive sensing theory: design of the measurement matrix and development of an efficient sparse recovery algorithm. These algorithms are usually classified into three categories: convex relaxation, non-convex optimization techniques, and greedy algorithms. This paper intends to supply a comprehensive study and a state-of-the-art review of these algorithms to researchers who wish to develop and use them. Moreover, a wide range of compressive sensing theory applications is summarized and some open research challenges are presented.

/pdf/a-review-of-sparse-recovery-algorithms-3a8ag4tbpe.pdf

A Review of Sparse Recovery Algorithms

Linear codes with complementary duals (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. Nonbinary 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 a Euclidean LCD code and any linear code over $\mathbb F_{q^{2}} (q>2)$ is equivalent to a Hermitian LCD code. Consequently an $[n,k,d]$ -linear Euclidean LCD code over $\mathbb F_{q}$ with $q>3$ exists if there is an $[n,k,d]$ -linear code over $\mathbb F_{q}$ and an $[n,k,d]$ -linear Hermitian LCD code over $\mathbb F_{q^{2}}$ with $q>2$ exists if there is an $[n,k,d]$ -linear code over $\mathbb F_{q^{2}}$ . Hence, when $q>3$ (resp. $q>2$ ) $q$ -ary Euclidean (resp. $q^{2}$ -ary Hermitian) LCD codes possess the same asymptotical bound as $q$ -ary linear codes (resp. $q^{2}$ -ary linear codes). This gives a direct proof that every triple of parameters $[n,k,d]$ which is attainable by linear codes over $\mathbb F_{q}$ with $q>3$ (resp. over $\mathbb F_{q^{2}}$ with $q>2$ ) is attainable by Euclidean LCD codes (resp. by Hermitian LCD codes). In particular there exist families of $q$ -ary Euclidean LCD codes ( $q>3$ ) and $q^{2}$ -ary Hermitian LCD codes ( $q>2$ ) exceeding the asymptotical Gilbert–Varshamov bound. Further, we give a second proof of these results using the theory of Grobner bases. Finally, we present a new approach of constructing LCD codes by extending linear codes.

Linear Codes Over $\mathbb F_q$ Are Equivalent to LCD Codes for $q>3$

Compressed sensing (CS) is a technique that is suitable for compressing and recovering signals having sparse representations in certain bases. CS has been widely used to optimize the measurement process of bandwidth and power constrained systems like wireless body sensor network. The central issues with CS are the construction of measurement matrix and the development of recovery algorithm. In this paper, we propose a simple deterministic measurement matrix that facilitates the hardware implementation. To control the sparsity level of the signals, we apply a thresholding approach in the discrete cosine transform domain. We propose a fast and simple recovery algorithm that performs the proposed thresholding approach. We validate the proposed method by compressing and recovering electrocardiogram and electromyogram signals. We implement the proposed measurement matrix in a MSP-EXP430G2 LaunchPad development board. The simulation and experimental results show that the proposed measurement matrix has a better performance in terms of reconstruction quality compared with random matrices. Depending on the compression ratio, it improves the signal-to-noise ratio of the reconstructed signals from 6 to 20 dB. The obtained results also confirm that the proposed recovery algorithm is, respectively, 23 and 12 times faster than the orthogonal matching pursuit (OMP) and stagewise OMP algorithms.

Compressed Sensing: A Simple Deterministic Measurement Matrix and a Fast Recovery Algorithm

Three-weight cyclic codes and their weight distributions

In addition to their applications in data storage, communications systems, and consumer electronics, linear complementary dual (LCD) codes—a class of linear codes—have been employed in cryptography recently. LCD cyclic codes were referred to as reversible cyclic codes in the literature. The objective of this paper is to construct several families of reversible cyclic codes over finite fields and analyze their parameters. The LCD cyclic codes presented in this paper have very good parameters in general, and contain many optimal codes. A well rounded treatment of reversible cyclic codes is also given in this paper.

pdf/lcd-cyclic-codes-over-finite-fields-21ulaj7kjc.pdf

LCD Cyclic Codes Over Finite Fields

Historically, LCD cyclic codes were referred to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In this paper, we explore two special families of LCD cyclic codes, which are both BCH codes. The dimensions and the minimum distances of these LCD BCH codes are investigated.

Two Families of LCD BCH Codes

Cyclic codes are widely employed in communication systems, storage devices, and consumer electronics, as they have efficient encoding and decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. A subclass of good BCH codes are the narrow-sense BCH codes over $ {\mathrm {GF}}(q)$ with length $n=(q^{m}-1)/(q-1)$ . Little is known about this class of BCH codes when $q>2$ . The objective of this paper is to study some of the codes within this class. In particular, the dimension, the minimum distance, and the weight distribution of some ternary BCH codes with length $n=(3^{m}-1)/2$ are determined in this paper. A class of ternary BCH codes meeting the Griesmer bound is identified. An application of some of the BCH codes in secret sharing is also investigated.

Narrow-Sense BCH Codes Over $ {\mathrm {GF}}(q)$ With Length $n=\frac {q^{m}-1}{q-1}$

Due to wide applications of BCH codes, the determination of their minimum distance is of great interest. However, this is a very challenging problem for which few theoretical results have been reported in the last four decades. Even for the narrow-sense primitive BCH codes, which form the most well studied subclass of BCH codes, there are very few theoretical results on the minimum distance. In this paper, we present new results on the minimum distance of narrow-sense primitive BCH codes with special Bose distance. We prove that for a prime power $q$, the $q$-ary narrow-sense primitive BCH code with length $q^m-1$ and Bose distance $q^m-q^{m-1}-q^i-1$, where $\frac{m-2}{2} \le i \le m-\lfloor \frac{m}{3} \rfloor-1$, has minimum distance $q^m-q^{m-1}-q^i-1$. This is achieved by employing the beautiful theory of sets of quadratic forms, symmetric bilinear forms, and alternating bilinear forms over finite fields, which can be best described using the framework of association schemes.

The Minimum Distance of Some Narrow-Sense Primitive BCH Codes

Symbol-pair code is a new coding framework which is proposed to correct errors in the symbol-pair read channel. In particular, maximum distance separable (MDS) symbol-pair codes are a kind of symbol-pair codes with the best possible error-correction capability. Employing cyclic and constacyclic codes, we construct three new classes of MDS symbol-pair codes with minimum pair-distance five or six. Moreover, we find a necessary and sufficient condition which ensures a class of cyclic codes to be MDS symbol-pair codes. This condition is related to certain property of a special kind of linear fractional transformations. A detailed analysis on these linear fractional transformations leads to an algorithm, which produces many MDS symbol-pair codes with minimum pair-distance seven.

Constructions of maximum distance separable symbol-pair codes using cyclic and constacyclic codes

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