Shuxing Li

Deterministic Construction of Compressed Sensing Matrices via Algebraic Curves

Compressed sensing is a sampling technique which provides a fundamentally new approach to data acquisition. Comparing with traditional methods, compressed sensing makes full use of sparsity so that a sparse signal can be reconstructed from very few measurements. A central problem in compressed sensing is the construction of sensing matrices. While random sensing matrices have been studied intensively, only a few deterministic constructions are known. Inspired by algebraic geometry codes, we introduce a new deterministic construction via algebraic curves over finite fields, which is a natural generalization of DeVores construction using polynomials over finite fields. The diversity of algebraic curves provides numerous choices for sensing matrices. By choosing appropriate curves, we are able to construct binary sensing matrices which are superior to Devores ones. We hope this connection between algebraic geometry and compressed sensing will provide a new point of view and stimulate further research in both areas.

Deterministic Sensing Matrices Arising From Near Orthogonal Systems

Compressed sensing is a novel sampling theory, which provides a fundamentally new approach to data acquisition. It asserts that a sparse or compressible signal can be reconstructed from much fewer measurements than traditional methods. A central problem in compressed sensing is the construction of the sensing matrix. While random sensing matrices have been studied intensively, only a few deterministic constructions are known. Among them, most constructions are based on coherence, which essentially generates matrices with low coherence. In this paper, we introduce the concept of near orthogonal systems to characterize the matrices with low coherence, which lie in the heart of many different applications. The constructions of these near orthogonal systems lead to deterministic constructions of sensing matrices. We obtain a series of m×n binary sensing matrices with sparsity level k=Θ(m(1/2)) or k=O((m/logm)(1/2)). In particular, some of our constructions are the best possible deterministic ones based on coherence. We conduct a lot of numerical experiments to show that our matrices arising from near orthogonal systems outperform several typical known sensing matrices.

The Weight Distribution of a Class of Cyclic Codes Related to Hermitian Forms Graphs

The determination of weight distribution of cyclic codes involves the evaluation of Gauss sums and exponential sums. Despite some cases where a neat expression is available, the computation is generally rather complicated. In this note, we determine the weight distribution of a class of reducible cyclic codes whose dual codes may have arbitrarily many zeros. This goal is achieved by building an unexpected connection between the corresponding exponential sums and the spectra of Hermitian forms graphs.

LCD Cyclic Codes Over Finite Fields

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.

Some New Results on the Cross Correlation of

The determination of the cross correlation between an m-sequence and its decimated sequence has been a longstanding research problem. Considering a ternary m-sequence of period 3^3r - 1, we determine the cross correlation distribution for decimations d = 3^r + 2 and d = 3^2r + 2, where gcd(r, 3) = 1. Meanwhile, for a binary m-sequence of period 2^2lm - 1, we make an initial investigation for the decimation d = (2^2lm - 1)/(2^m + 1) + 2^s, where l ≥ 2 is even and 0 <; s <; 2m - 1. It is shown that the cross correlation takes at least four values. Furthermore, we confirm the validity of two famous conjectures due to Sarwate et al. and Helleseth in this case.

m

Recently, there has been intensive research on the weight distributions of cyclic codes. In this paper, we compute the weight distributions of three classes of cyclic codes with Niho exponents. More specifically, we obtain two classes of binary three-weight and four-weight cyclic codes and a class of nonbinary four-weight cyclic codes. The weight distributions follow from the determination of value distributions of certain exponential sums. Several examples are presented to show that some of our codes are optimal and some have the best known parameters.

-Sequences

Constacyclic codes which generalize the classical cyclic codes have played important roles in recent constructions of many new quantum maximum distance separable (MDS) codes. However, the mathematical mechanism may not have been fully understood. In this paper, we use pseudo-cyclic codes, which is a further generalization of constacyclic codes, to construct the quantum MDS codes. We can not only provide a unified explanation of many previous constructions, but also produce some new quantum MDS codes.

On the Weight Distribution of Cyclic Codes With Niho Exponents

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.

Pseudo-cyclic Codes and the Construction of Quantum MDS Codes

The generalized Hamming weights (GHWs) of linear codes are fundamental parameters, the knowledge of which is of great interest in many applications. However, to determine the GHWs of linear codes is difficult in general. In this paper, we study the GHWs for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases. This is achieved by extending the idea of Yang et al. into higher dimension and by employing some interesting combinatorial arguments. It shall be noted that these cyclic codes may have arbitrary number of nonzeros.

Two Families of LCD BCH Codes

Partitioned difference families are an interesting class of discrete structures which can be used to derive optimal constant composition codes. There have been intensive researches on the construction of partitioned difference families. In this paper, we consider the combinatorial approach. We introduce a new combinatorial configuration named partitioned relative difference family, which proves to be very powerful in the construction of partitioned difference families. In particular, we present two general recursive constructions, which not only include some existing constructions as special cases, but also generate many new series of partitioned difference families. As an application, we use these partitioned difference families to construct several new classes of optimal constant composition codes.