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Compressed sensing (CS) is a new data acquisition theory taking full use of the sparsity of signals. It reveals that higher-dimensional sparse signals can be reconstructed from fewer nonadaptive linear measurements. The construction of CS matrices in CS is the key problem. In this paper, the deterministic CS matrices from optimal codebooks are constructed. Furthermore, the maximum sparsity of recovering the sparse signals by using our CS matrices are obtained. Meanwhile, a comparison is made with the CS matrices constructed by DeVore based on polynomials over finite fields. In the numerical simulations, our CS matrix outperforms DeVore ′ s matrix in the process of recovering sparse signals. [ABSTRACT FROM AUTHOR]

Linear code with locality r and availability t is that the value at each coordinate i can be recovered from t disjoint repairable sets each containing at most r other coordinates. This property is particularly useful for codes in distributed storage systems because it permits local repair of failed nodes and parallel access of hot data. In this paper, two constructions of (r , t) -locally repairable linear codes based on totally isotropic subspaces in symplectic space 𝔽 q (2 ν) over finite fields 𝔽 q are presented. Meanwhile, comparisons are made among the (r , t) -locally repairable codes we construct, the direct product code in Refs. [8], [11] and the codes in Ref. [9] about the information rate k n and relative distance d n . [ABSTRACT FROM AUTHOR]

In this paper, the Sphere-packing bound, Wang-Xing-Safavi-Naini bound, Johnson bound and Gilbert-Varshamov bound on the subspace code of length 2 ν + δ , size M , minimum subspace distance 2 j based on m -dimensional totally singular subspace in the (2 ν + δ) -dimensional orthogonal space 𝔽 q (2 ν + δ) over finite fields 𝔽 q of characteristic 2, denoted by (2 ν + δ , M , 2 j , m) q , are presented, where ν is a positive integer, δ = 0 , 1 , 2 , 0 ≤ m ≤ ν , 0 ≤ j ≤ m. Then, we prove that (2 ν + δ , M , 2 j , m) q codes attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in ℳ (m , 0 , 0 ; 2 ν + δ) , where ℳ (m , 0 , 0 ; 2 ν + δ) denotes the collection of all the m -dimensional totally singular subspaces in the (2 ν + δ) -dimensional orthogonal space 𝔽 q (2 ν + δ) over 𝔽 q of characteristic 2. Finally, Gilbert-Varshamov bound and linear programming bound on the subspace code (2 ν + δ , M , d) q in ℳ (2 ν + δ) are provided, where ℳ (2 ν + δ) denotes the collection of all the totally singular subspaces in the (2 ν + δ) -dimensional orthogonal space 𝔽 q (2 ν + δ) over 𝔽 q of characteristic 2. [ABSTRACT FROM AUTHOR]

Abstract In a number of applications, the codebook that minimizes the maximal cross-correlation amplitude (I max) is often desirable. There are two major ingredients in this paper. These two contents are to propose two constructions of codebooks. The codebooks generated by these constructions asymptotically meet the Welch bound. What we find is the codebook constructed in Section 2 can also asymptotically meet the Levenshtein bound. The parameters of these codebooks are new. [ABSTRACT FROM AUTHOR]

A symmetric informationally complete positive operator-valued measure (SIC-POVM) is a POVM in ℂ K consisting of K 2 positive operators of rank one such that all of whose Hermite inner products are equal. SIC-POVMs are important in quantum information theory, which have many applications in quantum state tomography, quantum cryptography and basic research in quantum mechanics. However, it is very difficult to construct SIC-POVMs. Therefore, many scholars have focused on approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs) for which the Hermite inner products are close to equal. In this paper, two new constructions of ASIC-POVMs are provided by using character sums and some special functions over finite fields. [ABSTRACT FROM AUTHOR]

Abstract Compressed sensing theory provides a new approach to acquire data as a sampling technique and makes sure that a sparse signal can be reconstructed from few measurements. The construction of compressed sensing matrices is a main problem in compressed sensing theory. In this paper, the deterministic compressed sensing matrices are provided using optimal codebooks and codes. Using specific linear and nonlinear codes, we present deterministic constructions of compressed sensing matrices, which are generalizations of DeVore′s construction and Li et al.′s construction. Compared with DeVore′s matrices and Li et al.′s matrices, by using appropriate optimal codebooks and specific codes, the compressed sensing matrices we construct are superior to DeVore′s matrices and Li et al.′s matrices. [ABSTRACT FROM AUTHOR]

Subspace code is very useful in the error correction for random network coding. In this paper, subspace code based on flats in affine space over finite fields is constructed. Sphere-packing bound, Wang–Xing–Safavi-Naini bound, anticode bound, Ahlswede–Aydinian bound and Gilbert–Varshamov bound for the size of code based on flats in affine space over finite fields are provided. Finally, we introduce another method of obtaining the anticode bound for the size of code based on flats in affine space from Erdős–Ko–Rado theorem. [ABSTRACT FROM AUTHOR]

Compressed sensing theory provides a new approach to acquire data as a sampling technique and makes sure that an original sparse signal 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 compressed sensing matrices with characters of finite fields are constructed and the coherence of the matrices are computed. Furthermore, the maximum sparsity of recovering the original sparse signals by using our compressed sensing matrices is obtained. Meanwhile, a comparison is made with the compressed sensing matrices constructed by DeVore based on polynomials over finite fields. In the numerical simulations, our compressed sensing matrix outperforms DeVore's matrix in the process of recovering original sparse signals. [ABSTRACT FROM AUTHOR]

Two orthonormal bases B and B′ of a K-dimensional complex inner-product space ℂK are called mutually unbiased bases (MUBs) if and only if |〈b|b′〉|2=1K holds for all b∈B and b′∈B′. The size of any set containing pairwise MUBs of ℂK cannot exceed K+1. If K is a power of a prime, then extremal sets containing K+1 MUBs are known to exist, which are called the complete MUBs of ℂK. We have not known whether there exist complete MUBs when K is not a power of a prime so far. Therefore, many researchers focus their attention on approximately mutually unbiased bases (AMUB). In this paper, two new constructions of AMUB of ℂK are provided based on the mixed character sum of two special kinds of functions over finite fields. [ABSTRACT FROM AUTHOR]

Recently, Luo et al. proposed two new codebooks based on the operations of some given sets, which are asymptotically optimal. In the paper, we present new construction of codebooks. We determine maximal cross-correlation amplitude of the constructed codebooks. The constructed codebooks are near optimal to the Welch bound. This new codebooks in the paper generalize Luo et al.′s constructions. The parameters of constructed codebooks are nimble and new under some circumstances. [ABSTRACT FROM AUTHOR]