Deterministic Sensing Matrices Arising from Near Orthogonal Systems

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 4, APRIL 2014

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Deterministic Sensing Matrices Arising from Near Orthogonal Systems Shuxing Li and Gennian Ge

Abstract— 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 = (m(1/2)) or k = O (m/log m)(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. Index Terms— Compressed sensing, deterministic construction, coherence, near orthogonal system.

I. I NTRODUCTION

C

OMPRESSED sensing (CS) is a new data acquisition theory exploiting the sparsity or compressibility of signals. It reveals that a sparse or compressible signal can be recovered from much fewer samples than traditional sampling theory [13], [29]. We can view CS as a two-stage scheme. Firstly, the structure of a sparse or approximately sparse signal is captured by employing nonadaptive linear projections. This stage combines both the sampling and the compression process. Secondly, the signal can be recovered by solving an optimization problem. Consider a discrete-time signal x ∈ Rn , we use m measurements in the sampling process. In matrix notation, y = x + e,

Manuscript received October 14, 2012; revised August 7, 2013; accepted November 28, 2013. Date of publication January 31, 2014; date of current version March 13, 2014. G. Ge was supported in part by the National Natural Science Foundation of China under Grant 61171198 and in part by Zhejiang Provincial Natural Science Foundation of China under Grant LZ13A010001. S. Li is with the Department of Mathematics, Zhejiang University, Hangzhou 310027, China (e-mail: [email protected]). G. Ge is with the School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China, and also with the Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing 100048, China (e-mail: [email protected]). Communicated by Y. Ma, Associate Editor for Signal Processing. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIT.2014.2303973

where  is an m × n sensing matrix with m < n and e is an unknown noise term. When x is sparse or approximately sparse, the CS theory asserts that by using an appropriate sensing matrix , y essentially contains enough information to recover x, even if m  n. x is called k-sparse if it has no more than k nonzero entries. To decide what matrix is appropriate, we need some criteria. Restricted isometry property (RIP) is a well-known criterion proposed in [14]. A matrix  is said to satisfy the RIP of order k if there exists a constant 0 ≤ δk < 1, such that (1 − δk )x22 ≤ x22 ≤ (1 + δk )x22

(1)

holds for any k-sparse signal x. δk is called restricted isometry constant (RIC) of order k, which is the smallest nonnegative number such that (1) holds for every k-sparse signal. RIP is a sufficient condition which guarantees reliable reconstruction of sparse or approximately sparse signals via 1 -minimization [12]. If a sensing matrix satisfies the RIP and its RIC is small enough, greedy algorithms such as Iterative Thresholding [8], [31], Orthogonal Matching Pursuit (OMP) [42], [53] and its modifications [22], [44], [45] also guarantee to recover sparse or approximately sparse signals. In a word, by choosing an appropriate sensing matrix, we can reduce the reconstruction of sparse or approximately sparse signals to an optimization problem with efficient algorithms available. The construction of sensing matrices is a central problem in CS. Suppose a k-sparse signal x ∈ Rn can be recovered uniformly and stably from m measurements. An upper bound of the possible sparsity is k ≤ Cm/ log(n/k), where C is a constant [18]. Random matrices achieving this bound are presented in [15], which promise to recover sparse signals with high probability. In fact, if the entries of a matrix are randomly drawn from certain probability distributions, then the matrix satisfies the RIP of order k with high probability, where k ≤ Cm/ log(n/k) for some constant C [4]. Although random matrices serve as good candidates, there are several reasons in favor of deterministic sensing matrices. Firstly, there is no efficient algorithm testing the RIP of a random matrix, even though it does satisfy the RIP with overwhelming probability. Secondly, random matrices require considerable storage space when the size of signal is large. Comparing with random sensing matrices, deterministic ones can overcome these drawbacks. The RIP of deterministic matrices is guaranteed by their constructions. Exploiting the structure of

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deterministic matrices, it is possible to save them in a more storage-saving manner. Moreover, the structure helps to devise specific fast recovery algorithm, which is crucial in the realtime applications. Therefore, we focus on the constructions of deterministic sensing matrices in this paper. For a matrix A with columns a1 , a2 , . . . , an , the coherence of A is defined as |ai , a j | , for 1 ≤ i, j ≤ n. μ(A) = max i = j ai 2 · a j 2 Coherence plays a central role in the deterministic constructions, because matrices with low coherence satisfy the RIP. Lemma I.1 ([9, Proposition 1]). Suppose  is a matrix with coherence μ. Then  satisfies the RIP of order k with δk ≤ μ(k − 1), whenever k < 1/μ + 1. Lemma I.1 says the matrices with low coherence are natural candidates for sensing matrices. On the other hand, for an m × n matrix , we have the well-known Welch bound [55]:  n−m . (2) μ() ≥ m(n − 1) This bound implies that the deterministic constructions based on coherence can only generate sensing matrices with the RIP 1 of order k = O(m 2 ). In recent years, several deterministic constructions using the RIP as a criterion have been proposed. Most of them are based on coherence. In [24], DeVore uses polynomials over finite field F p to construct binary sensing matrices of size p2 × pr+1 , where p is a prime power. These matrices are with coherence r/ p and satisfy the RIP of order k < p/r + 1. A generalization of DeVore’s construction is proposed in [41], using algebraic curves over finite fields. In [1], Bose, Chaudhuri, and Hocquenghem (BCH) codes with large minimum (l− j )

ln j j

) bipolar distance are used to construct (2l − 1) × 2 O(2 l− j l matrices with coherence μ ≤ (2 − 1)/(2 − 1) and the RIP of order k ≤ 2 j + 1. This construction is generalized to an analogy in the complex field by using p-ary BCH codes [2]. (l−r)

log p r

r ) ( pl − 1) × pO( p complex matrices are obtained with l−r coherence μ ≤ p( p − 1)/(2( p − 1)( pl − 1)). Methods of additive combinatorics lead to m × n matrices satisfying the 1 RIP of order k ≥ m 2 + for some  > 0 and n 1− ≤ m ≤ n [9]. It is remarkable that this construction overcomes the natural 1 barrier k = O(m 2 ) of coherence-based methods. On the other hand, some non-RIP deterministic constructions have been presented. In [3], chirp sequences are used to form the columns of a complex sensing matrix and a fast recovery algorithm is proposed. Realizing the connection between CS theory and coding theory, real sensing matrices are constructed from the second order Reed-Muller codes and its subcodes [34]. Calderbank et al. [11] summarize these two constructions by showing that these sensing matrices satisfy the statistical RIP. The statistical RIP is weaker than the RIP and guarantees recovery of all but an exponentially small fraction of sparse signals. A modified version of RIP named RIP-1 is proposed in [7]. Unbalanced expanders, which are bipartite graphs with good expansion property, generate binary sensing matrices satisfying RIP-1. In [35], binary matrices are

constructed by exploiting hash functions and extractor graphs. An iterative construction using hash families is proposed in [19]. Since the orthogonal matrix has zero coherence, we name matrices with low coherence as near orthogonal systems. Despite different backgrounds and motivations, near orthogonal systems essentially lie in the heart of many different scenes, with various names. In this paper, we provide an overview of near orthogonal systems originated from distinct applications. The constructions of these near orthogonal systems are actually deterministic constructions of sensing matrices based on coherence. More specifically, we obtain several classes of deterministic  m ×1 n matrices with sparsity 1 level k = (m 2 ) or k = O ( logmm ) 2 . A lot of experimental results show that our matrices have good recovery performance. Comparing with Gaussian matrices, random Discrete Fourier Transform matrices, Bernoulli matrices and matrices formed by BCH codes [1], [2], our matrices outperform them in many numerical simulations. The rest of this paper is organized as follows: Section II discusses near orthogonal systems arising from Maximum Welch-Bound-Equality sequence sets. Section III discusses near orthogonal systems arising from signal sets. Section IV discusses near orthogonal systems arising from mutually unbiased bases and approximately mutually unbiased bases. In each of these sections, we provide necessary explanations for the corresponding near orthogonal systems and mathematical tools involved. Section V is devoted to the numerical experiments. Section VI concludes this paper. II. N EAR O RTHOGONAL S YSTEMS A RISING FROM MWBE S EQUENCE S ETS A Maximum Welch-Bound-Equality (MWBE) sequence set can be viewed as a matrix whose coherence meets the Welch lower bound (2). Therefore, we regard an (N, K ) MWBE sequence set as a K × N matrix with coherence KN−K (N−1) . MWBE sequence sets are also known as equiangular tight frames [52] or optimal Grassmannian frames [51]. The applications of MWBE sequence sets in communications and coding theory are presented in [51]. Because of the strong constraints on the existence of MWBE sequence sets [52], the construction of them is very hard [48]. In recent years, new MWBE sequence sets are generated from difference sets [25], [26], [57] and Steiner systems [30]. In the following subsections, we focus on the near orthogonal systems arising from these new MWBE sequence sets. The constructions of sensing matrices from MWBE sequence sets are the best possible deterministic ones based on coherence, since they achieve the Welch bound (2). A. MWBE Sequence Sets Generated from Difference Sets Codebooks were used in code-division multiplex-access (CDMA) systems to maximize the incoherence of different signals. In [25], [26], [57], complex codebooks which are actually MWBE sequence sets are constructed from difference sets. Below, we review the construction in [26].

LI AND GE: DETERMINISTIC SENSING MATRICES ARISING FROM NEAR ORTHOGONAL SYSTEMS

Suppose G is a finite abelian group of order v. A subset D of size k in G is called a (v, k, λ) difference set if each nonidentity element of G can be represented as d1 − d2 , d1 , d2 ∈ D in exactly λ ways. Please refer to [36] for a survey of difference sets. A character χ of G is a homomorphism from G into the multiplicative group U of complex numbers of absolute value 1. The following elementary facts about group characters are well known: 1) χ(0) = 1; 2) χ(g) is a root of unity; 3) χ(g −1 ) = χ(g)−1 = χ(g), where the bar denotes the complex conjugate. All characters of G forms a group, which is called the  character group of G and denoted by G. For any finite abelian group G, its character group can be determined as follows. Given any positive integer n, let ζn be 2π i the n-th root of unity e n . For the cyclic group Zn , we have n = {χi | 0 ≤ i ≤ n − 1}, where Z ja

χ j (a) = ζn , ∀a ∈ Zn . Note that any finite abelian group is a direct sum of cyclic   Zn2 · · · Znt . Then, the character groups, i.e., G = Zn1  = {χ j1, j2 ,..., jt | ji ∈ Zni , 1 ≤ i ≤ t}, where group G χ j1 , j2 ,..., jt ((a1 , a2 , . . . , at )) =

t

ja

ζnii i , ∀(a1 , a2 , . . . , at ) ∈ G.

i=1

 = |G|. Thus, we have |G|  = Let G be an abelian group of order N and G {χ0 , χ1 , . . . , χ N−1 }. Suppose D = {d1 , d2 , . . . , d K } is a K-subset of G. We define a K × N matrix  := (G, D) whose i -th column is φi = (χi−1 (d1 ), χi−1 (d2 ), . . . , χi−1 (d K ))T .  is an MWBE sequence set if D is a difference set in G. Proposition II.1 ( [26, Theorem 3]).  is an (N, K ) MWBE sequence set if and only if the set D is an (N, K , λ) difference set in G, where K > 1. As a direct consequence, deterministic sensing matrices are obtained from difference sets. Theorem II.1 Given an (n, m, λ) difference set in an abelian group of order n, there exists an m × n matrix  with μ() =  n−m m(n−1) . Therefore, the specific construction of sensing matrices follows by choosing proper difference sets. The first construction uses the Singer difference set [36]. Construction II.1 (Singer matrix). Suppose q is a prime power and d is an integer with d ≥ 3. Let α be a primitive element of F∗q d and trq d /q (x) =

d−1

xq

i

i=0

be the trace function from Fq d to Fq . Then the set of integers {i | 0 ≤ i < (q d −1)/(q −1), trq d /q (α i ) = 0} forms a (v, k, λ)

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Singer difference set in the group Zv , where qd − 1 , q −1 q d−1 − 1 k= , q −1 q d−2 − 1 λ= . q−1 v =

By Theorem II.1, we obtain an m × n sensing matrix  with q d−1 − 1 , q −1 qd − 1 n= , q −1

m =

d−2 2

(q − 1) . −1 We name the above matrix as a Singer matrix hereafter. Using the McFarland difference set [43], we have the following similar construction. Construction II.2 (McFarland matrix). Suppose q is a prime power and d is a positive integer. Let G be a group of order v = q d+1 (q d + · · · + q 2 + q + 2) that contains an elementary abelian subgroup E of order q d+1 in its center. View E as the additive group of Fd+1 q , which is a (d + 1)-dimensional vector space over Fq . Set s = (q d+1 − 1)/(q − 1). There are exactly s subspaces of E with dimension d, which are denoted by H1, H2 , . . . , Hs . If g0 , . . . , gs are distinct coset representatives of E in G, then D = (g1 + H1) ∪ (g2 + H2 ) ∪ · · · ∪ (gs + Hs ) is a (v, k, λ) McFarland difference set with  d+1 −1 d+1 q v =q +1 , q −1  d+1 q −1 k = qd , q −1  d q −1 λ = qd . q −1 μ() =

q

q d−1

In particular, we choose a positive integer d such that q d+1 −1 +1 is a prime power. (Generally, it seems hard to q−1

determine which pair of (q, d) ensures that q q−1−1 + 1 is a prime power. A computer program shows that in the set {(q, d) | 2 ≤ q ≤ 100, q prime power, 1 ≤ d ≤ 10}, there are 41 suitable pairs of (q, d) among the total of d+1 350 pairs.) Now, we suppose q q−1−1 + 1 = r l , where r is a prime with gcd(r, q) = 1 and l is a positive integer. l Define (G 1 , +) = (Fd+1 q , +) and (G 2 , +) = (Fr , +). Then (G, +) = (G 1 × G 2 , +) is an abelian group of order v and E = G 1 × {0} is an elementary abelian subgroup of order q d+1 . By Theorem II.1, we obtain an m × n sensing matrix  with  d+1 q −1 , m = qd q −1  d+1 q −1 −1 q + 1 , μ() = d+1 . n = q d+1 q −1 q −1 d+1

We name the above matrix as a McFarland matrix hereafter.

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B. Steiner MWBE Sequence Sets The Steiner system is one of the main themes in combinatorial design theory [20]. In [30], several new infinite families of MWBE sequence sets are constructed from Steiner systems. We review this construction below. A (2, k, v) Steiner system is a pair (X, B), where X is a v-set of elements (points) and B is a collection of k-subsets of X (blocks), such that every 2-subset of points occurs in exactly one block in B. Let (X, B) be a (2, k, v) Steiner system where X = {x 1 , . . . , x v } and B = {B1 , . . . , Bb }. Direct calculation shows b = v(v−1) k(k−1) . The incidence matrix of (X, B) is a v × b binary matrix M defined by

1 if x i ∈ B j , Mi, j = 0 if x i ∈ B j . Proposition II.2 ([30, Theorem 1]). Every (2, k, v) Steiner system generates an (N, K ) MWBE sequence set with N = v(v−1) v(1 + v−1 k−1 ) and K = k(k−1) . Specifically, the K × N matrix  may be constructed as follows: 1) Let A be the transpose of the incidence matrix of a (2, k, v) Steiner system, which is with size v(v−1) k(k−1) × v. 2) For each j = 1, . . . , v, let H j be any (1 + v−1 k−1 ) × ) matrix that has orthogonal rows and uni(1 + v−1 k−1 modular entries, such as a possibly complex Hadamard matrix. v−1 3) For each j = 1, . . . , v, let  j be the v(v−1) k(k−1) × (1 + k−1 ) matrix obtained from the j -th column of A by replacing the one-valued entries with distinct rows of H j , and every zero-valued entry with a row of zeros. Note that there are v−1 k−1 one-valued entries in each column of A, hence only v−1 k−1 rows of H j are chosen to form  j . 4) Concatenate and rescale the  j ’s to form  = k−1 12 ( v−1 ) [1 · · · v ]. Remark 1. The above construction can be regarded as a special case of the mixing technique introduced in [1] and [2], which generates nonbinary near orthogonal systems from binary matrices. With the help of the Steiner system, we have the following theorem. Theorem II.2. Given a (2, k, v) Steiner system, there exists a v(v−1) v−1 k−1 k(k−1) × v(1 + k−1 ) matrix  with μ() = v−1 . Hence, the specific construction of sensing matrices follows by choosing proper Steiner systems. In particular, we use the following four infinite families of Steiner systems. Construction II.3 (Affine matrix). Let q be a prime power. For d ≥ 2, there exists a (2, q, q d ) Steiner system originated from affine space [20]. By Theorem II.2, we obtain an m × n sensing matrix  with  d q −1 m = q d−1 , q −1  d q−1 q −1 + 1 , μ() = d . n = qd q −1 q −1 We name the above matrix as an affine matrix hereafter.

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Construction II.4 (Projective matrix). Let q be a prime d+1 power. For d ≥ 2, there exists a (2, q + 1, q q−1−1 ) Steiner system originated from projective space [20]. By Theorem II.2, we obtain an m × n sensing matrix  with (q d − 1)(q d+1 − 1) , (q + 1)(q − 1)2  q d+1 − 1 q d − 1 +1 , n= q −1 q−1 q −1 μ() = d . q −1 We name the above matrix as a projective matrix hereafter. Construction II.5 (Unital matrix). Let q be a prime power. There exists a (2, q +1, q 3 +1) Steiner system originated from unital [20]. By Theorem II.2, we obtain an m × n sensing matrix  with m =

q 2 (q 3 + 1) , q +1 n = (q 2 + 1)(q 3 + 1), 1 μ() = 2 . q We name the above matrix as a unital matrix hereafter. Construction II.6 (Denniston matrix). For any 2 ≤ r < s, there exists a (2, 2r , 2r+s + 2r − 2s ) Steiner system originated from a Denniston design [20]. By Theorem II.2, we obtain an m × n sensing matrix  with m=

(2s + 1)(2r+s + 2r − 2s ) , 2r n = (2s + 2)(2r+s + 2r − 2s ), 1 . μ() = s 2 +1 We name the above matrix as a Denniston matrix hereafter. m =

III. N EAR O RTHOGONAL S YSTEMS A RISING FROM S IGNAL S ETS For an m × n matrix , the Welch bound (2) is not tight when n is large. More precisely, the equality in (2) holds when  is a real matrix, and only if only if n ≤ m(m+1) 2 n ≤ m 2 when  is a complex matrix [51]. When n is large, the following bounds developed by Levenstein [40] are better than the Welch bound. , then If  is a real matrix with n > m(m+1) 2  3n − m 2 − 2m . (3) μ() ≥ (m + 2)(n − m) If  is a complex matrix with n > m 2 , then  2n − m 2 − m . μ() ≥ (m + 1)(n − m)

(4)

An (N, K ) signal set can be regarded as a K × N matrix. In synchronous CDMA applications, a signal set is used to distinguish between the signals of different users. For this purpose, the coherence of the matrix should be as small as possible. The signal set meets (resp. nearly meets) the Levenstein

LI AND GE: DETERMINISTIC SENSING MATRICES ARISING FROM NEAR ORTHOGONAL SYSTEMS

bound is called an optimal (resp. a near optimal) signal set. In [27], optimal and near optimal signals are constructed from specific classes of planar functions and almost bent functions. We review this construction below. Suppose q = p t , where p is a prime and t is a positive integer. We use x 0 , x 1 , . . . , x q−1 to denote all of the elements of the finite field Fq . For any positive integer l, let ζl be the 2π i l-th root of unity e l . Let trq/ p be the absolute trace function from Fq to F p . For any x ∈ Fq , define tr

ψ(x) = ζ p q/ p

(x)

.

Then ψ is an additive group character of Fq . (q) Let ei be a vector in the q-dimensional Hilbert space with the i -th entry being one and the other entries being zeros. (q) Define E (q) = {ei | 1 ≤ i ≤ q}, which is the signal set formed by the standard basis. Let f be a function from Fq to Fq . For each pair (a, b) ∈ F2q , we define the unit-norm vector 1 C f (a, b) = √ (ψ(a f (x 0 ) + bx 0 ), . . . , ψ(a f (x q−1 )+bx q−1)). q Then, we define the signal set C f = {C f (a, b) | (a, b) ∈ F2q } ∪ E (q) . The signal set C f is optimal or near optimal with respect to the Levenstein bound if f is planar or almost bent. We introduce these concepts respectively in the following subsections. A. Optimal Signal Sets Generated from Planar Functions Suppose q = pt , where p is an odd prime and t is a positive integer. Suppose f is a function from abelian group A to abelian group B. The measure of nonlinearity of f is defined by P f = max max 0 =a∈A b∈B

|{x ∈ A | f (x + a) − f (x) = b}| . |A|

1 Clearly, P f ≥ |B| . A function f : A → B has perfect 1 . A perfect nonlinear function from nonlinearity if P f = |B| a finite abelian group to a finite abelian group of the same order is called a planar function in finite geometry. Planar functions were introduced by Dembowski and Ostrom for the construction of affine planes [23]. For a survey of highly nonlinear functions, we refer to [17]. Below, we list some known planar functions from F pt to F pt . 1) f (x) = x 2 ; k 2) f (x) = x p +1 , where t/ gcd(t, k) is odd [23]; k 3) f (x) = x (3 +1)/2 , where p = 3, k is odd and gcd(t, k) = 1 [21]; 4) f (x) = x 10 − ux 6 − u 2 x 2 , where p = 3, t is odd and u ∈ F3t [21], [28]. For a more comprehensive enumeration of planar functions, please refer to [47]. We can obtain optimal signal sets from planar functions as follows. Proposition III.1 ( [27, Theorem 4]). Let f be a planar function from Fq to Fq . Then, C f is a (q 2 + q, q) optimal signal set.

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Take m = q, n = q 2 + q, the Levenstein bound (4) is  1 2n − m 2 − m = √ . (m + 1)(n − m) q Consequently, we have the following theorem. Theorem III.1. Given a planar function from Fq to Fq , there exists a q × (q 2 + q) matrix  with μ() = √1q . Hence, the specific construction of sensing matrices follows by choosing proper planar functions. Construction III.1 (matrices formed by optimal signal sets). Let q = pt where p is an odd prime and t is a positive integer. f 1 (x) = x 2 is a planar function from Fq to Fq . By Theorem III.1., we obtain an m × n sensing matrix  with m = q, n = q 2 + q, 1 μ() = √ . q Suppose k is a positive integer such that t/ gcd(t, k) is odd. k Then f2 (x) = x p +1 is also a planar function from Fq to Fq . We can similarly obtain a sensing matrix with the same parameters as those of the above one. Construction III.2 (matrices formed by optimal signal sets). Let q = 3t where t is a positive integer. Suppose k is odd and k gcd(t, k) = 1. Then f3 (x) = x (3 +1)/2 is a planar function from Fq to Fq . By Theorem III.1., we obtain an m × n sensing matrix  with m = 3t , n = 32t + 3t , 1 μ() = √ . 3t If t is odd, for any u ∈ Fq , f 4 (x) = x 10 − ux 6 − u 2 x 2 is also a planar function from Fq to Fq . We can similarly obtain a sensing matrix with the same parameters as those of the above one. Note that the sensing matrices constructed from optimal signal sets are the best possible deterministic ones based on coherence, since they achieve the Levenstein bound (4). B. Near Optimal Signal Sets Generated from Almost Bent Functions Suppose q = 2t , where t is a positive integer. In this case, there is no planar function from Fq to Fq . For a function f from Fq to Fq , we define

(−1)tr2t /2 (a f (x)+bx), λ f (a, b) = x∈Fq

where (a, b) ∈ F2q and tr2t /2 is the trace function from F2t to F2 . f is called almost bent if λ f (a, b) = 0 or ±2(t +1)/2 for every pair (a, b) with a = 0. Let t be odd. Below, we list some known almost bent functions from F2t to F2t . i 1) Gold function: f (x) = x 2 +1 , where gcd(i, t) = 1 [32], [46];

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2) Kasami function: f (x) = x 2 −2 +1 , where gcd(i, t) = 1 [37]; (t−1)/2 +3 3) Welch function: f (x) = x 2 [16]; l l/2 4) Niho function: Suppose t = 2l + 1. f (x) = x 2 +2 −1 , l +2(3l+1)/2 −1 2 if l is even; f (x) = x , if l is odd [33]. We can obtain near optimal signal sets from almost bent functions as follows. Proposition III.2 ([27, Theorem 4]). Let f be an almost bent function from F2t to F2t . Then, C f is a (22t + 2t , 2t ) signal 1 set with coherence 2(t−1)/2 . t Take m = 2 , n = 22t + 2t , the Levenstein bound (3) is   1 3n − m 2 − 2m 2t +1 + 1 = ≈ (t −1)/2 . t t (m + 2)(n − m) 2 (2 + 2) 2 2i

i

Therefore, the signal sets constructed from almost bent functions are near optimal. Consequently, we have the following theorem. Theorem III.2. Given an almost bent function from F2t to F2t , there exists a 2t × (22t + 2t ) matrix  with 1 μ() = 2(t−1)/2 . Hence, the specific construction of sensing matrices follows by choosing proper almost bent functions. Construction III.3 (matrices formed by near optimal signal (t−1)/2 +3 is an sets). Let q = 2t where t is odd. f 1 (x) = x 2 almost bent function from F2t to F2t . By Theorem III.2., we obtain an m × n sensing matrix  with m = 2t , n = 22t + 2t , 1 μ() = (t −1)/2 . 2 2i

i

IV. N EAR O RTHOGONAL S YSTEMS A RISING FROM MUB S AND AMUB S The notion of mutually unbiased bases (MUBs) emerged from the work of Schwinger [49]. Two orthonormal bases B and B  of the vector space Cd are called mutually unbiased if and only if 1 (5) d holds for all b ∈ B and all b ∈ B  , where ·|· is the usual inner product in Hilbert space Cd . For a quantum system which is prepared in a basis state from B  , no information can be retrieved when it is measured with respect to the basis B. Therefore, MUB plays an important role in quantum information theory and quantum cryptography [5], [6], [10]. Any collection of pairwise mutually unbiased bases of Cd has cardinality no more than d + 1 [56]. Let N(d) denote the maximum cardinality of any set containing pairwise mutually |b|b|2 =

|b|b|2 =

1 + o(1) d

or |b|b|2 =

1 + o(log d) , d

the concept of approximately mutually unbiased bases (AMUBs) is proposed in [39]. For any non-prime-power integer d, d + 1 AMUBs are obtained in [39] and [50]. The corresponding constructions of MUBs and AMUBs give rise to near orthogonal systems. We review these constructions in the following subsections. A. MUBs in Hilbert Space With Prime Power Dimension Let Fq be the finite field with q elements which has odd characteristic p. Let trq/ p be the absolute trace function from Fq to F p . The following proposition constructs q + 1 MUBs in Cq with q an odd prime power. Proposition IV.1 ( [38, Theorem 2]). Let Ba = {v a,b | b ∈ Fq } be a set of vectors given by   tr (ax 2 +bx) v a,b = q −1/2 ζ p q/ p . x∈Fq

If gcd(i, k) = 1, then f 2 (x) = x 2 +1 and f 3 (x) = x 2 −2 +1 are almost bent functions from F2t to F2t . We can obtain sensing matrices with the same parameters as those of the l l/2 above one. Suppose t = 2l + 1. Take f 4 (x) = x 2 +2 −1 if l is l (3l+1)/2 −1 if l is odd. Similarly, we even and take f 5 (x) = x 2 +2 obtain sensing matrices with the same parameters as those of the above one. i

unbiased bases of Cd . It is known that N(d) = d + 1 holds when d is a prime power [56]. When d is not a prime power, determining N(d) is an open problem. It is believed that no d + 1 MUBs exist in this case [50]. Therefore, by relaxing the condition (5) to

The standard basis and the sets Ba , with a ∈ Fq , form a set of q + 1 MUBs of Cq . Remark 2. The near orthogonal systems derived above are special cases of the near orthogonal systems constructed from optimal signal sets. More precisely, the signal set C f with planar function f (x) = x 2 from Fq to Fq , gives the same near orthogonal system. When q is an even prime power, i.e., q = 2t for some positive t, the construction of q + 1 MUBs is over the Galois rings. We give a brief introduction on Galois rings, see [54] for more details. Let Z4 denote the residue class ring of integers modulo 4. Denote by 2 the ideal generated by 2 in Z4 [x]. A monic polynomial h(x) ∈ Z4 [x] is called basic primitive if and only if its image in Z4 [x]/2 ∼ = Z2 [x] under the canonical map is a primitive polynomial in Z2 [x]. Let h(x) be a monic basic primitive polynomial of degree t. The ring GR(4, t) = Z4 [x]/h(x) is called the Galois ring of degree t over Z4 . The construction for GR(4, t) ensures it has 4t elements. The element ξ = x + h(x) is of order 2t − 1. Define the t Teichmüller system Tt = {0, 1, ξ, . . . , ξ 2 −2 }. Any element r ∈ GR(4, t) can be uniquely written in the form r = a + 2b, where a, b ∈ Tt . The automorphism σ : GR(4, t) → GR(4, t) defined by σ (a + 2b) = a 2 + 2b 2 is called the Frobenius automorphism. This map leaves the elements of the prime ring Z4 fixed. All automorphisms of GR(4, t) are of the form σ k for some integer k ≥ 0. The  −1tracek map T r : GR(4, t) → Z4 is defined by T r (x) = tk=0 σ (x).

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Proposition IV.2 ([38, Theorem 3]). Let GR(4, t) be a Galois ring with Teichmüller system Tt . For a ∈ Tt , denote by Ma = {v a,b | b ∈ Tt } the set of vectors given by   v a,b = 2−t /2 i T r((a+2b)x) , x∈Tt

√ where i = −1. The standard basis and the sets Ma , with t a ∈ Tt , form a set of 2t + 1 MUBs of C2 . Therefore, we have the following construction. Construction IV.1 (matrices formed by MUBs). Let q be a prime power. Let  be the concatenation of the q + 1 MUBs constructed in Proposition IV.1 or Proposition IV.2. Then  is an m × n sensing matrix with m = q, n = q 2 + q, 1 μ() = √ . q Remark 3. When q is an odd prime power, Construction III.1 generates the same sensing matrices with planar function f 1 (x) = x 2 . This construction also includes the case when q is an even prime power. Remark 4. The above construction is a generalization of that for chirp matrices [11]. In fact, for an odd prime p, a chirp matrix is the concatenation of p MUBs constructed in Proposition IV.1 without the standard basis. B. AMUBs in Hilbert Space With Non-Prime-Power Dimension In Hilbert space Cl where l is a non-prime-power, l + 1 AMUBs are constructed in [50], as an analogy of MUBs in prime power dimension. Let p be the smallest prime with p ≥ l. For each a = 1, . . . , l, we consider the basis Ba = {u a,1 , . . . , u a,l }, where l 1  2 u a,b = √ ζ pax ζlbx x=1 l for b = 1, . . . , l. Proposition IV.3 ( [50, Theorem 1]). The standard basis B0 = {u 0,1 , . . . , u 0,l } and l bases Ba for a = 1, . . . , l are orthogonal and satisfy   1  log l  12 1 |u a,i | u b, j | ≤ 2π − 2 + O , log l l where a, b = 0, . . . , l, a = b and 1 ≤ i, j ≤ l. Therefore, we have the following construction. Construction IV.2 (matrices formed by AMUBs). Let l be a non-prime-power. Let  be the concatenation of l + 1 AMUBs constructed in Proposition IV.3. Then  is an m × n sensing matrix  with m = l, n = l 2 + l,  log l  1  2 . μ() = O l

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Note that if l is a prime power, this construction is the same as Construction IV.1. The number of measurements in the sensing matrices generated from MUBs must be a prime power, while the sensing matrices generated from AMUBs provide non-prime-power measurements. To sum up, we list the sensing matrices obtained from near orthogonal systems in Table 1, where q is a prime power, l is a non-prime-power, d is a positive integer, μ is the coherence and k is the highest possible sparsity of noiseless signals which can be exactly recovered. The sparsity level is the order of k comparing with m. V. N UMERICAL S IMULATIONS In this section, matrices arising from near orthogonal systems are compared with several typical matrices. Among them, Gaussian and complex-valued Gaussian matrices are widely-used random matrices. Given an n ×n Discrete Fourier Transform (DFT) matrix, an m × n random Discrete Fourier Transform (RDFT) matrix is formed by randomly choosing m rows from the DFT matrix, with m < n. The Bernoulli matrix is a random matrix in which each entry takes 1 or −1 with equal probability. Matrices formed by BCH codes [1] and p-ary BCH codes [2] are deterministic sensing matrices. Our matrices outperform them in many experiments. In the simulations, we use k-sparse vectors whose k nonzero entries obey standard Gaussian distribution as testing signals. If the sensing matrices are complex-valued, we use k-sparse complex-valued testing signals in which the real and imaginary parts of their nonzero entries obey standard Gaussian distribution. For noisy recovery, a sparse signal x is contaminated with additive Gaussian noise e, where the SNR is 30 dB. We use OMP as the recovery algorithm and run 1000 experiments for each sparsity order. For a signal x, suppose x ∗ is the recovered signal from OMP. The reconstruction Signal to Noise Ratio (SNR) of x is defined to be SNR(x) = 20 · log10



x2  dB. x − x ∗ 2

Firstly, we consider sensing matrices arising from MWBE sequence sets. For a signal x, if SNR(x) is no less than 100 dB, we say the recovery of x is perfect. Fig. 1 presents the perfect recovery percentage of noiseless k-sparse 7381×1 signals with 170 ≤ k ≤ 340. By Construction II.1, an 820 × 7381 Singer matrix is obtained from the (7381, 820, 91) Singer difference set. The BCH matrix is formed by choosing the first 7381 columns from a 29-ary 840 × 24389 BCH matrix. The Singer matrix outperforms the complex-valued Gaussian and BCH matrices while it performs equally well as the RDFT matrix. It is worthy to note that the BCH matrix used in the simulation is formed by choosing a part of columns form the original BCH matrix. The comparison is unfair to the BCH matrix since it is designed to sample signals with much larger size. However, for a deterministic matrix, some columns have to be discarded to fit the size of signals. Hence, we think it is a reasonable compromise to choose a part of columns from the original matrix.

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TABLE I T HE m × n B INARY S ENSING M ATRICES O BTAINED FROM N EAR O RTHOGONAL S YSTEMS

Fig. 1. The perfect recovery percentage of noiseless 7381 × 1 signals. The 29-ary BCH matrix is with size 840×7381 and others are with size 820×7381.

Similarly, by Construction II.2, a 132 × 1573 McFarland matrix is obtained from the (1573, 132, 11) McFarland difference set. The BCH matrix used here is formed by choosing the first 1573 columns from a 5-ary 124 × 78125 BCH matrix. Fig. 2 presents the perfect recovery percentage of noiseless k-sparse 1573 × 1 signals with 24 ≤ k ≤ 70. The McFarland matrix outperforms the others. Fig. 3 presents the reconstruction SNR of noisy k-sparse 1870 × 1 signals with 100 ≤ k ≤ 220. By Construction II.4, a 357 × 1870 projective matrix is obtained from a (2, 5, 85) Steiner system. The BCH matrix is formed by choosing the

Fig. 2. The perfect recovery percentage of noiseless 1573 × 1 signals. The 5-ary BCH matrix is with size 124×1573 and others are with size 132×1573.

first 1870 columns from a 19-ary 360×6859 BCH matrix. The projective matrix outperforms the complex-valued Gaussian and BCH matrices while it performs equally well as the RDFT matrix. Fig. 4 presents the perfect recovery percentage of noiseless k-sparse 3276 × 1 signals with 110 ≤ k ≤ 250. By Construction II.5, a 525 × 3276 unital matrix is obtained from a (2, 6, 126) Steiner system. The BCH matrix is constructed by choosing the first 3276 columns from a 23-ary 528 × 12167 BCH matrix. The unital matrix outperforms the others. Secondly, we focus on sensing matrices arising from signal sets. By Construction III.2, a 729 × 532170 sensing matrix

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Fig. 3. The reconstruction SNR of 1870 × 1 noisy signals with 30 dB SNR. The 19-ary BCH matrix is with size 360 × 1870 and others are with size 357 × 1870.

Fig. 5. The perfect recovery percentage of noiseless 6561 × 1 signals. The 3-ary BCH matrix is with size 728×6561 and others are with size 729×6561.

Fig. 4. The perfect recovery percentage of noiseless 3276 × 1 signals. The 23-ary BCH matrix is with size 528×3276 and others are with size 525×3276.

Fig. 6. The reconstruction SNR of noisy 2187 × 1 signals with 30 dB SNR. The 3-ary BCH matrix is with size 242 × 2187 and others are with size 243 × 2187.

can be obtained via planar function f 1 (x) = x (3 +1)/2 from F36 to F36 . We generate the first 6561 columns to form our sensing matrix. We use a BCH matrix which is obtained by choosing the first 6561 columns from a 3-ary 728 × 19873 BCH matrix. Fig. 5 presents the perfect recovery percentage of noiseless k-sparse 6561 × 1 signals with 130 ≤ k ≤ 310. The matrix generated from optimal signal set outperforms the complex-valued Gaussian and BCH matrices while it performs equally well as the RDFT matrix. Similarly, a 243 × 59292 sensing matrix can be obtained via planar function f 2 (x) = x 10 − x 6 − x 2 from F35 to F35 . We generate the first 2187 columns to form our sensing matrix. We use a BCH matrix which is obtained by choosing the first 2187 columns from a 3-ary 242 × 59049 BCH matrix. Fig. 6 presents the reconstruction SNR of noisy k-sparse 2187 × 1 signals with 50 ≤ k ≤ 140. The matrix generated from optimal signal set outperforms the Gaussian and BCH matrices while it performs equally well as the RDFT matrix.

By Construction III.3, a 128 × 16512 sensing matrix can be 3 obtained via Gold function f3 (x) = x 2 +1 from F27 to F27 . We generate the first 1536 columns to form our sensing matrix. The BCH matrix used here is obtained by choosing the first 1536 columns from a 127×16384 BCH matrix. Fig. 7 presents the reconstruction SNR of noisy k-sparse 1536×1 signals with 10 ≤ k ≤ 60. The matrix generated from near optimal signal set outperforms the others. Similarly, a 512 × 262656 sensing (9−1)/2 +3 matrix can be obtained via Welch function f4 (x) = x 2 from F29 to F29 . We generate the first 5120 columns to form our sensing matrix. We use a BCH matrix which is obtained by choosing the first 5120 columns from a 511 × 262144 BCH matrix. Fig. 8 presents the perfect recovery percentage of noiseless k-sparse 5120 ×1 signals with 80 ≤ k ≤ 170. The matrix generated from near optimal signal set outperforms the others. At last, we consider sensing matrices arising from MUBs and AMUBs. By Construction IV.1, a 169 × 28730 sensing

5

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Fig. 7. The reconstruction SNR of noisy 1536 × 1 signals with 30 dB SNR. The BCH matrix is with size 127 × 1536 and others are with size 128 × 1536.

Fig. 9. The perfect recovery percentage of noiseless 2197 × 1 signals. The 13-ary BCH matrix is with size 168×2197 and others are with size 169×2197.

Fig. 8. The perfect recovery percentage of noiseless 5120 × 1 signals. The BCH matrix is with size 511 × 5120 and others are with size 512 × 5120.

Fig. 10. The reconstruction SNR of noisy 2916 × 1 signals with 30 dB SNR. The 7-ary BCH matrix is with size 342 × 2916 and others are with size 324 × 2916.

matrix can be obtained from 170 MUBs in C169 . We generate the first 2197 columns to form our sensing matrix. The BCH matrix used here is a 13-ary 168 × 2197 BCH matrix. Fig. 9 presents the perfect recovery percentage of noiseless k-sparse 2197×1 signals with 35 ≤ k ≤ 85. The matrix generated from MUBs outperforms the complex-valued Gaussian and RDFT matrices while it performs equally well as the BCH matrix. By Construction IV.2, a 324 × 105300 sensing matrix can be obtained from 325 AMUBs in C324 . We generate the first 2916 columns to form our sensing matrix. The BCH matrix used here is formed by choosing the first 2916 columns form a 7-ary 342 × 823543 BCH matrix. Fig. 10 presents the reconstruction SNR of noisy k-sparse 2916 × 1 signals with 100 ≤ k ≤ 190. The matrix generated from AMUBs outperforms the complex-valued Gaussian matrix and performs equally well as the RDFT matrix. Meanwhile, it is slightly weaker than the BCH matrix. However, since the construction based on AMUBs produces sensing matrices with any non-prime-power measurements, it is still valuable in practice.

VI. C ONCLUSION Deterministic construction of sensing matrices is a vital problem in CS. In this paper, we introduce the concept of near orthogonal system which essentially lies in the heart of many distinct applications. We apply these near orthogonal systems which root in different scenes to the area of CS and obtain several classes of deterministic sensing matrices. In particular, the constructions from the MWBE sequence sets and the optimal signal sets are the best possible deterministic ones based on coherence, since they achieve the Welch bound or the Levenstein bound. A lot of numerical simulations are conducted to show the good recovery performance of our matrices. They outperform several typical sensing matrices in many experiments. As an advantage of deterministic sensing matrices, specific fast recovery algorithms can be devised to accelerate the reconstruction process. In this sense, devising more efficient recovery algorithms than the generic ones for our matrices is a further research problem.

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ACKNOWLEDGMENT The authors express their gratitude to the two anonymous reviewers for their detailed and constructive comments which greatly improve the presentation of this paper, and to Professor Yi Ma, the associate editor, for his excellent editorial job. R EFERENCES [1] A. Amini and F. Marvasti, “Deterministic construction of binary, bipolar and ternary compressed sensing matrices,” IEEE Trans. Inf. Theory, vol. 57, no. 4, pp. 2360–2370, Apr. 2011. [2] A. Amini, V. Montazerhodjat, and F. Marvasti, “Matrices with small coherence using p-ary block codes,” IEEE Trans. Signal Process., vol. 60, no. 1, pp. 172–181, Jan. 2012. [3] L. Applebaum, S. Howard, S. Searle, and R. Calderbank, “Chirp sensing codes: Deterministic compressed sensing measurements for fast recovery,” Appl. Comput. Harmon. Anal., vol. 26, no. 2, pp. 283–290, 2009. [4] R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Construct. Approx., vol. 28, no. 3, pp. 253–263, 2008. [5] H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A, vol. 61, no. 6, pp. 062308-1–062308-6, 2000. [6] C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proc. IEEE Int. Conf. Comput., Syst. Signal Process., Dec. 1984, pp. 175–179. [7] R. Berinde, A. Gilbert, P. Indyk, H. Karloff, and M. Strauss, “Combining geometry and combinatorics: A unified approach to sparse signal recovery,” in Proc. 46th Annu. Allerton Conf. Commun., Control, Comput., Sep. 2008, pp. 798–805. [8] T. Blumensath and M. E. Davies, “Iterative hard thresholding for compressed sensing,” Appl. Comput. Harmon. Anal., vol. 27, no. 3, pp. 265–274, 2009. [9] J. Bourgain, S. Dilworth, K. Ford, S. Konyagin, and D. Kutzarova, “Explicit constructions of RIP matrices and related problems,” Duke Math. J., vol. 159, no. 1, pp. 145–185, 2011. [10] D. Bruss, “Optimal eavesdropping in quantum cryptography with six states,” Phys. Rev. Lett., vol. 81, no. 14, pp. 3018–3021, 1998. [11] R. Calderbank, S. Howard, and S. Jafarpour, “Construction of a large class of deterministic sensing matrices that satisfy a statistical isometry property,” IEEE Trans. Inf. Theory, vol. 4, no. 2, pp. 358–374, Apr. 2010. [12] E. Candès, “The restricted isometry property and its implications for compressed sensing,” Comp. Rendus Math. Acad. Sci. Paris, vol. 346, nos. 9–10, pp. 589–592, 2008. [13] E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006. [14] E. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4203–4215, Dec. 2005. [15] E. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory, vol. 52, no. 12, pp. 5406–5425, Dec. 2006. [16] A. Canteaut, P. Charpin, and H. Dobbertin, “Binary m-sequences with three-valued crosscorrelation: A proof of Welch’s conjecture,” IEEE Trans. Inf. Theory, vol. 46, no. 1, pp. 4–8, Jan. 2000. [17] C. Carlet and C. Ding, “Highly nonlinear mappings,” J. Complex., vol. 20, nos. 2–3, pp. 205–244, 2004. [18] A. Cohen, W. Dahmen, and R. DeVore, “Compressed sensing and best k-term approximation,” J. Amer. Math. Soc., vol. 22, no. 1, pp. 211–231, 2009. [19] C. J. Colbourn, D. Horsley, and C. McLean, “Compressive sensing matrices and hash families,” IEEE Trans. Commun., vol. 59, no. 7, pp. 1840–1845, Jul. 2011. [20] C. J. Colbourn and R. Mathon, “Steiner systems,” in The CRC Handbook of Combinatorial Designs, 2nd ed. Boca Raton, FL, USA: CRC Press, 2007, pp. 58–71. [21] R. Coulter and R. Matthews, “Planar functions and planes of LenzBarlotti class II,” Des. Codes Cryptograph., vol. 10, no. 2, pp. 167–184, 1997. [22] W. Dai and O. Milenkovic, “Subspace pursuit for compressive sensing signal reconstruction,” IEEE Trans. Inf. Theory, vol. 55, no. 5, pp. 2230–2249, May 2009.

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Shuxing Li is currently a Ph.D. student at Zhejiang University, Hangzhou, Zhejiang, P. R. China. His research interests include combinatorial design theory, coding theory, algebraic combinatorics, and their interactions.

Gennian Ge received the M.S. and Ph.D. degrees in mathematics from Suzhou University, Suzhou, Jiangsu, P. R. China, in 1993 and 1996, respectively. After that, he became a member of Suzhou University. He was a postdoctoral fellow in the Department of Computer Science at Concordia University, Montreal, QC, Canada, from September 2001 to August 2002, and a visiting assistant professor in the Department of Computer Science at the University of Vermont, Burlington, Vermont, USA, from September 2002 to February 2004. He was a full professor in the Department of Mathematics at Zhejiang University, Hangzhou, Zhejiang, P. R. China, from March 2004 to February 2013. Currently, he is a full professor in the School of Mathematical Sciences at Capital Normal University, Beijing, P. R. China. His research interests include the constructions of combinatorial designs and their applications to codes and crypts. Dr. Ge is on the Editorial Board of Journal of Combinatorial Designs, SCIENCE CHINA Mathematics, Applied Mathematics− A Journal of Chinese Universities. He received the 2006 Hall Medal from the Institute of Combinatorics and its Applications.