COHERENCE OF SENSING MATRICES COMING FROM ALGEBRAIC-GEOMETRIC CODES

. Compressed sensing is a technique which is to used to reconstruct a sparse signal given few measurements of the signal. One of the main problems in compressed sensing is the deterministic construction of the sensing matrix. Li et al. introduced a new deterministic construction via algebraic-geometric codes (AG codes) and gave an upper bound for the coherence of the sensing matrices coming from AG codes. In this paper, we give the exact value of the coherence of the sensing matrices coming from AG codes in terms of the minimum distance of AG codes and deduce the upper bound given by Li et al. We also give formulas for the coherence of the sensing matrices coming from Hermitian two-point codes.


Introduction
Consider a discrete-time signal x ∈ R n . Let y ∈ R m (m < n) be the measurement vector with y = Φx, where the m × n matrix Φ consists of m linear projections of x. The m × n matrix Φ is called a sensing matrix. Compressed sensing is a technique which is used to reconstruct the original signal x from the measurement vector y. Since m < n, y = Φx is an underdetermined linear system. Nevertheless, Candès et al. [3] and Donoho [6] show that a sparse signal can be reconstructed from few measurements by solving an optimization problem. In [4], Candès and Tao provide a criterion for sensing matrices to guarantee a unique reconstruction of sparse signals which is named restricted isometry property (RIP). A signal x is said to be k-sparse if x has at most k nonzero entries. An m × n matrix Φ is said to satisfy the RIP of order k if there is a constant 0 ≤ δ k < 1 such that holds for all k-sparse signals x ∈ R n . The smallest nonnegative number δ k is called the restricted isometry constant (RIC). If the sensing matrix Φ satisfies the RIP, then a unique reconstruction of the sparse signal is possible. In [2], Bourgain et al.
give a criterion for a matrix to satisfy the RIP using the notion of coherence. The coherence of a matrix Φ with columns a 1 , . . . , a n is We now give the criterion that was given in [2].
. If the coherence of the matrix Φ is µ, then the matrix Φ satisfies the RIP of order k with constant δ = (k − 1)µ.
Although a random matrix satisfies the RIP of order k with high probability, there is no efficient algorithm that verifies whether a random matrix satisfies the RIP or not. Also, storing the random matrix requires a lot of storage space. To overcome these drawbacks, deterministic construction is inevitable. In [5], DeVore introduce a deterministic construction of sensing matrices using polynomials over finite fields. In [12], Li et al. generalize DeVore's construction to algebraic curves over finite fields. They show that where P is a set of rational points on an algebraic curve X and G is a divisor of X such that supp(G) ∩ P = ∅, and by computing the upper bound of coherence for appropriate curves obtain better sensing matrices than DeVores sensing matrices.
In this paper, we prove that where d(C) is the minimum distance of the AG code C, that is, we give the exact value of the coherence in terms of the minimum distance of the AG code and deduce the upper bound given in [12]. Finding the minimum distance of a code is a classical problem in coding theory and extensive research has been done to find the minimum distance of AG codes. In particular, the minimum distance of codes on Hermitian curve is given in [1,7,8,9,10,11,14]. Based on the formulas given in [14], we give formulas for the coherence of the sensing matrices coming from Hermitian two-point codes.
The organization of the paper is as follows. In Section 2, we give some background on AG codes that will be needed in the sequel sections. In Section 3, we review the construction of sensing matrices using algebraic curves over finite fields and give the exact value of the coherence of the sensing matrices. From the exact value of the coherence, we deduce the upper bound of the coherence given by Li et al. [12]. In Section 4, we give formulas for the coherence of the sensing matrices coming from Hermitian two-point codes.

Background on AG codes
We give the definition of the AG code and some properties. Let X/F be an algebraic curve (absolutely irreducible, smooth, projective) of genus g over a finite field F. Let F(X) be the function field of X/F and let Ω(X) be the module of rational differentials of X/F. Given a divisor A on X defined over F, let L(A) denote the vector space over F of functions f ∈ F(X)\{0} with (f ) + A ≥ 0 together with the zero function. The dimension of L(A) is denoted by l(A). Let Ω(A) denote the vector space over F of differentials ω ∈ Ω(X)\{0} with (ω) ≥ A together with the zero differential. Let K represent the canonical divisor class. For n distinct rational points P 1 , . . . , P n on X and for disjoint divisors D = P 1 + · · · + P n and G, the algebraic-geometric codes C L (D, G) and C Ω (D, G) are defined as the images of the maps The next theorem gives the parameters of C L (D, G).

Main result
We review the construction of sensing matrices using algebraic curves over finite fields and find the values of the coherence of the sensing matrices coming from algebraic curves. Let q be a prime power and let X be an algebraic curve over a finite field F q . Let P be a set of rational points on X and let G be a divisor of X such that deg G < |P| and supp(G) ∩ P = ∅. For each f ∈ L(G), define the column vector v f whose entries are indexed by the pairs in P × F q . Denote the entry indexed by (P, a) ∈ P × F q as f P,a . Then we take f P,a as follows : The column vectors {v f | f ∈ L(G)} form an m × n matrix Φ 0 , where m = |P| × q and n = q l(G) . Now we state the theorem given by Li et al.
The following theorem gives the exact value of the coherence in terms of the minimum distance of AG codes.
Proof. Let f, g ∈ L(G) be two distinct functions. Then 1 √ . Hence Note that Theorem 3.1 can be deduced from Theorem 3.2 and Theorem 2.2.

Coherence of sensing matrix coming from Hermitian curve
Matthews show that for codes on Hermitian curve, there are two-point codes that have better parameters than any comparable one-point codes [13]. For this reason, we will consider Hermitian two-point codes. Let X be a Hermitian curve defined by y q + y = x q+1 over F q 2 . Then X has q 3 + 1 rational points and the genus is q(q − 1)/2. We denote by P 0 the point (0, 0) and by P ∞ the point at infinity. The canonical divisor K of a Hermitian curve is K = (q − 2)H, where H ∼ (q+1)P ∞ ∼ (q+1)P 0 . The action of the automorphism group of the Hermitian curve on the set of its rational points is doubly transitive, and therefore without loss of generality it can be assumed that the support of a two-point divisor consists of P 0 and P ∞ . Let G = aP ∞ + bP 0 and D = P 1 + · · · + P n be divisors of X, where supp(G) ∩ supp(D) = ∅ and P 1 , . . . , P n are pairwise distinct. Then C L (D, G) and C Ω (D, G) are Hermitian two-point codes. We state the formulas for the minimum distance of Hermitian two-point codes given in [14]. The formulas give the minimum distance of the Hermitian two-point codes for all ranges of G. The ranges are divided into two parts as follows: Suppose that G satisfies either except for the case when (a 0 + b 0 < a 1 , b 1 and a 1 = q, b 1 = q), for which We give the formulas for the coherence of the sensing matrices coming from the Hermitian two-point code.
Suppose that G satisfies either (1) deg G > deg K + q or (2) deg K ≤ deg G ≤ deg K + q and G sP ∞ and G tP 0 for all s, t ∈ Z, then the coherence of the sensing matrix Φ coming from C L (D, G) ⊥ is except for the case when (a 0 + b 0 < a 1 , b 1 and a 1 = q, b 1 = q), for which Suppose that G satisfies either for some s, t ∈ Z, then the coherence of the sensing matrix Φ coming from C L (D, G) ⊥ is We give a toy example in the following. In [12], Li et al. construct sensing matrices from Hermitian one-point codes and improve the upper bound of the coherence of the DeVore's sensing matrices. They compute the upper bound of the coherence using Theorem 3.1. In the next example, we use Theorem 3.2 to compute the coherence of the matrices constructed in [12] and compare the values of coherence obtained by both theorems. Theorem 3.2 states that we can improve the upper bound of the coherence of the sensing matrices coming from AG codes by improving the lower bound for the minimum distance of the AG codes. In [16], Xing and Chen show that by choosing specific divisors for the Hermitian codes, it is possible to improve the minimum distance of the Hermitian one-point codes. In Table 1, we give the parameters for the Hermitian one-point codes used in Example 4.6 and the parameters for the Improved Hermitian codes given by Xing  the Improved Hermitian codes, we improve the bound of the coherence given by Li et al in [12]. In Table 2, we give the upper bound or the exact value of the coherence. For comparison we use the matrix of the same size. Also we denote the theorems used to compute the upper bound of the coherence.
Size of the matrix Hermitian one-point codes Improved Hermitian codes Theorem 3.  Table 2. Comparison of the coherence of the matrices

Conclusion
The deterministic constructions of the sensing matrices are far from being optimal and to close this gap we need to find sensing matrices with smaller coherence. In this paper, we give the exact value of the coherence of the sensing matrices coming from AG codes in terms of the minimum distance of AG codes. Thus the value of the coherence can be computed by computing the minimum distance of the AG code. To get a small value of the coherence we need to find an AG code with large minimum distance which is a central problem in coding theory.