Designs in finite metric spaces: a probabilistic approach
aa r X i v : . [ m a t h . C O ] F e b Designs in finite metric spaces: aprobabilistic approach ∗ Minjia Shi , Olivier Rioul , Patrick Sol´e , Key Laboratory of Intelligent Computing & Signal Processing of Ministry of Education,School of Mathematical Sciences, Anhui University,Hefei 230601, China Telecom ParisTech, Palaiseau, France I2M,(Aix-Marseille Univ., Centrale Marseille, CNRS), Marseille, France Corresponding Author
Abstract
A finite metric space is called here distance degree regular if its distancedegree sequence is the same for every vertex. A notion of designs in such spacesis introduced that generalizes that of designs in Q -polynomial distance-regulargraphs. An approximation of their cumulative distribution function, basedon the notion of Christoffel function in approximation theory is given. As anapplication we derive limit laws on the weight distributions of binary orthogonalarrays of strength going to infinity. An analogous result for combinatorialdesigns of strength going to infinity is given. Keywords: distance-regular graphs, designs, orthogonal polynomials, Christoffelfunction ∗ This research is supported by National Natural Science Foundation of China (12071001,61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province(1808085J20), the Academic Fund for Outstanding Talents in Universities (gxbjZD03). MS Math Sc. Cl. (2010):
Primary 05E35, Secondary O5E20, 05E24
In a celebrated paper [21] Sidelnikov proved that the weight distribution of binarycodes of dual distance d ⊥ going to infinity with the length is close deviates from thenormal law up to a term in inverse square root of the dual distance [19, Chap. 9, § d ⊥ − Q -polynomial association schemes [2, 4, 5, 19]. The name designscomes from the Johnson scheme where this notion coincides with that of classicalcombinatorial designs [5]. Later a similar connection was found between designs inassociation schemes and designs in lattices [6, 23]. These kinds of generalized designsare popular now in view of the applications in random network coding [9]. In view ofthis deep connection it is natural to seek to extend Sidelnikov’s theorem to designsin other Q -polynomial association schemes than the Hamming scheme. This is a vastresearch program which might take several years to accomplish.In the present paper our contribution is twofold.Firstly, we develop a theory of designs in finite metric spaces that replaces theconcept of designs in Q -polynomial association schemes, when the considered met-ric space does not afford that structure. We observe that, in contrast with Delsartedefinition of a design in a Q -polynomial association scheme, our definition (Defini-tion 1 below), has an immediate combinatorial meaning in terms of distribution ofdistances. To wit, the combinatorial meaning of designs in certain Q -polynomialassociation schemes was only derived in [23] in 1986, thirteen years after Delsarteintroduced designs in Q -polynomial association schemes in [5]. In particular, the ex-ample of permutations with distance the Hamming metric cannot be handled in thecontext of Q -polynomial association schemes, but can be treated within our frame-work. This space had been studied extensively in the context of permutations codes[3, 25]. The notion of t -design in that space is related to t -transitive permutationgroups (Theorem 8).Secondly, we use the technique of Chebyshev-Markov-Stieltjes inequalities in con-junction with orthogonal polynomials to control the difference between the cumulativedistribution function of weights in designs with that of weights in the whole space.2hile this technique has been applied by Bannai to the weight distribution of spher-ical designs [1], it has not appeared in the literature of algebraic combinatorics sofar. The bounding quantity in that setting is the Christoffel function, the inverseof the confluent Christoffel-Darboux kernel. While it is easy to make this quantityexplicit in low strength cases, it is difficult to find asymptotic bounds. In the presentpaper, we will use the bounds of Krasikov on the Christoffel-Darboux kernel of bi-nary Krawtchouk polynomials [15, 17] to derive an alternative proof of the SidelnikovTheorem. We will give a proof of an analogous result for combinatorial designs byusing the limiting behavior of Hahn polynomials.The material is organized as follows. The next section contains background mate-rial on metric spaces, distance-regular graphs and Q -polynomial association schemes.Section 3 introduces the definitions that are essential to our approach. Section 4 con-tains the main equivalences of our notion of designs in finite metric spaces. Section5 develops the bounds on the cumulative distribution functions of designs. Section6 contains some asymptotic results. Section 7 recapitulates the results obtained andgives some significant open problems. Throughout the paper we write X for a finite set equipped with a metric d, thatis to say a map X × X → N verifying the following three axioms1. ∀ x, y ∈ X, d ( x, y ) = d ( y, x )2. ∀ x, y ∈ X, d ( x, y ) = 0 iff x = y ∀ x, y, z ∈ X, d ( x, y ) ≤ d ( x, z ) + d ( z, y ) . In particular if X is the vertex set of a graph the shortest path distance on X isa metric.The diameter of a finite metric space is the largest value the distance may take.A finite metric space is Distance Degree Regular (DDR) if for every integer i less thanthe diameter the number |{ y ∈ X | d ( x, y ) = i }| is a constant v i that does not dependof the choice of x ∈ X. xample: Consider the symmetric group on n letters S n with metric d S ( σ, θ ) = n − F ( σθ − ) , where F ( ν ) denotes the number of fixed points of ν. The space ( S n , d S ) is a DDR met-ric space. If D m = m ! P mj =1 ( − j j ! denotes the number of fixed-point-free permutationsof S m , ( the so-called d´erangement number), then v i = (cid:18) ni (cid:19) D i . Note that d S is not a shortest path distance since d S ( σ, θ ) = 1 is impossible. Codes in( S n , d S ) were studied in [25] by using the conjugacy scheme of the group S n . However,in contrast with the next two subsections, this scheme is neither induced by a graphnor Q -polynomial. All graphs in this article are finite, undirected, connected, without multiple edges.The neighborhood Γ( x ) is the set of vertices connected to x . The degree of a vertex x is the size of Γ( x ). A graph is regular if every vertex has the same degree. The i -neighborhood Γ i ( x ) is the set of vertices at geodetic distance i to x . The diameter of the graph, denoted by d is the maximum i such that for some vertex x the setΓ i ( x ) is nonempty. A graph is distance degree regular (DDR for short) if all graphsΓ i , for i = 1 , . . . , d are regular. A graph is distance regular (DR for short) if for everytwo vertices u and v at distance i from each other the values b i = | Γ i +1 ( u ) ∩ Γ( v ) | , c i = | Γ i − ( u ) ∩ Γ( v ) | depend only on i and do not depend on the choice of u and v . In this case, the graphs Γ i are regular of degree v i and we will refer to the v i sas the valencies of Γ; the sequence { b , . . . , b diam − ; c , . . . , c diam } is usually called the intersection array of Γ . Thus every DR graph is DDR but not conversely.
Examples
For background material on the following two examples we refer to[4, 5, 19].1. The
Hamming graph H ( n, q ) is a graph on F nq two vertices being connected ifthey differ in exactly one coordinate. This graph is DR with valencies v i = (cid:18) ni (cid:19) ( q − i .
4. The
Johnson graph J ( ν, d ) is a graph on the subsets of cardinality d of a set ofcardinality ν. (Assume 2 d < ν ). Two subsets are connected iff they intersect inexactly d − v i = (cid:18) di (cid:19)(cid:18) ν − di (cid:19) . Note that J ( ν, d ) can be embedded in H ( ν,
2) by identifying subsets and char-acteristic vectors. Q -polynomial association schemes An association scheme on a set X with s classes is a partition of the cartesianproduct X × X = ∪ si =0 R i with the following properties1. R = { ( x, x ) | x ∈ X }
2. ( x, y ) ∈ R k , iff ( y, x ) ∈ R k ,
3. if ( x, y ) ∈ R k , the number of z ∈ X such that ( x, z ) ∈ R i , and ( z, y ) ∈ R j , is aninteger p kij that depends on i, j, k but not on the special choice of x and y A consequence of axiom 3 is that each graph R i is regular of degree v i , say. It canbe shown that the adjacency matrices D k of the relations R k span a commutativealgebra over the complex with idempotents J j [19, Chap. 21]. Let µ j =rank( J j ) . The first eigenvalues p k ( i ) of the scheme are defined by D k J i = p k ( i ) J i . Considering thematrix P = ( p k ( i )) and writing P Q = | X | I, with I an identity matrix defines the second eigenvalues q k ( i ) of the scheme by the relation Q ik = q k ( i ) . A scheme is said tobe Q -polynomial if there are numbers z , z , . . . , z s such that q k ( i ) = Φ k ( z i ) for somepolynomials Φ k of degree k. In view of the orthogonality relation [19, Chap 21, (17)] s X k =0 v k q i ( k ) q j ( k ) = | X | µ i δ ij , we see that the Φ i ( z ) √ µ i form a system of orthonormal polynomials for the scalar product h f, g i = s X k =0 v k | X | f ( z k ) g ( z k ) . Examples:
In both Hamming and Johnson schemes we have z k = k.
5. If Γ = H ( n, q ) then Φ k ( z ) = K k ( z ) √ v k , where K k is the Krawtchouk polynomial ofdegree k given by the generating function n X k =0 K k ( x ) z k = (1 + ( q − x ) n − x (1 − z ) x .
2. If Γ = J ( ν, n ) then Φ k ( z ) = H k ( z ) √ v k , where H k is the Hahn polynomial of degree k given, as per [8, (19) p.2481], by the formula H k ( z ) = m k k X j =0 ( − j (cid:0) kj (cid:1)(cid:0) ν +1 − kj (cid:1)(cid:0) nj (cid:1)(cid:0) ν − nj (cid:1) (cid:18) zj (cid:19) , where m k = (cid:0) νk (cid:1) − (cid:0) νk − (cid:1) . For any integer N > , denote by [0 .. N ] the set of integers in the range [0 , N ] . A finite metric space (
X, d ) is distance degree regular (DDR) if its distance degreesequence is the same for every point. Assume (
X, d ) to be of diameter n. In that case(
X, d ) is DDR iff for each 0 ≤ i ≤ n the graph Γ i = ( X, E i ) which connects vertices atdistance i in ( X, d ) is regular of degree v i . Thus E = { ( x, x ) | x ∈ X } is the diagonalof X . Note that the E i ’s form a partition of X . If D is any non void subset of X we define its frequencies as ∀ i ∈ [0 ..n ] , f i = | D ∩ E i || D | . Thus f = | D | , and n P i =0 f i = 1 . Note also that if D = X, then f i = v i | X | . Considerthe random variable a D defined on D with values in [0 ..n ] which affects to anequiprobably chosen ( x, y ) ∈ D the only i such that ( x, y ) ∈ E i . Thus the frequencies f i = P rob ( a D = i ) . Denote by E () mathematical expectation. Thus E ( a iD ) = n X j =0 f j j i , E ( a iX ) = n X j =0 v j v j i . Definition 1.
The set D ⊆ X is a t -design for some integer t if E ( a iD ) = E ( a iX )6or i = 1 , . . . , t. (Note that trivially E (1) = 1 so that we do not consider i = 0 . ) Thus, distancesin t -designs are very regularly distributed. For a 2-design, for instance, the averageand variance of the distance coincide with that of the whole space. We will see in thenext section that in the case of Hamming and Johnson graphs, we obtain classicalcombinatorial objects: block designs, orthogonal arrays. Definition 2.
We define a scalar product on R [ x ] attached to D by the relation h f, g i D = n X i =0 f i f ( i ) g ( i ) . Thus, in the special case of D = X we have h f, g i X = 1 | X | n X i =0 v i f ( i ) g ( i ) . We shall say that a sequence Φ i ( x ) of polynomials of degree i is orthonormal of size N + 1 if it satisfies ∀ i, j ∈ [0 ..N ] , h Φ i , Φ j i X = δ ij , where N ≤ n, the letter δ denotes the Kronecker symbol. That sequence is uniquelydefined if we assume the leading coefficient of all Φ i ( x ) for i = 0 , , . . . , N to bepositive.For a given DDR metric space ( X, d ) , we shall denote by N ( X ) the largest possiblesuch N. For instance if X is an n -class P - and Q -polynomial association scheme, itis well-known that N ( X ) = n. This fact is extended to DDR graphs in the nextProposition.
Proposition 3.
If none of the v i ’s are zero, then h , i X admits an orthonormal systemof polynomials of size n + 1 . In particular, the metric space of a DDR graph admitsan orthonormal system of polynomials of size n + 1 . Proof.
By Lagrange interpolation we see that the functions 1 , x, . . . , x n are linearlyindependent on [0 ..n ] . The sequence of the Φ i ’s for i = 0 , , . . . , n is then constructedby the usual Gram-Schmidt orthogonalization process. Note that this is possiblebecause the bilinear form h , i X is then nondegenerate: h f, f i X = 0 ⇒ f = 0 . Byproperties of the shortest path distance, the property of non vanishing of the v i ’sholds in particular for the metric space of a DDR graph.7 efinition 4. For a given D ⊆ X the dual frequencies are defined for i = 0 , , . . . , N ( X )as b f i = n X k =0 Φ i ( k ) f k . Definition 5.
For a given D ⊆ X the cumulative distribution function (c.d.f.) isdefined as F D ( x ) = P rob ( a D ≤ x ) = X i ≤ x f i . Examples:
1. If D is a linear code of H ( n, q ) , with weight distribution A i = |{ x ∈ D | w H ( x ) = i }| , then F D ( x ) = P i ≤ x A i | D | .
2. If D is a set of points in J ( ν, k ) , with Hamming distance distribution B i in H ( ν, , then F D ( x ) = P i ≤ x B i | D | . First, we give a characterization of t -designs in terms of dual frequencies. Proposition 6.
Let t be an integer ∈ [1 ..N ( X )] . The set D ⊆ X is a t -design iff b f i = 0 for i = 1 , . . . , t. Proof.
Note first that E ( a iD ) = h x i , i D , E ( a iX ) = h x i , i X . Moving the basis of R [ x ] from the Φ i ’s to the basis of monomials we see that D is a t -design if and only if for i = 1 , , . . . , t, we have h Φ i , i D = h Φ i , i X . Now, by definition, the dual frequency b f i = h Φ i , i D . By orthogonality of the Φ i ’sfor the scalar product h ., . i X , we see that h Φ i , i X = 0 for i = 1 , . . . , t. Thus, thecondition b f i = 0 for i = 1 , , . . . , t, is equivalent to the fact that D is a t -design.8ext, we connect the notion of designs in Q -polynomial association schemes withour notion of designs in metric spaces. Theorem 7. If ( X, d ) is the metric space induced by a Q -polynomial DR graph Γ , with z k = k for k = 0 , , . . . , n, then a t -design in ( X, d ) is exactly a t -design in theunderlying association scheme of Γ . Proof.
In that situation the frequencies are proportional to the inner distribution (see[4, p.54]) of D in the scheme of the graph, and the dual frequencies are proportionalto the dual inner distribution since the second eigenvalues of the scheme, by the Q -polynomiality condition, are orthogonal polynomials w.r.t. the distribution v i | X | . Theresult follows.
Examples:
The following two examples of interpretation of t -designs as classicalcombinatorial objects were observed first in [5] and can be read about in [19, chap.21].1. If Γ is the Hamming graph H ( n, q ) then a t -design is an orthogonal array ofstrength t. That means that every row induced by a t -uple columns of D seesthe q t possible values a constant number of times.2. If Γ is the Johnson graph J ( ν, n ) then a t -design D is a combinatorial design ofstrength t. This means the following. Consider D as a collection of subsets ofsize n , traditionally called blocks. That means that every t -uple of elements ofthe groundset is contained in the same number η of blocks. One says that D isa t − ( ν, n, η ) design.Now, we give an example of t -design in a metric space that is not a DR graph, oreven a DDR graph. Theorem 8. If D ⊆ S n is a t -transitive permutation group then it is a t -design in ( S n , d S ) . Proof.
The moments of order i ≤ t of the number of fixed points of the permutationsin D coincide with those of a Poisson law of parameter one. This is a result ofFrobenius (1904). A modern exposition is in [14, Chap. 5.5]. The result follows bythe definition of d S . Distribution functions
In this section we show that the distribution function of designs are close to thatof the whole space. The proof of the following result follows the philosophy of [1].
Theorem 9.
Let D be a t -design in ( X, d ) , with t ≤ N ( X ) . Put κ = ⌊ t ⌋ . Denote by λ ( x ) the Christoffel function given by λ ( x ) = ( κ P i =0 Φ i ( x ) ) − . Then we have the bound | F D ( x ) − F X ( x ) | ≤ λ ( x ) . Proof.
By Definition 1, we have h x i , i X = h x i , i D for i = 0 , , , . . . , t. The orthonormal polynomials for h , i X exist for degrees ≤ t by the hypothesis t ≤ N ( X ) . A coincidence of moments up to order t entails a coincidence of orthonormalpolynomials up to degree κ by Chebyshev determinant for orthonormal polynomials[13, Lemma 2.1] (see also [24, (2.2.6), p. 27]). By the same formula, the orthonormalpolynomials for h , i D are well-defined for degrees ≤ κ, since the orthonormal polyno-mials up to degree κ attached to X exist. By [13, Th. 4.1] or [21, Th. 7.2] we havethe Markov-Stieltjes inequalities X x i 7] given by K = 1 , K ( x ) = n − x, K ( x ) = 2 x − nx + (cid:18) n (cid:19) . We are seeking a lower bound for1 + K ( x ) n + K ( x ) (cid:0) n (cid:1) , when x ∈ [0 , n ] . Making the change of variable y = n − x ∈ [ − n, n ] , we obtain K ( x ) = y − n , and, therefore 1 /λ = 1 + y n + ( y − n ) n ( n − 1) = 3 n − n + ( y − − n ( n − , an increasing function of y that takes its minimum over [0 , n ] at y = 0 . Example: If D is the extended Hamming code of length n = 16 , dual distance8 , the weight distribution is, in Magma notation [18], equal to[ < , >, < , >, < , >, < , >, < , >, < , >, < , > ] . For x = 8 , we get F D ( x ) = ≈ . , and F X ( x ) = P j =0 ( nj ) ≈ . . The difference is ≈ . < × ≈ . . We give three bounds that are not uniform in x. First, for orthogonal arrays. Corollary 11. If D is a q -ary orthogonal array of strength at least two, then its c.d.f.is as close to that of the binomial distribution as | F D ( x ) − F X ( x ) | < nn + ( n ( q − − qx ) . Proof. Immediate from the data of the first two Krawtchouk polynomials: K = 1 , K ( x ) = n ( q − − qx. xample: If D is a binary Simplex code of length n = 2 m − , there is a uniquenonzero weight, namely n +12 that appears | D | − n times. If we compute thebound for x = n +12 , its right hand side is nn +1 , which is also the value of F D ( x ) while F X ( x ) > . . Next, we consider combinatorial designs. Corollary 12. If D is a − ( ν, n, λ ) design, then its c.d.f. is as close to that of thehypergeometric distribution as | F D ( x ) − F X ( x ) | < ( ν − n ) ( ν − n ) + n ( ν − . Proof. From the data of the first two Hahn polynomials: H = 1 , H ( x ) = ( ν − − νxn ( ν − n ) ) , we obtain 1 /λ = 1 + H ( x ) n ( ν − n ) = 1 + ( ν − ( n − nν + νx ) n ( ν − n ) , a monotonic function of x. Eventually, we consider permutation groups. Exceptionally, we do not considerthe distance but the codistance n − d s . Corollary 13. If D is a -transitive permutation group on n letters, then the c.d.f.of its fixed points G D ( x ) is as close to that of the Poisson law of parameter one P ( x ) = P ≤ i ≤ x i ! as | G D ( x ) − P ( x ) | < nn + (1 − x ) . Proof. Immediate from the data of the first two Charlier polynomials C = 1 , C ( x ) =1 − x, obtained from the generating series e t (1 − t ) x = ∞ X n =0 C n ( x ) t n n !of [20, (1.12.11)]. 12 Asymptotic results In this section we give an alternative proof of a result of Sidelnikov on the weightenumerator of long codes [21]. We prepare for the proof by a form of the CentralLimit Theorem for the binomial distribution. Denote by Ψ( x ) = √ π R x −∞ exp( − t ) dt the cumulative distribution function of the centered normal law of variance unity. Let B n ( x ) = P i ≤ x ( ni ) n denote the cumulative distribution function of the binomial law(sum of n Bernoulli trials). Theorem 14. For some absolute constant C > , we have | B n ( x ) − Ψ( x ) | ≤ C √ n . Proof. Immediate by Berry-Essen theorem [11].Recall the binary entropy function [19] defined as H ( x ) = − x log x − (1 − x ) log (1 − x ) . A tedious but straightforward consequence of Stirling formula is (cid:18) NαN (cid:19) ∼ NH ( α ) p πα (1 − α ) (3)for N → ∞ and 0 < α < . See (1) in [12]. Theorem 15. Let n → ∞ , and let k be an integer such that k ∼ θn, with < θ < a real constant. Assume x = n + O ( √ n ) . Then any binary orthogonal array D with n columns, of strength ≥ k + 1 , satisfies | F D ( x ) − B n ( x ) | = O ( 1 √ n ) . Proof. (sketch) We use [15, Th. 1.1] or [17, Lemma 4] to claim the lower bound1 λ ( x ) = Ω (cid:0) ( k +1) (cid:0) nk (cid:1) G k ( x ) (cid:1) , G k ( x ) = 2 k (2 x + p k − n )( n − k ) Γ( x )Γ( n − x ) n ( p k + 2)Γ( n + 1)Γ( n − (cid:18) n/ k/ (cid:19) , where p k = 2 p k ( n − k ) (note that µ k tends to a constant in n. )To derive the said bound, divide numerator and denominator by n and simplify.Observe that p k ∼ n p θ (1 − θ ) . For the term (cid:0) n/ k/ (cid:1) we use the entropic estimatementioned above. We write Γ( x )Γ( n − x ) = ( n − ( n − x − ) . We use the Moivre-Laplace formulato get (cid:0) nx (cid:1) n ∼ exp( − ( x − n/ n/ ) p πn/ O ( 1 √ n ) , where the constant implied by O () is independent of x. The result follows after tediousbut straightforward manipulations.We are now ready for the main result of this section. Theorem 16. Let n → ∞ , and let k be an integer such that k ∼ θn, with < θ < a real constant. Assume x = n + O ( √ n ) . Then any binary orthogonal array D with n columns, of strength ≥ k + 1 , satisfies | F D ( x ) − Ψ( x ) | = O ( 1 √ n ) . Proof. Immediate by combining Theorem 14 with Theorem 15. Note, before doing asymptotics on the strength of designs, that t designs existfor all t [26]. Let H k ( x ) denote the Hahn polynomial of degree k of the variable x ,as defined in [8]. Let v k = (cid:0) nk (cid:1)(cid:0) ν − nk (cid:1) be the valency of order k of the Johnson graph J ( ν, n ) . We normalize \ H k ( x ) = H k ( x ) √ v k . Theorem 17. Assume both ν and n go to infinity with n/ν → p ∈ (0 , . Put q = 1 − p. Let z = nx with x ∈ (0 , . Then, we have for fixed k, and n → ∞ the limit \ H k ( z ) → (1 − x/q ) k p p k q k . roof. First, note that m k ∼ ν k k ! ∼ n k p k k ! . Next, observe that v k ∼ n k ( ν − n ) k k ! ∼ n k ( k !) ( q/p ) k . This yields √ v k ∼ n k k ! ( p q/p ) k . Combining we obtain m k √ v k ∼ / p p k q k . Similar calcu-lations give the term of order j of H k ( xn ) to have the limit( − j (cid:18) kj (cid:19) ( ν j ) n j x j n j ( ν − n ) j → ( − j (cid:18) kj (cid:19) ( x/q ) j , and, summing on j yield H k ( xn ) → k X j =0 (cid:18) kj (cid:19) ( − x/q ) j = (1 − x/q ) k . The result follows upon writing \ H k ( xn ) = m k √ v k H k ( xn ) . We can now derive the main result of this section. Theorem 18. Let D be a t − ( ν, n, η ) design with ν, n, t → ∞ , and t fixed and n ∼ pν with < p < real constants. Put q = 1 − p, and k = ⌊ t/ ⌋ . Let J ( ν, n ; x ) = P i ≤ x v i ( νn ) . Then | F D ( x ) − J ( ν, n ; x ) | ≤ λ k ( n ) , where lim n →∞ λ k ( n ) = 1 − a ( x )1 − a ( x ) k +1 . and a ( x ) = (1 − x/q ) √ pq . Proof. Immediate by taking the limit of the Christoffel-Darboux kernel of order k given by k X j =0 \ H j ( xn ) = k X j =0 H j ( xn ) v j and summing the geometric series of ratio a ( x ) coming from Theorem 17. In this paper we have used a probabilistic approach to approximate the c.d.f.of designs in various finite metric spaces. The key tool is the Christoffel-Darbouxkernel attached to the orthonormal polynomials w.r.t. the valencies of the space.15e have used some strong analytic bounds on this quantity for binary Krawtchoukpolynomials derived in [15, 17]. 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