Bounds for discrepancies in the Hamming space
aa r X i v : . [ m a t h . M G ] A ug BOUNDS FOR DISCREPANCIES IN THE HAMMING SPACE
ALEXANDER BARG AND MAXIM SKRIGANOV A BSTRACT . We derive bounds for the ball L p -discrepancies in the Hamming space for < p < ∞ and p = ∞ . Sharp estimates of discrepancies have been obtained for many spaces such as the Euclidean spheres andmore general compact Riemannian manifolds. In the present paper, we show that the behavior of discrepanciesin the Hamming space differs fundamentally because the volume of the ball in this space depends on its radiusexponentially while such a dependence for the Riemannian manifolds is polynomial.
1. I
NTRODUCTION
Basic definitions.
Let X n = { , } n be the binary Hamming space which can be also thought of as alinear space F n over the finite field F . The cardinality | X n | = 2 n . Denote by B ( x, t ) the ball with center at x ∈ X n and radius t ≥ , i.e., the set of all points y ∈ X n with d ( x, y ) ≤ t, where d ( x, y ) is the Hammingdistance. The volume of the ball v ( t ) := | B ( x, t ) | = P ti =0 (cid:0) ni (cid:1) is independent of x ∈ X n . It is convenient toassume that B ( x, t ) = ∅ and v ( t ) = 0 for t < , and B ( x, t ) = X n and v ( t ) = 2 n for t > n .For an N -point subset Z N ⊂ X n and a ball B ( y, t ) define the local discrepancy as follows: D ( Z N , y, t ) = | B ( y, t ) ∩ Z N | − N − n v ( t ) . (1)We note that D ( Z N , y, n ) = 0 for any Z N , y, and thus below we limit ourselves to the values ≤ t ≤ n − . Define the weighted L p -discrepancy by D p ( G, Z N ) = (cid:16) X n − t =0 g t X y ∈ X n − n | D ( Z N , y, t ) | p (cid:17) /p , < p < ∞ , (2)where G = ( g , . . . , g n − ) is a vector of nonnegative weights normalized by X n − t =0 g t = 1 . (3)With such a normalization, we have D p ( G, Z N ) ≤ D q ( G, Z N ) 0 < p < q < ∞ . (4)The L ∞ -discrepancy is defined by D ∞ ( I, Z N ) = max t ∈ I max y ∈ X n | D ( Z N , y, t ) | , (5)where I ⊆ { , . . . , n − } is a subset of the set of the radii.We also introduce the following extremal discrepancies D p ( G, n, N ) = min Z N ⊂ X n D p ( G, Z N ) , < p < ∞ , Department of ECE and Institute for Systems Research, University of Maryland, College Park, MD 20742, USA and Inst. for Probl.Inform. Trans., Moscow, Russia. Email: [email protected]. Research of this author was partially supported by NSF grants CCF1618603and CCF1814487. St. Petersburg Department of Steklov Institute of Mathematics, Russian Academy of Sciences, nab. Fontanki 27, St. Petersburg,191023, Russia. Email: [email protected]. and D ∞ ( I, n, N ) = min Z N ⊂ X n D ∞ ( I, Z N ) . These quantities can be thought of as geometric characteristics of the Hamming space.It is useful to keep in mind the following simple observations:(i) If Z cN = X n \ Z N is the complement of Z N ⊆ X n , then D ( Z N , y, t ) = − D ( Z cN , y, t ) , and we have D p ( G, Z N ) = D p ( G, Z cN ) and D p ( G, n, N ) = D p ( G, n, n − N ) , for all < p ≤ ∞ . Hence, generally it suffices to consider only subsets Z N with N ≤ n − . Togetherwith results of [1] on quadratic discrepancies this gives rise to the next claim: Let Z N be a perfect codein X n , then the set Z cN attains the minimum value D ( G , n, n − N ) , where G = (1 /n, /n, . . . , /n ) .For instance, for n = 2 m − and N = 2 n (1 − − m ) , m ≥ the code Z N formed of spheres of radiusone around the codewords of the Hamming code (i.e., the union of the n cosets of the Hamming code) is aminimizer of quadratic discrepancy. Another family of minimizers is given by X n \{ y, ¯ y } for any y ∈ X n ,where ¯ y := 1 n + y is a point antipodal to y and n ∈ X n denotes the all-ones vector. Some other examples canbe also given; see [1]. For the reader’s convenience, we emphasize that the quadratic discrepancy D L ( Z N ) in [1] is related with our definition (2) by D L ( Z N ) = 2 n n N − ( D ( G , Z N )) .(ii) Without loss of generality we can restrict the range of summation on t in (2) from { , . . . , n } to { , . . . , ν } , where ν = ⌊ ( n − / ⌋ , limiting ourselves to a half of the full range. More precisely, we have D p ( G, Z N ) = D p ( G ∗ , Z N ) and D p ( G, n, N ) = D p ( G ∗ , n, N ) , where G ∗ = ( g ∗ , . . . , g ∗ ν ) with g ∗ t = g t + g n − t +1 .Indeed, notice that B ( y, t ) = X n \ B (¯ y, n − − t ) , and therefore D ( Z N , y, t ) = D ( Z N , ¯ y, n − − t ) .Also, obviously, X y ∈ X n | D ( Z N , ¯ y, t ) | p = X y ∈ X n | D ( Z N , y, t ) | p , and thus D p ( G, Z N ) = (cid:16)X νt =0 (cid:16) g t − n X y ∈ X n | D ( Z N , y, t ) | p + g n − − t − n X y ∈ X n | D ( Z N , ¯ y, t ) | p (cid:17)(cid:17) /p = (cid:16) − n X νt =0 ( g t + g n − t − ) X y ∈ X n | D ( Z N , y, t ) | p (cid:17) /p . We conclude that limiting the summation range of t amounts to changing the weights in definition (2). Similararguments hold true for the L ∞ -discrepancy (5).1.2. Earlier results.
Discrepancies in compact metric measure spaces have been studied for a long time,starting with basic results in the theory of uniform distributions [2, 3, 14]. In particular, quadratic discrepancyof finite subsets of the Euclidean sphere is related to the structure of the distances in the subset through a well-known identity called Stolarsky’s invariance principle [19]. Stolarsky’s identity expresses the L -discrepancyof a spherical set as a difference between the average distance on the sphere and the average distance in the set.Recently it has been a subject of renewed attention in the literature. In particular, papers [9, 15, 4] gave new,simplified proofs of Stolarsky’s invariance, while [18] extended Stolarsky’s principle to projective spaces andderived asymptotically tight estimates of discrepancy. Sharp bounds on quadratic discrepancy were obtainedin [6, 8, 15, 16]. Finally, paper [17] introduced new asymptotic upper bounds on L p -discrepancies of finitesets in compact metric measure spaces.A recent paper [1] initiated the study of Stolarsky’s invariance in finite metric spaces, deriving an explicitform of the invariance principle in the Hamming space X n as well as bounds on the quadratic discrepancy of OUNDS FOR DISCREPANCIES IN THE HAMMING SPACE 3 subsets (codes) in X n . Explicit formulas were obtained for the uniform weights G = (1 /n, /n, . . . , /n ) .Namely, let x, y ∈ X n be two points with d ( x, y ) = w. Define λ ( x, y ) = λ ( w ) := 2 n − w w (cid:18) w − ⌈ w ⌉ − (cid:19) , w = 0 , . . . , n. As shown in [1, Eq. (23)], Stolarsky’s identity for Z N ⊂ X n can be written in the following form: n nD ( G , Z N ) = nN n +1 (cid:18) nn (cid:19) − N X i,j =1 λ ( d ( z i , z j )) . (6)Using this representation, [1, Cor.5.3, Thm.5.5] further showed that c n − / N / (cid:16) − N n (cid:17) / ≤ D ( G , n, N ) ≤ C n − / N / , where c, C are some universal constants. Here the upper bound is proved by random choice and the lowerbound by linear programming. The method of linear programming, well known in coding theory [11, 12],is applicable to the problem of bounding the quadratic discrepancy because it can be expressed as an energyfunctional on the code with potential given by λ. Moreover, there exist sequences of subsets (codes) Z N ⊂ X n , n = 2 m − whose quadratic discrepancy meets the lower bound. Observe also that if N = o (2 n ) , thenthe bounds differ only by a factor of n : for example, if N ≃ αn , < α < , then N / (log N ) − / . D ( G , n, N ) . N / (log N ) − / , (7)In this short paper we develop the results of [1], proving bounds on D p ( G, n, N ) , p ∈ (0 , ∞ ] . We alsoconsider a restricted version of the discrepancy D p ( G, Z N ) , limiting ourselves to the case of hemispheres in X n . In other words, we take local discrepancy for t = ( n − / in (1) ( n odd) and average its value overthe centers of the balls. For the case of the Euclidean sphere, quadratic discrepancy for hemispheres waspreviously studied in [4, 16], which established a version of Stolarsky’s invariance for this case.2. B OUNDS ON D p ( G, n, N ) We are interested in universal bounds for discrepancies (2)–(5) for given n, N and p ∈ (0 , ∞ ] withoutaccounting for the structure of the subset. For the case of finite subsets in compact Riemannian manifoldsthis problem was recently studied in [17], and we draw on the approach of this paper in the derivations below.2.1. The case < p < ∞ . We shall consider random subsets Z N ⊂ X n , using the following standard resultto handle discrepancies of such subsets. Lemma 2.1 (Marcinkiewicz–Zygmund inequality; [10], Sec.10.3) . Let ζ j , j ∈ J, | J | < ∞ , be a finitecollection of real-valued independent random variables with expectations E ζ j = 0 and E ζ j < ∞ , j ∈ J .Then, we have E | X j ∈ J ζ j | p ≤ p ( p + 1) p/ E ( X j ∈ J ζ j ) p/ , ≤ p < ∞ . In our first result we construct a random subset Z N by uniform random choice. Later we will refine thisprocedure, obtaining a more precise bound on D p . Theorem 2.2.
For all N ≤ n − , we have D p ( G, n, N ) ≤ ( p + 1) / N / for ≤ p < ∞ , / N / for < p < . (8) A. BARG AND M. SKRIGANOV
Remark 2.1.
Bounds of the type (8) hold true for arbitrary compact metric measure spaces. Theorem 2.2 isgiven here to compare it with Theorem 2.3 below. Notice also that the upper bound (7) is better than (8) with p = 2 and G = G by a logarithmic factor. Such an improvement is obtained in [1] because of the explicitformula (6) for the quadratic discrepancy with the uniform weights G . Proof.
Choose a subset Z N by selecting the points { z i } N independently and uniformly in X n . The probabil-ity that such a point falls into a subset E ∈ X n equals to | E | / | X n | . Therefore, for the local discrepancy (1) ofthis random subset Z N we have D ( Z N , y, t ) = X Ni =1 ζ i ( y, t ) , (9)where ζ i ( y, t ) = B ( y,t ) ( z i ) − v ( t ) | X n | , where E is the indicator function of a subset E ⊆ X n . The quantities ζ i ( y, t ) are independent randomvariables that satisfy | ζ i ( y, t ) | ≤ and E ζ i ( y, t ) = 0 .Applying the Marcinkiewicz–Zygmund inequality to the sum (9), we obtain E | D ( Z N , y, t ) | p ≤ p ( p + 1) p/ N p/ , ≤ p < ∞ , and, therefore, in view of (3), E D ( G, Z N ) p ≤ p ( p + 1) p/ N p/ , ≤ p < ∞ . Thus, there exists a subset Z N = Z N ( p ) ⊂ X n , ≤ p < ∞ , whose discrepancy is bounded above as in thisinequality. For < p < , in view of (4), we can put Z N ( p ) = Z N (1) to complete the proof. (cid:3) In some situations the bound of this theorem can be improved relying on the method of jittered (or strat-ified ) sampling, which uses a partition of the metric space into subsets of small diameter and equal volume.This idea goes back to classical works on discrepancy theory [2, 3, pp.237-240] and it was used more re-cently in [5, 6, 7] for the case of the Euclidean sphere and in [17] for general metric spaces. Below we followthe approach of [17]. In the case of the Hamming space the natural way to proceed is to partition X n intosub-hypercubes of a fixed dimension.In our analysis bounds on the volume of ball v ( t ) are crucial. For large n and t = λn , ≤ λ ≤ , thewell-known bound on v ( t ) (cf. [13, p. 310]), can be written in the form v ( λn ) ≤ nH ( λ ) , (10)where H ( λ ) = ( h ( λ ) , if ≤ λ ≤ / , , if / < λ ≤ , (11)and h ( λ ) = − λ log λ − (1 − λ ) log (1 − λ ) is the standard binary entropy, and in general, the bound (10)can not be improved. Formally speaking, the statement (10) requires λn be integer, but this does not matterfor the asymptotic arguments that we employ. Theorem 2.3.
Let N = 2 αn , < α < , be a power of . Suppose that the weights g t = 0 for t > βn, <β < / . Then D p ( G, n, N ) ≤ ( p + 1) / N (1 − κ ) / , for ≤ p < ∞ , / N (1 − κ ) / , for < p < , (12) where κ = κ ( α, β ) = 1 − H (1 + β − α ) α ≥ . (13) If α > + β , then the exponent κ ( α, β ) > , and the bound (12) is better than (8) . OUNDS FOR DISCREPANCIES IN THE HAMMING SPACE 5
Proof.
Let V ⊂ X n be the k -dimensional subspace, k = γn, < γ < , consisting of all vectors ( x , . . . , x n ) with x i = 0 if i > k . Let N = 2 n − k = 2 αn , α = 1 − γ. The affine subspaces V i = V + s i , s i ∈ X n /V form a partition of the Hamming space X n = [ Ni =1 V i , V i ∩ V j = ∅ , where | V i | = 2 γn , diam V i = γn, where diam E = max { d ( x , x ) : x , x ∈ E } denotes the diameter of asubset E ⊆ X n .We consider a subset Z N = { z i } N with z i ∈ V i , i = 1 , . . . , N . For such a subset, the local discrepancy(1) can be written as follows D ( Z N , y, t ) = X Ni =1 ζ i ( y, t ) , (14)where ζ i ( y, t ) = { B ( y,t ) ∩ V i } ( z i ) − N | ( B ( y, t ) ∩ V i || X n | . Notice that if V i ⊂ B ( y, t ) , then ζ i ( y, t ) ≡ (recall that x i ∈ V i ). Therefore, the sum (14) takes the form D ( Z N , y, t ) = X Ni ∈ J ζ i ( y, t ) , where J is a subset of indices i such that V i ∩ B ( y, t ) = ∅ but V i B ( y, t ) ( V i is not either completely insideor completely outside B ( y, t ) ). Since diam V i = k , we conclude that all V i , i ∈ J, are contained in the ball B ( y, t + k ) and do not intersect the ball B ( y, t − k − . Therefore, | J | | V i | ≤ v ( t + k ) − v ( t − k − ≤ v ( t + k ) . Here we estimate J from above by the number of sets V i such that B ( y, t ) ⊂ V i . We note that discardingthe term v ( t − k − entails no significant loss in the asymptotics because this term is exponentially smallcompared to v ( t + k ) . For t ≤ βn , using the bound (11) and α + γ = 1 , we obtain | J | ≤ nH ( β + γ ) − γn = 2 αn (1 − κ ) = N − κ , where κ is defined in (13).Now consider a random subset Z N = { z i } N in which each point z i is selected independently and uni-formly in V i . For a subset E ∈ V i we have Pr( z i ∈ E ) = | E | / | V i | = N | E | / | X n | . The quantities ζ i ( y, t ) are bounded independent random variables that satisfy | ζ i ( y, t ) | ≤ and E ζ i ( y, t ) = 0 . Applying theMarcinkiewicz–Zygmund inequality to the sum (14), we obtain E | D ( Z N , y, t ) | p ≤ p ( p + 1) p/ N p (1 − κ ) / and, therefore, in view of (3), E | D ( G, Z N ) | p ≤ p ( p + 1) p/ N p (1 − κ ) / . (15)Thus, there exists a subset Z N = Z N ( p ) ⊂ X n , ≤ p < ∞ , whose discrepancy is bounded above as in thisinequality. For < p < , in view of (4), we can put Z ( p ) = Z (1) to complete the proof. (cid:3) Remark 2.2.
We conjecture that the improvement of the discrepancy estimate for weights equal to zero inthe neighborhood of t = n/ takes place also for d -dimensional Euclidean spheres S d ⊂ R d +1 in the casethat the dimension d grows in proportion to the cardinality N . Indeed, the sphere S d and the Hamming space X n share the property that for large dimensions the invariant measure concentrates around the “equator”. Thisinteresting problem deserves a separate study. A. BARG AND M. SKRIGANOV
The case p = ∞ . The following statement is analogous to [17, Prop.2.2]. For ≤ p < ∞ and anysubset Z N ⊆ X n , we have D ∞ ( I, Z N ) ≤ | I | /p n/p D p ( G I , Z N ) , (16)where D p ( G I , Z N ) = (cid:16) X νt =0 | I | − X y ∈ X n − n | D ( Z N , y, t ) | p (cid:17) /p , is a special L p -discrepancy with G I = ( g , . . . , g ν ) , where g t = | I | − for t ∈ I and g t = 0 otherwise.Indeed, for y ∈ X n and t ∈ I we have | D ( Z N , y, t ) | ≤ (cid:16) X t ∈ I X y ∈ X n | D ( Z N , y, t ) | p (cid:17) /p = | I | /p n/p (cid:16) X t ∈ I | I | − X y ∈ X n − n | D ( Z N , y, t ) | p (cid:17) /p . Theorem 2.4. (i) Let I ⊆ { , , . . . , n } be an arbitrary subset of the set of radii, and N ≤ n − . Then D ∞ ( I, Z N ) ≤ n ) / N / . (17) If N increases exponentially, N ∼ = 2 αn , then D ∞ ( I, n, N ) = O ((log N ) / N / ) . (ii) Let I ⊆ { , , . . . , βn } be an arbitrary subset of the set of radii t ≤ βn, < β < / , and let N = 2 αn ≤ n − be a power of . Then D ∞ ( I, n, N ) ≤ (cid:16) Nα (cid:17) / N (1 − κ ) / , (18) where the exponent κ = κ ( α, β ) is given in (13) . If α > + β , then the exponent κ ( α, β ) > , and the bound (18) is better than (17) .Proof. Substituting the bounds (8) and (12) into inequality (16), we obtain D ∞ ( I, Z N ) ≤ n /p n/p p + 1) / N / (19)and D ∞ ( I, Z N ) ≤ n /p n/p p + 1) / N (1 − κ ) / . (20)Now, we put p = n in (19) and (20) to obtain, respectively, (17) and (18). (cid:3)
3. D
ISCREPANCY FOR HEMISPHERES
Let X n , n = 2 m + 1 be the Hamming space. In this section we consider a restricted version of discrepancywhere instead of all the ball radii in (2) we consider discrepancy only with respect to the balls of radius m ,calling them hemispheres. For any pair of antipodal points y, ¯ y X n = B ( y, m ) ∪ B (¯ y, m ) , B ( y, m ) ∩ B (¯ y, m ) = ∅ , hence − n v ( m ) = 2 − n | B ( y, m ) | = 1 / . For a subset Z N ⊂ X n define D ( m ) p ( Z N ) = (cid:16) − n X y ∈ X n | D ( Z N , y, m ) | p (cid:17) /p , < p < ∞ , (21)where D ( Z N , y, m ) = | B ( y, m ) ∩ Z N | − N OUNDS FOR DISCREPANCIES IN THE HAMMING SPACE 7 is the local discrepancy defined in (1). In the previous notation D ( m ) p ( Z N ) = D p ( G ( m ) , Z N ) , with weights G ( m ) = ( g , . . . , g n − ) , where g m = 1 and g t = 0 if t = m . Further, let D ( m ) ∞ ( Z N ) = max y ∈ X n | D ( Z N , y ) | . As before, define D ( m ) p ( n, N ) = min Z N ⊂ X n D ( m ) p ( Z N ) , p ∈ (0 , ∞ ] . First we address the question of global minimizers of discrepancy.
Theorem 3.1.
For the Hamming space X n with odd n = 2 m + 1 , we have the following.(i) Let N = 2 K be even, then for all subsets Z N ⊆ X n and p ∈ (0 , ∞ ] D ( m ) p ( Z N ) ≥ (22) with equality for subsets Z N consisting of K pairs of antipodal points.(ii) Let N = 2 K + 1 be odd, then for all subsets Z N ⊆ X n and p ∈ (0 , ∞ ] D ( m ) p ( Z N ) ≥ / (23) with equality for subsets Z N consisting of K pairs of antipodal points supplemented with a single point.In other words, for all p ∈ (0 , ∞ ] the extremal discrepancy D ( m ) p ( n, N ) = 0 if N is even and D ( m ) p ( n, N ) =1 / if N is odd.Remark 3.1. The phenomenon of such small discrepancies for hemispheres is also known for Euclideanspheres S d ⊂ R d +1 , see [4, 15, 16]. The sphere S d can be represented as a disjoint union of two antipodalhemispheres and the equator. But the equator in this partition is of zero invariant measure and has no effecton the discrepancy. A similar situation holds for the Hamming space X n with odd n , because in this case the“equator” with t = n/ is simply an empty set. Proof.
From (21) we conclude that N = | B ( y, m ) ∩ Z N | + | B (¯ y, m ) ∩ Z N | , and for any y ∈ X n the local discrepancy can be written as | D ( Z N , y, m ) | = (cid:12)(cid:12)(cid:12) | B ( y, m ) ∩ Z N | − N (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) | B ( y, m ) ∩ Z N | − | B (¯ y, m ) ∩ Z N | (cid:12)(cid:12)(cid:12) . (24)Let N = 2 K . Inequality (22) holds for all subsets Z N . If Z N is formed of K pairs of antipodal points,then | D ( Z N , y, m ) | = 0 for all y ∈ X n . This proves part (i).Let N = 2 K + 1 . It follows from (24) that | D ( Z N , y, m ) | ≥ , since N is odd and | B ( y, m ) ∩ Z N | is even. This implies inequality (23). Furthermore, it also follows from (24) that | D ( Z N , y, m ) | = 1 forall y ∈ X n if Z N consists of K pairs of antipodal points supplemented with a single point. This proves part(ii). (cid:3) Thus in particular, any linear code Z N ⊂ X n that contains the all-ones vector has discrepancy zero (suchcodes are called self-complementary). Many well-known families of binary linear codes such as the Hammingcodes, BCH codes, etc. possess this property.A minor generalization of the above proof implies the following useful relation. Let Z N = Z ′ N ∪ Z ′′ N bea union of two subsets, where Z ′ N contains all pairs of antipodal points in Z N then D ( m ) p ( Z N ) = D ( m ) p ( Z ′′ N ) , p ∈ (0 , ∞ ] . A. BARG AND M. SKRIGANOV
Quadratic discrepancy for hemispheres.
In this section we consider the discrepancy D ( m ) p ( Z N ) de-fined in (21) for the special case p = 2 . Let Z N ⊂ X n be a code, where n = 2 m + 1 . For a pair of points x, y ∈ X n such that d ( x, y ) = w let µ m ( x, y ) = µ m ( w ) = | B ( x ) ∩ B ( y ) | be the size of the intersection ofthe balls of radius t with centers at x and y. By abuse of notation we write µ m both as a kernel on X n × X n and as a function on { , , . . . , n } . This is possible because µ m ( x, y ) depends only on the distance between x and y . Note that µ m (0) = v ( m ) = 2 n − and µ m ( n ) = 0 . In this subsection we use some more specific facts of coding theory. We refer to [13] for details. For acode Z N ⊂ X n let A w = A w ( Z N ) = 1 N |{ ( z i , z j ) ∈ Z N | d ( z i , z i ) = w }| , w = 0 , , . . . , n be the normalized number of ordered pairs of points at distance w (the numbers A w , w = 0 , , . . . , n formthe distance distribution of Z N ). Recall that the dual distance distribution of the code Z N is given by A ⊥ i = 1 N X nw =0 A w K ( n ) i ( w ) , i = 0 , , . . . , n, (25)where K ( n ) i ( x ) be the binary Krawtchouk polynomial of degree k = 0 , . . . , n , defined as follows: K ( n ) i ( x ) = i X j =0 ( − j (cid:18) xj (cid:19)(cid:18) n − xi − j (cid:19) . (26)The vector ( A ⊥ i ) forms the MacWilliams transform of the distance distribution of the code Z N , and if Z N is alinear code, it coincides with the weight distribution of the dual code Z ⊥ N [13, pp. 129,138]. The MacWilliamstransform is an involution [11, Thm. 3], which enables us to invert relations (25): A i = 2 n N X nw =0 A ⊥ w K ( n ) i ( w ) , i = 0 , , . . . , n. (27)The following result is implied by [1], Lemma 4.1. Lemma 3.2.
The Krawtchouk expansion of the function µ m ( w ) , w = 0 , , . . . , n has the form µ m ( w ) = b µ + n X k =1 k odd b µ k K ( n ) k ( w ) where b µ = 2 n − and for all k = 1 , , . . . , n b µ k = 2 − n (cid:18) mm (cid:19) (cid:0) m ( k − / (cid:1) (cid:0) mk − (cid:1) . In the next proposition we establish a version of Stolarsky’s invariance principle for the quadratic discrep-ancy D ( m )2 ( Z N ) defined above in (21). Proposition 3.3.
We have n N − D ( m )2 ( Z N ) = 1 N X nw =0 A w µ m ( w ) − n − (28) = n X k =1 k odd b µ k A ⊥ k . (29) OUNDS FOR DISCREPANCIES IN THE HAMMING SPACE 9
Proof.
Starting with (21), we compute n D ( m )2 ( Z N ) = X y ∈ X n (cid:16) X Nj =1 B ( y,m ) ( z j ) − N (cid:17) = X y ∈ X n (cid:16) X Nj =1 B ( z j ,m ) ( y ) − N (cid:17) = X y ∈ X n (cid:16) X Ni,j =1 B ( z i ,m ) ( y ) B ( z j ,m ) ( y ) − N X Nj =1 B ( z j ,m ) ( y ) + N (cid:17) = X Ni,j =1 X y ∈ X n B ( z i ,m ) ( y ) B ( z j ,m ) ( y ) − n − N = X Ni,j =1 µ m ( z i , z j ) − n − N = N X nw =0 A w µ m ( w ) − n − N , where the last equality uses the definition of A w . This proves (28).To obtain (29), substitute the result of Lemma 3.2 into (28) and then use (25). (cid:3)
The size of the intersection of the balls can be written in a more explicit form: µ m ( w ) = X i,j (cid:18) wi (cid:19)(cid:18) n − wj (cid:19) , w = 0 , , . . . , n, where i + j ≤ m, ≤ w − i + j ≤ m ; in particular, µ m (0) = 2 n − . It is not difficult to show that for any l = 1 , , . . . , ⌊ n/ ⌋ we have µ m (2 l −
1) = µ m (2 l ) and otherwise µ m ( w ) is a decreasing function of w .Let h µ m i E be the average value of the kernel µ m ( x, y ) over the subset E ⊂ X n . Since h µ m i X n = b µ , wecan write (28) in the following form: n N − D ( m )2 ( Z N ) = h µ m i Z N − h µ m i X n . (30)Relations (30), (28) are similar to the invariance principle for hemispheres in the case of the Euclidean sphere,[4, Thm. 3.1]. At the same time, the concrete forms of the results for the Hamming space and the sphere aredifferent: while for the sphere the quadratic discrepancy is expressed via the average geodesic distance in Z N , in the Hamming case it is related to the average of the kernel µ m and is not immediately connected tothe average distance. Note that for quadratic discrepancy D ( G, Z N ) for the Hamming space defined abovein (2), results of this form were previously established in [1].Our final result in this section concerns a characterization of codes with zero discrepancy for hemispheresfor the case of even N . Theorem 3.4.
Let Z N be a code of even size N . Then D ( m )2 ( Z N ) = 0 if and only if the code Z N is formedof N/ antipodal pairs of points.Proof. The sufficiency part has been proved in Theorem 3.1. The proof in the other direction is a combinationof the following steps.
Step 1.
Since b µ k > for all k, expression (29) implies that a code Z N ⊂ X n has zero quadraticdiscrepancy for hemispheres if and only if its dual distance coefficients A ⊥ k = 0 only if k is even, Step 2.
A code Z N is formed of antipodal pairs if and only if its distance distribution is symmetric, i.e., A w = A n − w for all w = 0 , , . . . , m. Indeed, the distance distribution coefficients A w , w = 0 , . . . , n can be written as A w = X z ∈ Z N A w ( z ) , (31)where A w ( z ) = N |{ y : d ( z, y ) = w }| is the local distance distribution at the point z ∈ Z N . Suppose the code is formed of antipodal pairs. For every y ∈ Z N such that d ( z, y ) = w , the opposite point ¯ y satisfies d ( z, ¯ y ) = n − w, and thus, the pair ( y, ¯ y ) contributes to A w ( z ) and A n − w ( z ) in equal amounts.Therefore, from (31) also A w = A n − w . Now suppose that the distance distribution is symmetric. For any code A = 1 , and then also A n = 1 , butthis means that every code point has a diametrically opposite one, or otherwise (31) cannot be satisfied for w = n. Step 3.
The matrix Φ m = K ( n )1 (0) K ( n )1 (1) . . . K ( n )1 ( m ) K ( n )3 (0) K ( n )3 (1) . . . K ( n )3 ( m ) ... ... . . . ... K ( n )2 m +1 (0) K ( n )2 m +1 (1) . . . K ( n )2 m +1 ( m ) has rank m + 1 . This is shown as follows. Orthogonality of Krawtchouk polynomials [11], [13, Thm 5.16]implies that (cid:18) nk (cid:19) n δ j,k = m +1 X w =0 K ( n ) k ( w ) K ( n ) j ( w ) (cid:18) nw (cid:19) = m X w =0 K ( n ) k ( w ) K ( n ) j ( w ) (cid:18) nw (cid:19) + m +1 X w = m +1 ( − j + k K ( n ) k ( n − w ) K ( n ) j ( n − w ) (cid:18) nn − w (cid:19) = 2 m X w =0 K ( n ) k ( w ) K ( n ) j ( w ) (cid:18) nw (cid:19) . Here on the second line we used the relation K ( n ) k ( w ) = ( − k K ( n ) k ( n − w ) , ≤ k, w ≤ n. (32)which is immediate from (26). In other words, for odd j, k we have m X w =0 K ( n ) k ( w ) K ( n ) j ( w ) (cid:18) nw (cid:19) = δ k,j n − (cid:18) nk (cid:19) . (33)Rephrasing this relation, we obtain Φ m B Φ Tm = 2 n − diag (cid:16)(cid:18) n (cid:19) , (cid:18) n (cid:19) , . . . , (cid:18) n m + 1 (cid:19)(cid:17) , where B = diag ( (cid:0) nw (cid:1) , w = 0 , , . . . , m ) . This implies that rank(Φ m ) = m. Step 4.
To complete the proof, suppose that D ( m )2 ( Z N ) = 0 and thus from Step 1 above, A ⊥ k = 0 for allodd k . In particular, for k = 1 , , . . . , m + 1 , using (25) and (32), we obtain m +1 X w =0 A w K ( n ) k ( w ) = m X w =0 ( A w − A n − w ) K ( n ) k ( w ) = 0 . (34)Define the vector α = ( A w − A n − w , w = 0 , , . . . , m ) . From (34) and the definition of Φ m we obtain that Φ m α T = 0 . From Step 3), we conclude that α = 0 or A w = A n − w , w = 0 , , . . . , m. Now Step 2 implies our claim. (cid:3)
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