Bounds for discrepancies in the Hamming space☆
Introduction
Let be the binary Hamming space which can be also thought of as a linear space over the finite field . The cardinality . Denote by the ball with center at and radius , i.e., the set of all points with , where is the Hamming distance. The volume of the ball is independent of . It is convenient to assume that and for , and and for .
For an -point subset and a ball define the local discrepancy as follows: We note that for any , and thus below we limit ourselves to the values . Define the weighted -discrepancy by where is a vector of nonnegative weights normalized by With such a normalization, we have
The -discrepancy is defined by where is a subset of the set of the radii.
We also introduce the following extremal discrepancies and 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 is the complement of , then and we have for all . Hence, generally it suffices to consider only subsets with . Together with results of [1] on quadratic discrepancies, this remark enables us to identify some new examples of subsets that attain the minimum value of with uniform weights Let us give one example. A code with minimum pairwise distance is called perfect if spheres of radius around the codewords fill the entire space without overlapping. Perfect codes were shown in [1] to have the smallest quadratic discrepancy among all codes of the same cardinality. For instance, for the well-known Hamming code is perfect [13, p.23ff]. Therefore, for the code formed of spheres of radius one around the codewords of the Hamming code (i.e., the union of the cosets of the Hamming code) is a minimizer of quadratic discrepancy. Another family of minimizers is given by for any , where is a point antipodal to and denotes the all-ones vector. Some other examples can be also given; see [1]. For the reader’s convenience, we emphasize that the quadratic discrepancy in [1] is related to our definition (2) by .
(ii) Without loss of generality we can restrict the range of summation on in (2) from to , where , limiting ourselves to a half of the full range. More precisely, we have where with .
Indeed, notice that , and therefore . Also, obviously, and thus We conclude that limiting the summation range of amounts to changing the weights in definition (2). Similar arguments hold true for the -discrepancy (5).
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 discrepancy of 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 -discrepancy of 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 [4], [7], [15] gave new, simplified proofs of Stolarsky’s invariance, while [18] extended Stolarsky’s principle to projective spaces and derived asymptotically tight estimates of discrepancy. Sharp bounds on quadratic discrepancy were obtained in [6], [8], [15], [16]. Finally, paper [17] introduced new asymptotic upper bounds on -discrepancies of finite sets in compact metric measure spaces.
A recent paper [1] initiated the study of Stolarsky’s invariance in finite metric spaces, deriving an explicit form of the invariance principle in the Hamming space as well as bounds on the quadratic discrepancy of subsets (codes) in . Explicit formulas were obtained for the uniform weights . Namely, let be two points with . Define As shown in [1, Eq. (23)], Stolarsky’s identity for can be written in the following form: Using this representation, [1, Cor.5.3, Thm.5.5] further showed that where are some universal constants. Here the upper bound is proved by random choice and the lower bound 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 energy functional on the code with potential given by . Moreover, there exist sequences of subsets (codes) whose quadratic discrepancy meets the lower bound. Observe also that if , then the bounds differ only by a factor of : for example, if , then
In this short paper we develop the results of [1], proving bounds on . We also consider a restricted version of the discrepancy , limiting ourselves to the case of hemispheres in . In other words, we take local discrepancy for in (1) ( odd) and average its value over the centers of the balls. For the case of the Euclidean sphere, quadratic discrepancy for hemispheres was previously studied in [4], [16], which established a version of Stolarsky’s invariance for this case.
Section snippets
Bounds on
We are interested in universal bounds for discrepancies (2)–(5) for given and without accounting for the structure of the subset. For the case of finite subsets in compact Riemannian manifolds this problem was recently studied in [17], and we draw on the approach of this paper in the derivations below.
Discrepancy for hemispheres
Let be the Hamming space. In this section we consider a restricted version of discrepancy where instead of all the ball radii in (2) we consider discrepancy only with respect to the balls of radius , calling them hemispheres. For any pair of antipodal points hence .
For a subset define where is the local discrepancy defined in (1). In the previous
Acknowledgments
We are grateful to the anonymous reviewer for the careful reading of the manuscript and helpful remarks and to the handling editor for his attention to the paper.
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2022, Journal of Mathematical Analysis and ApplicationsCitation Excerpt :For equivalence of (1) and (4), see the discussion after Proposition 2.11. In the recent years, numerous authors (including the first two authors of this paper) revisited this fascinating fact, extended it, and applied it to various problems of discrete geometry and optimization: new proofs of the original Stolarsky Invariance Principle have been given in [6,12,25], it has been extended to geodesic distances and other rotationally invariant kernels on the sphere [5,6], to compact, connected, two-point homogeneous spaces [36,37] and to the Hamming cube [2,3], and applied to two problems of Fejes Tóth on sums of various distances on the sphere and in projective spaces [6,9]. We can now obtain the following general version of the Stolarsky Invariance Principle:
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Communicated by D. Bilyk.
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Research of this author was partially supported by NSF, USA grants CCF1618603 and CCF1814487.