On the ℓ4:ℓ2 ratio of functions with restricted Fourier support

Kirshner, Naomi; Samorodnitsky, Alex

doi:10.1016/j.jcta.2019.105202

Cited by 3 publications

(5 citation statements)

References 17 publications

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“…We now prove Theorem 4.1, assuming the four claims above to hold, and proceeding similarly to [19]. We will show by induction on n that for all n and for all 1 ≤ s ≤ n 2 holds R(n, s, p) ≤ 2 c n log(n) • r(n, s, p), for some constant c which may depend on p.…”

Section: Proposition 47mentioning

confidence: 94%

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A moment ratio bound for polynomials and some extremal properties of Krawchouk polynomials and Hamming spheres

Kirshner¹,

Samorodnitsky²

2019

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Let p ≥ 2. We improve the bound f p f 2 ≤ (p − 1) s/2 for a polynomial f of degree s on the boolean cube {0, 1} n , which comes from hypercontractivity, replacing the right hand side of this inequality by an explicit bivariate function of p and s, which is smaller than (p − 1) s/2 for any p > 2 and s > 0. We show the new bound to be tight, within a smaller order factor, for the Krawchouk polynomial of degree s.This implies several nearly-extremal properties of Krawchouk polynomials and Hamming spheres (equivalently, Hamming balls). In particular, Krawchouk polynomials have (almost) the heaviest tails among all polynomials of the same degree and ℓ 2 norm 1 . The Hamming spheres have the following approximate edge-isoperimetric property: For all 1 ≤ s ≤ n 2 , and for all even distances 0 ≤ i ≤ 2s(n−s) n , the Hamming sphere of radius s contains, up to a multiplicative factor of O(i), as many pairs of points at distance i as possible, among sets of the same size 2 . This also implies that Hamming spheres are (almost) stablest with respect to noise among sets of the same size. In coding theory terms this means that a Hamming sphere (equivalently a Hamming ball) has the maximal probability of undetected error, among all binary codes of the same rate.We also describe a family of hypercontractive inequalities for functions on {0, 1} n , which improve on the 'usual' "q → 2" inequality by taking into account the concentration of a function (expressed as the ratio between its ℓ r norms), and which are nearly tight for characteristic functions of Hamming spheres. p 2 . However, if we allow p to grow with n, the two bounds can be significantly different, even for small s. This will be important in estimates which take into account higher moments of polynomials, as is the cases we discuss below.Let us also observe that both bounds hold in somewhat higher generality -for all polynomials of degree at most s on {0, 1} n (see Corollary 1.4 below).We proceed with an informal description of several applications of (1). The formal statements and a more extensive discussion of these results will be given below, in Section 1.2. First, it will be shown that (1) is "nearly tight" (in the sense that will be clarified below) if f is the Krawchouk polynomial K s defined byRecalling that K s is proportional to the Fourier transform of the characteristic function of the Hamming sphere of radius s around zero, this says, alternatively, that Fourier transforms of Hamming spheres are nearly extremal with respect to (1). This will be shown to imply that Krawchouk polynomials and Hamming spheres have certain nearly extremal properties, compared to other objects with similar characteristics. Specifically, we will show that, up to at most polynomial in n error, the following facts hold for functions on {0, 1} n :• Krawchouk polynomials have (almost) the heaviest tails among all polynomials of the same degree and ℓ 2 norm. That is, for a polynomial f of degree s with f 2 = K s 2 , and for a threshold T > 0 holdswhere T ′ is not much larger than T . ...

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Section: Proposition 47mentioning

confidence: 94%

“…-A special case of (1), for p = 4, was shown in [19], where it was also conjectured that the Krawchouk polynomials actually attain the maximum for E f 4 (E f 2 ) 2 among all homogeneous polynomials of the same degree. This conjecture has been recently proved in [1], by a short and a very elegant argument (using compression).…”

Section: Related Workmentioning

confidence: 96%

A moment ratio bound for polynomials and some extremal properties of Krawchouk polynomials and Hamming spheres

Kirshner¹,

Samorodnitsky²

2019

Preprint

Self Cite

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“…Let f be a function on {0, 1} n , and let A be the Fourier support of f . By Proposition 1.1 in [15] we have f now D = D(x). The parameters d, i, t satisfy the conditions of Lemma 3.3, and hence, by the lemma, |C||D| 2 n ≤ 2 −αn , for some positive constant α = α(R).…”

Section: Lemma 41mentioning

confidence: 96%

“…-If f is a non-zero function whose Fourier transform is supported on the vectors of minimal weight in a linear code, we conjecture that the ratio f 4 f 2 is at most subexponential in n. Since the ratio of the fourth and the second norm of a function is upper-bounded by the fourth root of the cardinality of its Fourier support (see e.g., Proposition 1.1 in [15]) this conjecture is weaker than the corresponding conjecture of [13].…”

Section: Linear Codesmentioning

confidence: 96%

“…With that, this special case essentially captures the complexity of the conjecture in full generality. In fact, it is known (see e.g., Proposition 1.1 in [15]) that for any subset A ⊆ {0, 1} n the maximum of the ratio f 4 f 2 over non-zero functions f whose Fourier transform is supported on A is attained, up to a polylogarithmic in |A| factor, on the characteristic function of some subset B ⊆ A. Hence, it would suffice to prove the conjecture for the linear code C ′ spanned by the vectors in B and for the appropriate symmetric function.…”

Section: Some Commentsmentioning

confidence: 99%

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One more proof of the first linear programming bound for binary codes and two conjectures

Samorodnitsky

2021

Preprint

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We give one more proof of the first linear programming bound for binary codes, following the line of work initiated by Friedman and Tillich [9]. The new argument is somewhat similar to the one given in [23], but we believe it to be both simpler and more intuitive. Moreover, it provides the following 'geometric' explanation for the bound. A binary code with minimal distance δn is small because the projections of the characteristic functions of its elements on the subspace spanned by the Walsh-Fourier characters of weight up to 1 2 − δ(1 − δ) • n are essentially independent. Hence the cardinality of the code is bounded by the dimension of the subspace.We present two conjectures, suggested by the new proof, one for linear and one for general binary codes which, if true, would lead to an improvement of the first linear programming bound. The conjecture for linear codes is related to and is influenced by conjectures of Håstad and of Kalai and Linial. We verify the conjectures for the (simple) cases of random linear codes and general random codes.

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One more proof of the first linear programming bound for binary codes and two conjectures

Samorodnitsky

2023

Isr. J. Math.

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On the ℓ4:ℓ2 ratio of functions with restricted Fourier support

Cited by 3 publications

References 17 publications

A moment ratio bound for polynomials and some extremal properties of Krawchouk polynomials and Hamming spheres

A moment ratio bound for polynomials and some extremal properties of Krawchouk polynomials and Hamming spheres

One more proof of the first linear programming bound for binary codes and two conjectures

One more proof of the first linear programming bound for binary codes and two conjectures

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