Abstract
A set \(T\subset {GF(q)}\), \(q=p^h\) is a super-Vandermonde set if \(\sum _{y\in T} y^k=0\) for \(0< k <|T|\). We determine the structure of super-Vandermonde sets of size \(p+1\) (almost small) and size \(q/p-1\) (almost large).
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1 Introduction
A super-Vandermonde set (short: an sV-set) in GF(q), \(q=p^h\), p a prime, is a set T of size \(1<t<q\) such that
for \(0< k< t\). It follows from the non-singularity of the Vandermonde matrices \((y^k)_{yk}\), \(y\in T\) and \(k\in [0,t)\) resp. \(k\in (0,t]\) that \(0\not \in T\) and that \(\pi _t(T)\ne 0\) (in particular \(p\!\not \!|\,\,t\)). The Newton identities relating the power sums \(\pi _k(T)\) and the elementary symmetric polynomials \(\sigma _k(T)\) imply that in the polynomial
the only possible nonzero coefficients are the constant term \((-1)^t\sigma _t\) and the coefficient of \(Z^{t-k}\): \((-1)^k\sigma _k\) with \(k=0\mod p\). The Newton-identities are given by:
and we see that indeed \(\sigma _k=0\) if k is not divisible by p (and less than t).
In terms of the inverses of the elements in T, we get that being sV is equivalent to
with g a p-th power.
The underlying notion of Vandermonde set was introduced by Gács and Weiner in [1]. They appear at several places in the investigation of special point sets in finite projective planes. More about this, as well as many examples, can be found in Chapter 1 of the thesis of Takáts [2], or in her paper [3] with Péter Sziklai, which also classifies small and large sV-sets. Here small means \(t<p\), and small sV-sets are cosets of multiplicative subgroups of \({GF(q)}^*\): in this case the polynomial g is constant, so
where \(t\,|\,q-1\) and c is a t-th power, so that T is a coset of the group of t-th roots of unity.
By large we mean \(t>q/p\) and again we get cosets of multiplicative subgroups, corresponding to the case that \(g=-c\) is constant. The proof in this case is much more involved, but in the final section we will give a simpler proof.
2 Super-Vandermonde sets of size \(p+1\)
If T is an sV-set of size \(p+1\), then the polynomial \(\prod _{y\in T} (Z-y)\) is of the form \(f(Z)=Z^{p+1}+aZ+b\), so our problem is to classify the polynomials of this form that are fully reducible over GF(q). Notice that two different polynomials of this form have a gcd of degree at most one, so that two elements of GF(q) are contained in at most one sV-set of size \(p+1\). We will see in fact that two elements are contained in an sV-set of this size precisely when they have the same GF(p)-norm. We will prove in the next theorem that they can all be obtained from 2-dimensional GF(p)-vector subspaces of GF(q).
Theorem 2.1
Let T be an sV-set in GF(q), \(q=p^h\), p prime, of size \(p+1\). Then there exists \(\alpha \in {GF(q)}^*\) such that
where \(\{ x_1,\dots ,x_{p+1}\}\) represent the 1-dimensional subspaces of a 2-dimensional GF(p)-vector subspace of GF(q).
Conversely, every 2-dimensional GF(p)-vector subspace of GF(q) defines a family of \(q-1\) sV-sets of type (1). In particular, the elements of an sV-set of size \(p+1\) have the same norm over GF(p).
Proof
We first observe that if \(T=\{y_1,\dots ,y_t\}\) is an sV-set, then for each \(\gamma \in {GF(q)}^*\), the set \(\gamma T=\{\gamma y_1,\dots ,\gamma y_t\}\) is an sV-set as well (and of the same size of course). We first show that 2-dimensional subspaces give rise to sV-sets. Let U be a 2-dimensional GF(p)-vector subspace of GF(q), then U is the set of zeros of a polynomial of the form
for some \(a,b\in {GF(q)}\). If \(x_1\) and \(x_2\) are two nonzero roots of (2) which are not proportional over GF(p), then \(x_1^{p-1}\) and \(x_2^{p-1}\) are two different roots of the polynomial \(Z^{p+1}+aZ+b\), which turns out to be fully reducible over GF(q). It follows that for each \(\alpha \in {GF(q)}^*\)
is an sV-set of size \(p+1\).
On the other hand let \(T=\{y_1,\dots ,y_{p+1}\}\) be an sV-set of size \(p+1\) and let
be the associated polynomial. Then, there exist \(y_i,y_j\in T\), with the same GF(p)-norm \(\delta \). Let \(\alpha \) be an element of \({GF(q)}^*\) with norm \(N(\alpha )=\delta \) and set \(z_k:=y_k/\alpha \), for \(k\in \{1,\dots ,p+1\}\). Then
is an sV-set of size \(p+1\) with \(N(z_i)=N(z_j)=1\) and its associated polynomial is
Denoting by \(x_i\) and \(x_j\) the elements of \({GF(q)}^*\) such that \(z_i=x_i^{p-1}\) and \(z_j=x_j^{p-1}\), then \(x_i\) and \(x_j\) are independent over GF(p) and so \(U:=\langle x_i,x_j\rangle \) is a 2-dimensional GF(p)-vector subspace of GF(q), whose elements are the zeros of the polynomial
It follows that the elements of \(\frac{1}{\alpha }T\) are of the form \(x^{p-1}\). This completes the proof. \(\square \)
3 Super-Vandermonde sets of size \(q/p-1\)
Consider the polynomial \({\hbox {Tr}_{q\longrightarrow p}}(aZ)=aZ+a^pZ^p+\cdots +a^{p^{h-1}}Z^{p^{h-1}}\), the trace from GF(q) to GF(p). It is clearly fully reducible over GF(q), and we see that the nonzero roots form an sV-set of size \(q/p-1\). The aim of this section is to prove the converse:
Proposition 3.1
Let T be an sV-set in GF(q), of size \(q/p-1\), (\(q=p^h\)) then
for some \(a\in {GF(q)}^*\).
Proof
Consider as before the polynomial
where g is a p-th power. Let \(T_a\) be the sV-set corresponding to the hyperplane \(\hbox {Tr}(aZ)=0\) with
The greatest common divisor of \(\phi \) and \(\phi _a\) divides \((g(Y)-g_a(Y))^{1/p}\) of degree at most \(q/p^2-1\). So we find that T has at most \(q/p^2-1\) points in every hyperplane, unless it coincides with it. Since the average size of the intersection of T with a hyperplane equals
we see that for some a, T coincides with \(T_a\). \(\square \)
4 Large super-Vandermonde sets
Proposition 4.1
Let T be an sV-set in GF(q), \(q=p^h\) of size \(t>q/p\), then
for some t-th power \(c\in {GF(q)}^*\), so T is coset of a multiplicative subgroup.
Proof
As before \(\phi (Y)=\prod _{y\in T} (Y-y^{-1})=Y^t+g(Y)\), where g is a p-th power. Since this polynomial is fully reducible we may write:
where also the polynomials \(h_i\) are p-th powers. We now equate left and right the terms of degree d mod p, \(d=0,p-1,\dots ,1\), writing \(e=t-q/p\) and \(E=q/p\):
We look at the divisibility by Y. From the last equation we see that \(h_1\) is not divisible by Y, in particular \(h_1\ne 0\), then we see from the other equation involving \(h_1\) that \(h_{1+e}\) is divisible by \(Y^{E}\), next \(h_{1+2e}\) by \(Y^{2E}\), (where of course we take indices mod p) and so on until finally \(h_{1+(p-1)e}=h_{p-e+1}\) is divisible by \(Y^{(p-1)E}=Y^{q-q/p}\). If \(h_{p-e+1}\) is nonzero then the total degree of the left hand side will be at least \(t+1+q-q/p>q\), a contradiction, so \(h_{p-e+1}=0\) and now the last equation tells us that g is constant. \(\square \)
References
Gács A., Weiner Z.: On \((q + t, t)\)-arcs of type \((0, 2, t)\). Des. Codes Cryptogr. 29, 131–139 (2003).
Takáts M.: Directions and other topics in Galois Geometries. Thesis. Department of Computer Science, Institute of Mathematics Eötvös Loránd University (2014).
Sziklai P., Takáts M.: Vandermonde sets and super-Vandermonde sets. Finite Fields appl. 14, 1056–1067 (2008).
Acknowledgments
This research was supported by the Italian national research project Geometrie di Galois e Strutture di Incidenza (COFIN 2012), by the Dipartimento di Matematica e Fisica of the Seconda Università degli Studi di Napoli and by GNSAGA of the Italian Istituto Nazionale di Alta Matematica.
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This is one of several papers published in Designs, Codes and Cryptography comprising the special issue in honor of Andries Brouwer’s 65th birthday.
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Blokhuis, A., Marino, G., Mazzocca, F. et al. On almost small and almost large super-Vandermonde sets in GF(q). Des. Codes Cryptogr. 84, 197–201 (2017). https://doi.org/10.1007/s10623-016-0254-z
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DOI: https://doi.org/10.1007/s10623-016-0254-z