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Structure of Group Invariant Weighing Matrices of Small Weight This document is downloaded from DR‑NTU (https://dr.ntu.edu.sg) Nanyang Technological University, Singapore. Structure of group invariant weighing matrices of small weight Leung, Ka Hin; Schmidt, Bernhard 2017 Leung, K. H., & Schmidt, B. (2018). Structure of group invariant weighing matrices of small weight. Journal of Combinatorial Theory, Series A, 154, 114‑128. https://hdl.handle.net/10356/87721 https://doi.org/10.1016/j.jcta.2017.08.016 © 2017 Elsevier. This is the author created version of a work that has been peer reviewed and accepted for publication by Journal of Combinatorial Theory, Series A, Elsevier. It incorporates referee’s comments but changes resulting from the publishing process, such as copyediting, structural formatting, may not be reflected in this document. The published version is available at: [http://dx.doi.org/10.1016/j.jcta.2017.08.016]. Downloaded on 02 Oct 2021 23:53:02 SGT Structure of Group Invariant Weighing Matrices of Small Weight Ka Hin Leung ∗ Department of Mathematics National University of Singapore Kent Ridge, Singapore 119260 Republic of Singapore Bernhard Schmidt Division of Mathematical Sciences School of Physical & Mathematical Sciences Nanyang Technological University Singapore 637371 Republic of Singapore February 26, 2018 Abstract We show that every weighing matrix of weight n invariant under a finite abelian group G can be generated from a subgroup H of G with jHj ≤ 2n−1. Furthermore, if n is an odd prime power and a proper circulant weighing matrix of weight n ∗Research is supported by grant R-146-000-158-112, Ministry of Education, Singapore 1 and order v exists, then v ≤ 2n−1. We also obtain a lower bound on the weight of group invariant matrices depending on the invariant factors of the underlying group. These results are obtained by investigating the structure of subsets of finite abelian groups that do not have unique differences. 1 Introduction Let G be a finite multiplicative group of order v and let Z[G] denote the corresponding P integral group ring. Any X 2 Z[G] can be written as X = g2G agg with ag 2 Z. The P (−1) P −1 integers ag are the coefficients of X. We write jXj = g2G ag and X = agg . P We identify a subset S of G with the group ring element g2S g. For the identity element 1G of G and an integer s, we write s for the group ring element s1G. The set supp(X) = fg 2 G : ag 6= 0g is called the support of X. A weighing matrix of order v is a v ×v matrix M with entries 0; ±1 only such that MM T = nI where n is a positive integer and I is an identity matrix. The integer n is the weight of the matrix. Let G be a finite group and let H = (hf;g)f;g2G be a jGj × jGj matrix, indexed with the elements of G. We say that H is G-invariant if hfk;gk = hf;g for all f; g; k 2 G. Weighing matrices invariant under cyclic groups are called circulant weighing matrices. The existence of group invariant weighing matrices has been studied quite intensively, see [1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 17, 18], for instance. Interest in methods for the study of group invariant weighing matrices also stems from the multiplier conjecture for difference sets: The most powerful known approach to this conjecture due to McFarland [13] depends on nonexistence results for group ring elements which satisfy the same equation XX(−1) = n as group invariant weighing matrices. In fact, our results can be used to improve multiplier theorems. The following is well known, see [16, Lem. 1.3.9]. Lemma 1. Let G be a finite group of order v. The existence of G-invariant weighing matrix of weight n is equivalent to the existence of X 2 Z[G] with coefficients 0; ±1 only such that XX(−1) = n. 2 In view of Lemma 1, we will always view G-invariant weighing matrices as elements of Z[G]. The key to our results is the investigation of the support of group invariant weiging matrices in Section 3. As the support of such matrices does not contain a unique difference, we can use the Smith Normal Form of the matrix of the corresponding linear system to gain insights into the structure of the support. Many group invariant weighing matrices can be constructed as follows. Let H be a subgroup of a finite abelian group G and let g1; : : : ; gK 2 G be representatives of distinct cosets of H in G. Suppose that X1;:::;XK 2 Z[H] have coefficients 0; ±1 only and PK (−1) that i=1 XiXi = n and XiXj = 0 whenever i 6= j. It follows by straightforward computation ([2, Thm. 2.4]) that K X X = Xigi (1) i=1 is a G-invariant weighing matrix of weight n. If (1) holds, we say that X is generated from H. Note that, indeed, the main conditions that make (1) a weighing matrix only involve PK (−1) equations over the group ring of H. These conditions are i=1 XiXi = n and XiXj = 0 for i 6= j. The choice of the gi's only makes sure that the coefficients of X are 0; ±1. In fact, such gi's exist in any abelian group which contains H as a subgroup of index at least K. The following [1, Construction 3.10] provides examples of group invariant weighing matrices obtained via (1). Result 2. Let q = pa where p is a prime and a is a positive integer. Let d ≥ 2 be an integer and assume that d is even if p is odd. Set r = qd + qd−1 + ··· + 1. Let V be a (d + 1)-dimensional vector space over Fq and let U1;:::;Ur be the d-dimensional subspaces of V . Let G be any abelian group containing V as a subgroup such that the index of V in G is at least (r+1)=2. Finally, let g1; : : : ; g(r−1)=2 2 GnH be representatives of distinct cosets of H in G. Then (r−1)=2 X X = U1 + (U2i − U2i+1)gi (2) i=1 3 is a G-invariant weighing matrix of weight q2d. The main aim of our work is to show that group invariant weighing matrices necessarily have the form (1) if their weight is small compared to order of the underlying group. Moreover, we show that the order of the group H which contains the \building blocks" Xi is bounded by a constant only depending on n. Some results in this direction concerning circulant weighing matrices previously were obtained in [12]. The main result of [12] is the following. Result 3. For every positive integer n, there is a positive integer F (n), only depending on n, such that every circulant weighing matrix of weight n is generated from a cyclic group of order F (n). Though the constant F (n) can be computed for any given n, it is huge even for moderately sized n. In particular, all primes ≤ 4n + 1 are divisors of F (n). In Section 5, we prove the following result which substantially generalizes and improves Result 3. Theorem 4. Let n be a positive integer. Every weighing matrix of weight n invariant under an abelian group G is generated from a subgroup H of G with jHj ≤ 2n−1. A G-invariant weighing matrix X 2 Z[G] is called proper if hsupp(Xg)i = G for all g 2 G. Note that X is proper if and only if Xg 62 Z[U] for all proper subgroups U of G all g 2 G. Example 5. Let q = 2a where a is a positive integer. There exists a proper weighing matrix Y of weight q2 invariant under a cyclic group, say U, of order q2 + q + 1 (see [17]). Let G = U × hgi × hhi where g is an element of order 2 and the order of h is coprime to 2(q2 + q + 1). Note that G is a cyclic group of order 2(q2 + q + 1)k where k is the order of h. Set X = (1 + g)Y + (1 − g)hY: Using YY (−1) = q2, (1 + g)(1 − g) = 0, and (1 + g)2 + (1 − g)2 = 4, it is straightforward to verify that X is a G-invariant weighing matrix of weight 4q2 = 22a+2. Furthermore, the fact that Y is proper implies that X is proper, too. 4 Note that the order of h in Example 5 can be arbitrarily large. Hence Example 5 shows that, for any fixed a, there exist proper circulant weighing matrices of weight 22a+2 invariant under groups of arbitrarily large order. We will show that this cannot happen if the weight is an odd prime power. In fact, Theorem 4, together with results from [12], yields the following result. Corollary 6. Let n be an odd prime power. If there exists a proper circulant weighing matrix of weight n and order v, then v ≤ 2n−1. Example 7. We show that the statement of Corollary 6 does not hold any more if \circulant" is replaced by \group invariant". Let p be an odd prime and 3 G = (Z=pZ) × hg1i × · · · × hg(r−1)=2i; 2 where r = p +p+1 and the gi's are elements of order at least 2. Then (2) defines a proper 4 G-invariant weighing matrix of weight p . As the order of the gi's can be arbitrarily large, this shows that, for fixed p, there exist proper weighing matrices of weight p4 invariant under arbitrarily large groups.
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