Let p be a prime and let h be a positive integer such that p is semiprimitive modulo h. We classify all Butson Hadamard matrices whose entries are complex hth roots of unity and which are invariant under the elementary abelian group of order p^2 .

A Hadamard difference set (HDS) D of order u^2 in an abelian group G satisfies |χ(D)|=u for every nontrivial character χ of G. We call such a character value naive if it is divisible by u, i.e., if it is equal to u times a root of unity. All previously known abelian HDSs only have naive character values. We show that for d≥ 1 and u=3d, a group Z_3^2× H, with H an abelian group of order 2^2d+2, contains a HDS of order u^2 with non-naive character values if and only if 8≤exp H≤ 2^d+2. All difference sets obtained are new. The proof rests on a specific HDS in Z_3^2× Z_8× Z_2, a covering extended building set on Z_3^2× Z_8× Z_4, and a variation of the Davis-Jedwab recursive construction.

Let q be a prime and let λ >1 be an integer coprime to q such that λ is self-conjugate modulo λ q and (λ ,q-1)=1 or 2. Suppose a (λ q,q,λ q,λ ) relative difference set D exists in an abelian group G. Then λ is a square and D admits a (q, q, q, 1) relative difference set as a sub-difference set. Moreover, q=3 and the Sylow q-subgroup of G is isomorphic to C_3× C_3 . If p is an odd prime dividing λ , then p^4b||λ for some positive integer b.

Suppose a (λ n,n,λ n, λ ) relative difference set exists in an abelian group G=S× H , where |S|=λ , |H|=n^2 , (λ ,n)=1 , and λ is self-conjugate modulo λ n . Then λ is a square, say λ =u^2 , and exp (S) divides u by Turyn’s exponent bound. We classify all such relative difference sets with exp (S)=u . We also show that n must be a prime power if an abelian (λ n, n, λ n, λ ) RDS with (λ ,n)=1 exists and λ is self-conjugate modulo n.

In this paper, we study semi-regular relative difference sets. We give some nonexistence results on abelian (mn,n,mn,m) relative difference sets. In particular, we focus on the case when m is prime and show that, for any fixed integer n≥2, there are at most finitely many primes p for which an abelian (pn,n,pn,p) relative difference set may exist. We illustrate our results by investigating the existence of (mn,n,mn,m) relative difference sets with m∈{2,3,4} in detail.

Let a and h be positive integers and let p be a prime. Let q1,..., qt be the distinct prime divisors of hand write Q(h) = {Sigma(t)(i) = 1c(i)q(i): c(i) is an element of Z, c(i) >= 0}. We provide constructions of group invariant Butson Hadamard matrices BH(G, h) in the following cases. 1. G =(Z(p))(2a) and at least one of the following conditions is satisfied. p(a) is an element of Q(h), p(a)+ 2 is an element of Q(h) and his even, p(a) + 1 = (q(1) - 1)(q(2) - 1) where q(1) and q(2) are distinct prime divisors of h. 2. G = Z(pa) x Z(pa) and p - 1, p is an element of Q(h). 3. G = (Z(p2))(a) and p(b) is an element of Q(h) for some divisor bof awith 1 <= b < a. 4. G = Px Z(pa) where Pis any abelian group of order p(a) and p is an element of Q(h). (C) 2021 Elsevier Inc. All rights reserved.

Let A,B be subsets of a finite abelian group G. Suppose that A+B does not contain a unique sum, i.e., there is no g∈G with a unique representation g=a+b, a∈A, b∈B. From such sets A,B, sparse linear systems over the rational numbers arise. We obtain a new determinant bound on invertible submatrices of the coefficient matrices of these linear systems. Under the condition that |A|+|B| is small compared to the order of G, these bounds provide essential information on the Smith Normal Form of these coefficient matrices. We use this information to prove that A and B admit coset partitions whose parts have properties resembling those of A and B. As a consequence, we improve previously known sufficient conditions for the existence of unique sums in A+B and show how our structural results can be used to classify sets A and B for which A+B does not contain a unique sum when |A|+|B| is relatively small. Our method also can be applied to subsets of abelian groups which have no unique differences.

Let $q$ be a power of a prime $p$, let $k$ be a nontrivial divisor of $q-1$ and write $e=(q-1)/k$. We study upper bounds for cyclotomic numbers $(a,b)$ of order $e$ over the finite field $\mathbb{F}_q$. A general result of our study is that $(a,b)\leq 3$ for all $a,b \in \mathbb{Z}$ if $p> (\sqrt{14})^{k/ord_k(p)}$. More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: $(0,0), (0,a), (a,0), (a,a)$ and $(a,b)$, where $a\neq b$ and $a,b \in \{1,\dots,e-1\}$. The main idea we use is to transform equations over $\mathbb{F}_q$ into equations over the field of complex numbers on which we have more information. A major tool for the improvements we obtain over known results is new upper bounds on the norm of cyclotomic integers.

Let p be an odd prime, let a be a positive integer, let m be an odd positive integer, and suppose that a generalized bent function from Z2pam to Z2pa exists. We show that this implies m≠1, p≤22m+2m+1, and ordp(2)≤2m−1. We obtain further necessary conditions and prove that p=7 if m=3 and p∈{7,23,31,73,89} if m=5. Our results are based on new tools for the investigation of cyclotomic integers of prescribed complex modulus, including “minimal aliases” invariant under automorphisms, and bounds on the ℓ2-norms of their coefficient vectors. These methods have further applications, for instance, to relative difference sets, circulant Butson matrices, and other kinds of bent functions.

This survey concerns the following closely related concepts. • Group invariant Butson matrices, • generalized bent functions, • cyclic n-roots, • generalized Hadamard matrices, • abelian splitting semiregular difference sets. We explain the connections between these notions and show that group invariant Butson matrices can be viewed as their “common denominator”. We also review the most relevant known results on these objects, some of which are quite recent.

Let K be a finite abelian group and let exp⁡(K) denote the least common multiple of the orders of the elements of K. A BH(K,h) matrix is a K-invariant |K|×|K| matrix H whose entries are complex hth roots of unity such that HH⁎=|K|I, where H⁎ denotes the complex conjugate transpose of H, and I is the identity matrix of order |K|. Let νp(x) denote the p-adic valuation of the integer x. Using bilinear forms on K, we show that a BH(K,h) exists whenever(i)νp(h)≥⌈νp(exp⁡(K))/2⌉ for every prime divisor p of |K| and(ii)ν2(h)≥2 if ν2(|K|) is odd and K has a direct factor Z2. Employing the field descent method, we prove that these conditions are necessary for the existence of a BH(K,h) matrix in the case where K is cyclic of prime power order.

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 |H|≤2n−1. Furthermore, if n is an odd prime power and a proper circulant weighing matrix of weight n 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.

Let p be a prime and let A be a subset of F p with A = -A and |A \ {0}| ≤ 2log 3 ( p ). Then there is an element of F p which has a unique representation as a difference of two elements of A .

The essential fact behind the so-called field-descent method is that certain cyclotomic integers necessarily are contained in relatively small fields and thus must have relatively small complex modulus. In this paper, we develop a method which reveals a complementary phenomenon: certain cyclotomic integers cannot be contained in relatively small fields and thus must have relatively large complex modulus.This method, in particular, yields progress towards the circulant Hadamard matrix conjecture. In fact, we show that such matrices give rise to certain "twisted cyclotomic integers" which often have small complex modulus, but are not contained in small fields. Hence our "anti-field-descent" method provides new necessary conditions for the existence of circulant Hadamard matrices. The application of the new conditions to previously open cases of Barker sequences shows that there is no Barker sequence of length l with 13 < l <= 4.10(33). Furthermore, 229,682 of the 237,807 known open cases of the Barker sequence conjecture are ruled out. (C) 2015 Elsevier Inc. All rights reserved.

We review the current status of the multiplier conjecture for difference sets, present some new results on it, and determine the open cases of the conjecture for abelian groups of order \(<\)10\(^6\). It turns out that for Paley parameters \((4n-1,2n-1,n-1,n)\), where \(4n-1\) is a prime power, the validity of the multiplier conjecture can be verified in the vast majority of cases, while for other parameter sets numerous cases remain open.

We show that the assumption n1>λ in the Second Multiplier Theorem can be replaced by a divisibility condition weaker than the condition in McFarland's multiplier theorem, thus obtaining significant progress towards the multiplier conjecture.

If a Hadamard difference set exists in H×K, where H is an abelian 2-group and K is a cyclic 3-group, then |H|>4|K|. Furthermore, Lander's conjecture holds for all Hadamard difference sets of order at most 529.

‘There exist normal (2m,2,2m,m) relative difference sets and thus Hadamard groups of order 4m for all m of the form m= x2^a+t+u+w+δ -ϵ +16^b 9^c 10^d 22^e 26^f ∏ _i=1^s p_i^4a_i∏ _i=1^t q_i^2 ∏ _i=1^u ( (r_i+1)/2)r_i^v_i) ∏ _i=1^w s_i under the following conditions: a,b,c,d,e,f,s,t,u,w are nonnegative integers, a_1,… ,a_r and v_1,… ,v_u are positive integers, p_1,… ,p_s are odd primes, q_1,… ,q_t and r_1,… ,r_u are prime powers with q_i≡ 1 (mod 4) and r_i≡ 1 (mod 4) for all i, s_1,… ,s_w are integers with 1≤ s_i ≤ 33 or s_i∈{39,43} for all i, x is a positive integer such that 2x-1 or 4x-1 is a prime power. Moreover, δ =1 if x>1 and c+s>0, δ =0 otherwise, ϵ =1 if x=1, c+s=0 , and t+u+w>0, ϵ =0 otherwise. We also obtain some necessary conditions for the existence of (2m,2,2m,m) relative difference sets in partial semidirect products of ℤ _4 with abelian groups, and provide a table cases for which m≤ 100 and the existence of such relative difference sets is open.

We obtain several new number theoretic results which improve the field descent method. We use these results to rule out many of the known open cases of the circulant Hadamard matrix conjecture. In particular, the only known open case of the Barker sequence conjecture is settled.

We classify all circulant weighing matrices whose order and weight are products of powers of 2 and 3. In particular, we show that proper CW(v, 36)'s exist for all v equivalent to 0 (mod 48), all of which are new. (C) 2012 Published by Elsevier Inc.