John E. Olson

On the sum of two sets in a group

Abstract Sums C = A + B of two finite sets in a (generally non-abelian) group are considered. The following two theorems are proved. 1. ∣C∣ ≥ ∣A∣ + 1 2 ∣B∣ unless C + (−B + B) = C; 2. There is a subset S of C and a subgroup H such that ∣S∣ ≥ ∣A∣ + ∣B∣ − ∣H∣, and either H + S = S or S + H = S.

An addition theorem for finite Abelian groups

Abstract If g1, g2, …, g2n−1 is a sequence of 2n − 1 elements in an Abelian group G of order n, it is known that there are n distinct indices i1, i2, …, in such that 0 = gi1 + gi2 + ⋯ + gin. In this paper a suitably general condition on the sequence is given which insures that every element g in G has a representation g = gi1 + gi2 + ⋯ + gin as the sum of n terms of the sequence.

On a combinatorial problem of Erdös, Ginzburg, and Ziv

Abstract The following theorem is proved. If g 1 ,…, g 2 n −1 is a sequence of 2 n − 1 elements in a finite group of order n (written additively), then there are n distinct indices i 1 ,…, i n such that g i 1 + … + g i n = 0.

Balancing families of sets

Abstract Given k finite sets S 1 ,…, S k , to what extent is it possible to partition their union into two parts A and B in such a way that, for each j , S j ∩ A and S j ∩ B contain approximately the same number of elements? Bounds are found for this and similar questions.

On the symmetric difference of two sets in a group

How to extend to non-abelian groups certain combinatorial theorems concerning sums of sets in an abelian group is the subject of this paper. A key result is that the following theorem—free of the hypothesis of normality of the subgroups—holds for the symmetric difference of two sets. Theorem . If X = X + H and Y = Y + K are two finite sets in a group, where H and K are subgroups, and if X + K ≠ X and Y + H ≠ Y, then | X \ Y | + | Y \ X | ⩾ | H | + | K | − 2 | H ∩ K .

A balancing strategy

Abstract Suppose x 1 , x 2 ,…, is a sequence of vectors in R k , ‖ X n ‖⩽1, where ‖( x 1 ,…, x k )‖ = max j | x j |. An algorithm is given for choosing a corresponding sequence e 1 , e 2 ,…, of numbers, e n = ±1, so that ‖ e 1 x 1 + … + e n x n ‖ remains small.

Application of coding theory to the construction of the Singer difference sets

In this paper we give a new and simple construction for the cyclic [(q m − 1)/(q − 1), q m−1, q m−2(q− 1)]—difference sets (q = p γ is a prime power) using the methods of coding theory. The construction is such that, in the case q = 2, the 2-ranks of both the incidence matrix and its complementary matrix are easily determined.

A note on constant weight cyclic codes

Abstract In this paper we show that the dual code of a g-ary Hamming single error-correcting code is a constant weight cyclic code with weight q(m-1). We give, as a consequence, a very simple alternative derivation of the weight enumerator formula for the q-ary Hamming single error-correcting code. We show also that a binary cyclic equidistant code can be derived from this dual code