Thomas Bier

Balanced magic rectangles

Abstract A magic rectangle is an m × n array the entries of which are the first mn positive integers, the rows of which have constant sum and the columns of which have constant sum; these two constants are the same just in case m = n when we have the famous magic squares (without diagonal conditions) of which magic rectangles are an obvious but apparently neglected generalization. A necessary condition for there to be such a magic rectangle is that m and n be both even, but not both 2; or be both odd. We investigate the sufficiency of this condition. We are also considering related questions concerning the existence of certain orthogonal pairs of quasi-Latin rectangles. We confirm that the condition is sufficient at least when m and n are both even, and more generally when m and n are not coprime, and also when n is a multiple of 3 and m is any odd positive integer greater or equal to three. Our main tool is the notion of a balanced magic rectangle.

Centrally symmetric and magic rectangles

Abstract We define a (4 × n )-rectangle R with ground set G ( R ) = ±[ c + 1, c + 2 n ] to be centrally symmetric with threshold c if all row sums and all column sums of R are equal to zero. A ( p × q )-rectangle A with ground set [1, pq ] is called magic if A has constant row sums and A has constant column sums, the two constants not necessarily equal. In this paper we solve the problem of the existence of centrally symmetric and magic rectangles by determining all pairs of integers ( n, c ) resp. ( p, q ), for which there exists a centrally symmetric (4 × n )-rectangle with threshold c resp. a magic ( p × q )-rectangle.

Remarks on Recent Formulas of Wilson and Frankl

By combining the methods in two recent papers of Wilson 5 and Frankl 3 we obtain some specific details on certain integral bases of the space null t -designs and its orthogonal.

Some bounds for the distribution numbers of an association scheme

We generalize the definition of distribution numbers of an association scheme (with symmetric classes). We then derive upper and lower bounds in terms of T-designs of the association scheme. The lower bound uses the assumption of transitivity of the automorphism group of the association scheme. We give examples to show that these bounds are not always best possible.

Embeddings of binary trees into hypercubes

The authors present a mathematical model of parallel computing in a hypercubical parallel computer. This is based on embedding binary trees or forests into the n-dimensional hypercube. They consider three different models corresponding to three different computing situations. First, they assume that the processing time at each level of the binary tree is arbitrary, and develop the corresponding mathematical model of an embedding of a binary tree into the hypercube. Then they assume that the processing time at each level of the binary tree is the same for all processors involved at that level, and for this they develop the mathematical model of a loop embedding of a binary tree into the hypercube. The most general case is that in which only certain neighboring levels are active. Here they assume for simplicity that only the processors corresponding to two neighboring levels are active at the same time, and correspondingly they develop the mathematical model of a level embedding of a binary tree into the hypercube to cover this case. Both loop embeddings and level embeddings allow the authors to use the same processor several times during the execution of a program.

A family of nonbinary linear codes

Abstract For any prime power q > 2 we construct a family of linear codes over an alphabet of q letters with the following parameters: length = ( q − 1) k − 1 ; dimension = k ; minimum distance = ( q − 1) k − 2 ·( q − 2). The weight distribution of these codes is explicitely described. The codes are subschemes of H (( q − 1) k − 1 , q ). Some properties of the dual codes are discussed.

Totient-numbers of the rectangular association scheme

In this paper we compute the totient numbers of all eigenspaces and of certain sums of eigenspaces of the rectangular association scheme. This is then used to obtain the totient numbers of certain sums of eigenspaces of the hypercubic association schemeH(n, q).

Totient numbers for cyclic group rings

Abstract In this note we compute the totient numbers for the rational group-ring of the cyclic groups. We also give a simplified proof of our product formula for the totients and mention some examples for elementary abelian p -groups.

Totient Numbers of an Arithmetic Progression in a Hypercube: the Coprime Case

For integers n , q ➮ 2 denote by H ( n , q ) the hypercube (i.e. association scheme) and by ℝ[ H ( n , q )] = E 0 ⊕ E 1 ⊕ ... ⊕ E n the decomposition of the q n -dimensional euclidean space ℝ[ H ( n , q )] into the eigenspaces E i of the (0,1)-adjacency matrix of the graph H ( n , q ) with associated eigenvalues ( n − i ) q − n for i = 0, 1, ... , n . For I ⊂ {1, 2, ... , n ) let E I , = ⊕ i ∈ I , E i and denote by φ( E I ) the smallest positive integer m that can be written as m = Σ x ∈ H ( n , q ) v x for an integral vector v ∈ ℤ[ H ( n , q )] ∩ (E 0 ⊕ E I ). It is proved that for I = { l , 2 l , ... , ml } with ml ⩽, n , l ⩾ 1 with complement I = {1, 2, ... , n } - I , φ( E I ) = q n − m , φ( E I ) = q m if q and l are coprime.

Some distribution numbers of the triangular association scheme

In line with the two preceding papers here we compute some distribution numbers of the triangular association scheme (or Johnson-scheme) J ( n , k ). In the first section we give some general upper and lower bounds. In the second section we show that these bounds are met with equality in case of the existence of Steiner systems. Here we heavily use theorem 4 of [1]. In the third section we compute vt 1 ( J ( n , k )) for I = {2, 3, . . . , k } and all values of n , k . Finally we conclude with an example where one of our (obvious) bounds is not attained with equality.