Vol. 15, No. 1, 1965

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Transitive groups of collineations on certain designs

Richard Earl Block

Vol. 15 (1965), No. 1, 13–18

Let M = (aij) be an m × n matrix with entries in {1,1}. Suppose that there is a positive integer d such that the inner product of every pair of distinct rows of M is n 2d; this is equivalent to assuming that any two distinct rows have Hamming distance d, i.e. differ in exactly d places. The rows of M form the code words of a binary code; such a code is called a (binary) constant-distance code, of length n and distance d. Special cases of matrices which may be taken to be M are the Hadamard matrices, which are defined by the condition that m = n = 2d, and the incidence matrices (written with ± 1) of balanced incomplete block designs, which are characterized by the property that all column sums are equal and all row sums are equal.

Suppose that π is a permutation of {1,,n} such that replacement, for i = 1,n, of the π(i)-th column of M by the i-th column of M sends each row of M into a row of M. Then π induces a permutation of the rows of M. Call such a pair of permutations of the columns and of the rows a collineation of M, or of the code. We shall examine constant-distance codes with a group G of collineations which is transitive on the columns. We shall show that G has at most two orbits on the rows (just one orbit if and only if M comes from a balanced incomplete block design), and that if G is nilpotent then at most one of these orbits contains more than a constant row.

Mathematical Subject Classification
Primary: 05.20
Received: 20 December 1963
Published: 1 March 1965
Richard Earl Block