Transitive groups of collineations on certain designs

Richard Earl Block

Vol. 15 (1965), No. 1, 13–18

DOI: 10.2140/pjm.1965.15.13

Abstract

Let $M=\left(a_{ij}\right)$ be an $m\times n$ matrix with entries in $\left\{1,-1\right\}$. 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 $\pm 1$) 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 $\pi$ is a permutation of $\left\{1,\cdots ,n\right\}$ such that replacement, for $i=1\cdots ,n$, of the $\pi \left(i\right)$-th column of $M$ by the $i$-th column of $M$ sends each row of $M$ into a row of $M$. Then $\pi$ 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

Milestones

Received: 20 December 1963

Published: 1 March 1965