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
$n2d$; 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)
constantdistance 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
$\left\{1,\cdots \phantom{\rule{0.3em}{0ex}},n\right\}$ such
that replacement, for
$i=1\cdots \phantom{\rule{0.3em}{0ex}},n$,
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 constantdistance 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.
