Some properties of nonbinary error-correcting codes

An error-correcting code may be thought of as a subset
S_0
of points belonging to a set
S
in which a metric is defined such that the distance between every pair of distinct points of
S_0
is larger than some given number. In Hamming's original formulation,
S
was taken to be the set of all
2^n
n
-bit binary numbers and the distance between a pair of binary numbers
s
and
t
was taken to be the number of bits of
s
which do not agree with the corresponding bits of
t
. In this note we shall take
S
to be the set of all
n
-tuples in which each coordinate of an
n
-tuple can assume one of
k
integral values:
0, 1, \cdots, k - 1
, with
k \geqq 2
. Properties of these nonbinary codes will be discussed.