Single error-correcting codes for nonbinary balanced channels

Close-packed, single error-correcting codes are studied, the letters of which are
N
-tuples of
M
-ary digits, where
M
is the power of a prime. The length
N
of the letters must be given by
N = (M^k - 1)/(M - 1)
, where
k
is an integer. A balanced communication channel, for which all errors in a transmitted digit are equally likely, is defined and a physical model given. The probability of correct reception of the code letters and the rate with which they transmit information in a balanced channel are calculated. This involves deriving formulas for the numbers of code letters having various numbers of 0's. Numerical results are given for a quaternary code and are extended to the case where the quaternary channel has a null zone, so that erasures as well as errors may occur. For the type of signals and noise assumed, the balanced channel without a null zone is found to yield the better performance.