Problems of Information Transmission, Vol. 39, No. 3, 2003, pp. 255–265. Translated from Problemy Peredachi Informatsii, No. 3, 2003, pp. 28–39.
Original Russian Text Copyright
2003 by Dodunekova.
INFORMATION THEORY AND CODING THEORY
Extended Binomial Moments of a Linear Code
and the Undetected Error Probability
Chalmers University of Technology and G¨oteborg University
Received June 3, 2002; in ﬁnal form, January 28, 2003
Abstract—Extended binomial moments of a linear code, introduced in this paper, are synony-
mously related to the code weight distribution and linearly to its binomial moments. In contrast
to the latter, the extended binomial moments are monotone, which makes them appropriate
for studying the undetected error probability. We establish some properties of the extended
binomial moments and, based on this, derive new lower and upper bounds on the probability of
undetected error. Also, we give a simpliﬁcation of some previously obtained suﬃcient conditions
for proper and good codes, stated in terms of the extended binomial moments.
The performance of a linear code in detecting errors on a symmetric memoryless channel is
characterized by the undetected error probability of the code. This probability is a function of ε,
the symbol error probability of the channel, and involves explicitly the weight distribution of the
code. Clearly, the best error detecting code for a particular channel would have the smallest
undetected error probability. However, most often the symbol error probability of the channel is
not known exactly, and even when it is, there is in general no eﬃcient method for ﬁnding a linear
code with the smallest undetected error probability, except through exhaustive search. As a result,
the concepts of a proper and a good code have been introduced (see, e.g. [1–3]), and it has become
accepted that codes with these properties perform suﬃciently well in error detection. A linear
code is proper if its undetected error probability is an increasing function of ε. A linear code is
good if its undetected error probability is as large as possible for the largest possible value of ε.
Thus, knowledge of the code weight distribution is necessary for ﬁnding out whether the code is
proper, good, or neither of these. Unfortunately, linear codes presently known with their weight
distributions are relatively few.
Establishing such properties as properness and goodness for a parametric class of codes known
with their weight distribution might seem to be an easy task but, in fact, it is not. Analytical
study of the undetected error probability function is shown to be eﬃcient in a small number of
cases only. One reason for this could be complexity of formulas for the code weight distribution.
For instance, analytical study of the derivative of the undetected error probability of maximum
distance separable (MDS) codes carried out in  shows that these codes are proper, but such
a study does not seem to work for near MDS codes. These latter codes were studied in  by
a diﬀerent approach, generalized later in [5, 6] and presenting discrete conditions suﬃcient for a
code to be proper or good. Not only codes known to be proper or good at that time, such as the
Hamming codes and MDS codes, turned out to satisfy these conditions, but also a number of other
well-known classes and subclasses of codes, such as maximum minimum distance (MMD) codes and
their duals, some near MDS codes, and some cyclic redundancy-check (CRC) codes were shown
to satisfy the suﬃcient conditions for goodness or properness, see [4, 7–14]. In particular, some
2003 MAIK “Nauka/Interperiodica”