On the properness of some optimal binary linear codes of dimension at most seven
by
Rossitza Dodunekova
Mathematical Sciences, Chalmers University of Technolöogy
Coauthors: S. M. Xiaolei Hu

A linear code is said to be proper for error detection over a symmetric memoryless channel if its undetected error probability is an increasing function of the channel symbol error probability. The error detection performance of a proper code is better in channels with smaller symbol error probability, which property makes the code sufficiently appropriate for use in channels where the symbol error probability is not known exactly.

Properness and other kinds of optimality seem to be closely related, since many codes known to be optimal in one sense or another, or close to optimal, such as Perfect codes over finite fields, Maximum Distance Separable (MDS) codes, Near MDS codes, and Maximum Minimum Distance codes, turn out to be proper as well.

Most of the above codes have been shown to be proper by using certain sufficient conditions for properness expressed in terms of the extended binomial moments of the code. These moments form a monotone sequence which makes them appropriate for the study of the undetected error probability. They admit certain upper and lower bounds expressed in terms of only the basic parameters of the code.

In this paper we study some optimal codes of dimension at most seven with regard to properness. Using the sufficient conditions mentioned above we show that the codes and their duals are proper. This result supports the idea of a positive correlation between properness and other kinds of optimality. One interesting observation based on computer graphs is that the extended binomial moments of these proper codes are rather close to the general lower bound mentioned above.

Date received: February 21, 2008