On the properness of some optimal binary linear codes and their dual codes

A linear code is said to be proper in error detection over a symmetric memoryless channel if its undetected error probability is an increasing function of the channel symbol error probability. A proper code performs well in error detection in the sense that the better the channel, the better the performance, which makes the code appropriate for use in channels where the symbol error probability is not known exactly. A q-ary linear code may be optimal in different ways. Of most interest are codes whose parameters are in some sense extremal. For example, Maximum Distance Separable (MDS) codes are distance-optimal among the q-ary linear codes of the same length and dimension. Codes may be also length-optimal and size-optimal. Studies have shown that many linear codes which are optimal in some sense, or close to optimal, are also proper, and most often their dual codes are proper, too. For example, proper are the Perfect codes over finite fields, MDS codes and some Near MDS codes, many Griesmer codes, and Maximum Minimum Distance codes and their duals. Could it be the case that properness and optimality are closely related? What kind of relation would this be? It is most natural to start the study of these questions by looking for optimal codes which are not proper. In this work we present some preliminary results in this direction. We have studied some binary linear codes of optimal length which cannot be obtained by shortening or puncturing other binary linear codes. The codes turn out to be proper, together with their dual codes. Moreover, like most of the codes listed above, these binary codes satisfy certain conditions that imply properness. These conditions are expressed in terms of the so called extended binomial moments, which are just linear combinations of the elements of the weight distribution of the codes. One interesting observation based on