Identification via Channels and Constant-Weight Codes

In the standard problem of transmission, the goal is to encode a message in a way such that after it passes through a noisy channel, the message can be successfully decoded at the other end. For this case, it turns out that one can send messages that scale exponentially with the blocklength and have the error probability go to 0. For this case, error control coding provides ways of adding redundancy into messages so one can still determine the intended message. In the problem of identification via channels, introduced by Ahlswede and Dueck [1], the receiver is only interested in testing whether a particular message was sent, but the encoder does not know which message the decoder wants. Errors are now considered in terms of false alarm and missed identification. It turns out that for this case, one can design systems such that the number of different messages one can identify grows doubly exponentially with the blocklength. The trick is that each message can map into a list of codewords and the encoder selects one randomly. As long as the fraction of the pairwise overlap of these lists is small, the error probabilities will be small. In this report, we will survey existing approaches to solve this problem. We first state the problem of identification via channels and motivate its study. To solve the problem, we consider the design of constant-weight codes using Reed-Solomon codes, which is based on the papers of Verdú and Wei [2] as well as Kurosawa and Yoshida [3]. Following this, we study constant-weight codes for finite blocklengths and compare the performance of codes constructed in [2] with those in [3]. Then, we examine how constantweight codes can be used for a special case of the one-way communication problem considered by Orlitksy in [4].