Undetected Error Probability of Linear Block Codes on Channels with Memory

This paper addresses the problem of determining the undetected error probability, P/sub u/ (/spl epsiv/), for linear (n,k) block codes on channels with memory. In the past, P/sub u/ (/spl epsiv/) was investigated mainly on memoryless channels, such as the binary symmetric channel (BSC). We present two techniques for determining P/sub u/ (/spl epsiv/), where both techniques employ trellis diagrams. The first technique is based upon a trellis diagram of the states of a channel model such as the Gilbert-Elliott or Fritchman channel models. The second technique involves taking the trellis diagram of the syndrome register of a code as well as the stationary and transition probabilities of any of the aforementioned channel models into account. Results indicate that in many cases P/sub u/ (/spl epsiv/) for codes on channels with memory, far exceeds that of P/sub u/ (/spl epsiv/) on memoryless channels for the same code. This fact therefore makes it very important to be able to calculate P/sub u/ (/spl epsiv/) on channels with memory, seeing that P/sub u/ (/spl epsiv/) on the BSC certainly does not represent an upperbound. We also show that the often assumed upperbound on P/sub u/ (/spl epsiv/), 2/sup -(n-k)/, is exceeded on channels with memory. The first technique that we present is applicable to short or low rate codes, while the second can be used with high rate or long codes.