Error recovery of variable length code over BSC with arbitrary crossover probability

The error recovery capability of variable length code (VLC) has been considered as an important performance and design criterion in addition to its coding efficiency. However, almost all of the existing methods for evaluating the error recovery capability of VLC assume that the transmission fault is a random single bit inversion. In this paper, we consider a more generalized problem of precisely evaluating the error recovery capability of VLC in the case that the encoded bit stream is transmitted over a BSC with arbitrary crossover probability. By making use of the Perron-Frobenius Theorem, we derive a very simple expression for the exact mean error propagation rate (MEPR), and show that the variance of error propagation rate (VEPR) is zero. We also prove that in the regime of very low crossover probability, the mean error propagation length (MEPL) derived for single inversion error case approaches a scaled value of the MEPR. Furthermore, we briefly discuss the problem of evaluating the error detection capability of non-exhaustive code over BSC.