The undetected error probability for Reed-Solomon codes

R.J. McEliece and L. Swanson (IEEE Tran. Inform. Theory, vol.IT-32, p.701-3, 1986) offered an upper bound on P/sub E/(u), the decoder error probability given u symbol errors occur. In the present study, by using a combinatoric technique such as the principle of inclusion and exclusion, an exact formula for P/sub E/(u) is derived. The P/sub E/(u) of a maximum distance separable code is observed to approach Q rapidly as u gets large, where Q is the probability that a completely random error pattern will cause decoder error. An upper bound for the expansion mod P/sub E/(u)/Q-1 mod is derived, and is shown to decrease nearly exponentially as u increases. This proves analytically that P/sub E/(u) indeed approaches Q as u becomes large, and that some laws of large number come into play.<>