Binomial and monotonic behavior of the probability of undetected error and the 2 -r -bound
H. D. Wacker, Josef Boercsoek · 2008
Proper linear codes play an important role in error detection. They are characterized by an increasing probability of undetected error pue(ε,C) and are considered “good for error detection”. A lot of CRCs commonly used to protect data transmission via a variety of field busses are known for being proper. In this paper the weight distribution of proper linear codes on a binary symmetric channel without memory is investigated. A proof is given that its components are upper bounded by the binomial coefficients in a certain sense. Secondly an upper bound of the tail of the binomial is given, and the results are then used to derive estimates of pue(ε,C). If a code is not proper, it would be desirable to have at least subintervals, where pue(ε,C) increases, or where it satisfies the 2-r-bound. It is for this reason that next the range of monotonicity and of the 2-r-bound is determined. Finally, applications on safety integrity levels are studied.