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On the undetected error probability of BCH codes Chong, William


The undetected error probability, Pu(ε), for a variety of Bose-Chaudhuri-Hocquenghem (BCH) codes, used solely for error detection on a binary symmetric channel (BSC) with cross-over probability, is studied. The undetected error probability can be evaluated if the weight distribution of the code or its dual is available. Unfortunately, no general expression for the weight distribution of BCH codes which correct more than 3 errors is known. A few BCH codes for which it is computationally feasible to determine the weight clisiribution are studied. A proper code is one for which Pu(ε) increases monotonically with for 0 ≤ε≤0.5 Algorithms for determining the properness of linear block codes based on Fourier’s and Sturm’s Theorems for finding the number of real roots of a polynomial in a given interval are investigated. They are useful in cases where the weight distribution is known. Kasami et al have studied linear programming methods for obtaining upper and lower bounds on the weight distributions of 4- and 5-error-correcting BCH codes. Here, two methods which can be used to improve these bounds are presented. The first method uses the minimum distance of a dual BCH code. The second method requires, in addition, the number of codewords at the minimum distance. Comparisons between the improved bounds and the Kasami bounds are given for several BCH codes.

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