Upper and lower bounds on the undetected error probability of binary codes derived from shortened Reed-Solomon codes

The number of codewords with small weights is investigated for the binary codes derived from shortened Reed-Solomon codes. A formula is shown for the exact number of codewords with weight 2 in the binary image of a shortened Reed-Solomon code generated by (X-α). This number does not depend on the choice of the primitive element α. Upper bounds on the number of codewords with small weights are also derived for the binary image of a shortened code generated by (X-α) and (X-1)(X-α) by using the relation between the Reed-Solomon code and Hamming code. The number of codewords with the minimum weight is also discussed for the binary images of other Reed-Solomon codes. By using the results, the undetected error probability is evaluated