On the Probability of Undetected Error for the Maximum Distance Separable Codes

In this paper we investigate the performance of maximum-distance-separable codes with symbols from
GF(q)
when they are used for pure error detection or for simultaneous error correction and detection over a
q
-input and
q
-output discret memoryless channel with symbol error probability ε. First we show that the probability of undetected error for an MDS code used for pure error detection is upper bounded by
q^{-r}
and decreases monotonically as εdecreases from
(q - 1)/q
to 0, where
r
is the number of parity-check symbols of the code. Then we show that the probability of undetected error for an MDS code used for correcting
t
or fewer symbol errors is upper bounded by
q^{-r} \Sum\min{i=0}\max{t}(\min{i} \max{n})(q - 1)^{i}
and decreases monotonically as ε decreases from
(q - 1)/q
to 0. These results show that the MDS codes are effective for both pure error detection and simultaneous error correction and detection.