Novel Burst Error Correction Algorithms for Reed-Solomon Codes

In this paper, we present three novel burst error correcting algorithms for an (
n
,
k
) Reed-Solomon code. The algorithmic complexities are of the same order as error-and-erasure decoding,
O
(
rn
), where
r
=
n
-
k
. In particular, their hardware implementation shares elements of Blahut error-and-erasure decoding. In contrast, all existing single-burst error correcting algorithms, which are equivalent to the proposed first algorithm, have complexity
O
(
r
²
n
). The first algorithm corrects the shortest single-burst with length
f
up to
r
-1. The algorithm follows the key characterization that the ending locations of all candidate bursts can be purely determined by the roots of a polynomial which is a linear function of syndromes, and moreover, the shortest burst is associated with the longest sequence of consecutive roots. The algorithmic miscorrection probability is bounded by
q
^-
(
r-1-f)
, where
q
denotes the field size. The second algorithm extends the first one to correct the shortest burst with length
f
≤
r
-3 and additionally a random symbol error. The algorithmic miscorrection probability is bounded by
q
^-(
r
-3-
f
). The third algorithm probabilistically corrects the shortest burst with length
f
≤
r
-1-2δ and additionally δ (a small constant) random symbol errors. The algorithmic miscorrection and failure probabilities are both bounded by
q
^-(r-1-2δ-f
). Our simulation results for (60, 40) and (30, 16) shortened Reed-Solomon codes verify that the miscorrection probability for three algorithms and the failure probability for the third algorithm all decay exponentially (at the rate of
q
^-1
) with respect to the length of burst.