U. Glaeser

The error event likelihoods are calculated as the difference in the squared Euclidean distances between
the signal and the convolution of maximum likelihood sequence estimate and the channel PR, versus that
between the signal and the convolution of an alternative data pattern and the channel PR. During each
clock cycle, the best M of them are chosen, and the syndromes for these error events are calculated.
Throughout the processing of each block, a list is maintained of the N most likely error events, along with
their associated error types, positions and syndromes. At the end of the block, when the list of candidate
error events is finalized, the likelihoods and syndromes are calculated for each of ( ) combinations of Lset
candidate error events that are possible. After disqualifying those L-sets of candidates, which overlap
in the time domain, and those candidates and L-sets of candidates, which produce a syndrome which does
not match the actual syndrome, the candidate or L-set of candidates, which remains and which has the
highest likelihood is chosen for correction. Finding the error event position and type completes decoding.
The decoder can make two types of errors: it fails to correct if the syndrome is zero, or it makes a
wrong correction if the syndrome is nonzero, but the most likely error event or combination of error
events does not produce the right syndrome. A code must be able to detect a single error from the list
of dominant error events and should minimize the probability of producing zero syndrome when more
than one error event occurs in a codeword. Consider a linear code given by an (n − k) × n parity check
matrix H. We are interested in capable of correcting or detecting dominant errors. If all errors from a
list were contiguous and shorter than m, a cyclic n − k = m parity bit code could be used to correct a
single error event [16]; however, in reality, the error sequences are more complex, and occurrence
probabilities of error events of lengths 6, 7, 8 or more are not negligible. Furthermore, practical reasons
(such as decoding delay, thermal asperities, etc.) dictate using short codes, and consequently, in order to
keep the code rate high, only a relatively small number of parity bits is allowed, making the design of
error event detection codes nontrivial. The code redundancy must be used carefully so that the code is
optimal for a given E.
The parity check matrix of a code can be created by a recursive algorithm that adds one column of H
at a time using the criterion that after adding each new column, the code error-event-detection capabilities
are still satisfied. The algorithm can be described as a process of building a directed graph whose vertices
are labeled by the portions of parity check matrix long enough to capture the longest error event, and
whose edges are labeled by column vectors that can be appended to the parity check matrix without
violating the error event detection capability [4]. To formalize code construction requirements, for each
error event from E, denote by si,l a syndrome of error vector σl(ei) (si,l = σl(ei) · H T N
L
), where σl(ei) is an
l-time shifted version of error event ei. The code should be designed in such a way that any shift of any
dominant error sequence produces a nonzero syndrome, i.e., that si,l ≠ 0 for any 1 ≤ i ≤ I and 1 ≤ l ≤ n.
In this way, a single error event can be detected (relying on error event likelihoods to localize the error
event). The correctable shifts must include negative shifts as well as shifts larger than n in order to cover
those error events that straddle adjacent codewords, because the failure to correct straddling events
significantly affects the performance. A stronger code could have a parity check matrix that guaranties
that syndromes of any two-error event-error position pairs ((i1, l1), (i2, l2)) are different, i.e., si1 ,l ≠ s .
1
i2 ,l2 This condition would result in a single error event correction capability. The codes capable of correcting
multiple error events can be defined analogously. We can even strengthen this property and require that
for any two shifts and any two dominant error events, the Hamming distance between any pair of
syndromes is larger than δ; however, by strengthening any of these requirements the code rate decreases.
If Li is a length of the ith error event, and if L is the length of the longest error event from E,
( L = max1≤i≤ I { Li} ), then it is easy to see that for a code capable of detecting an error event from E
that ends at position j, the linear combination of error events and the columns of H from j − L + 1 to j
has to be nonzero. More precisely, for any i and any j (ignoring the codeword boundary effects)
where e i,m is the mth element of the error event e i, and h j is the jth column of H.
© 2002 by CRC Press LLC
T ∑
ei,m ⋅ hj−Li +m
1≤m≤L i
≠ 0