IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 10, OCTOBER 2006 2573

High-Rate Error-Correction Codes TargetingDominant Error Patterns

Jihoon Park and Jaekyun Moon, Fellow, IEEE

Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA

We propose a high-rate error-pattern-correcting code constructed by a generator polynomial targeting a set of known dominant errorpatterns. This code is based on first constructing a low-rate cyclic code that possesses a distinct syndrome set for each target error pattern.This base code is then extended by simply applying the same generator polynomial to a larger message block. It is shown that the capturedsyndrome along with a soft metric can be used to correct a single occurrence of any target error pattern within the codeword with highprobability. The proposed scheme outperforms, by a significant margin, conventional post-Viterbi error correction based on high-rateerror-detection coding. The performance comparison is provided for a high-density perpendicular recording model.

Index Terms—Biased soft metric, dominant error patterns, generator polynomial, high-rate error-pattern-correcting cyclic code (CC).

I. INTRODUCTION

CONVENTIONAL block error-correction codes (ECCs),such as the Bose–Chaudhuri–Hocquengem (BCH) code,

are designed to have a certain minimum-distance property thatguarantees correction of up to errors within the received dataword. In interference-dominant channels, such as high-densitymagnetic recording, errors tend to occur in specific patterns.While conventional codes can also correct some of these fre-quently observed error patterns, they are not very effective inproviding immunity against these error patterns, some of whichmay have high weights. In this paper, we introduce a new ap-proach to designing error-control codes. Focusing on a cycliccode (CC), we aim at correcting any single occurrence ofknown error patterns, rather than bit errors.

Targeting a list of known dominant error patterns that makeup a very large percentage of all observed occurrences of errors,we first construct a generator polynomial that can produce dis-tinct syndrome sets for the targeted error patterns. No two errorpatterns within the list map to the same syndrome set, and thesingle occurrence of any targeted error patterns can be success-fully detected. Among the target error patterns, the captured syn-drome points uniquely to one error pattern, and either its preciseposition or a few possible positions. By tailoring the generatorpolynomial specifically to the known set of error polynomials,the code becomes highly effective in handling the frequently ob-served error patterns.

In the second step, the rate of the code is extended by simplyapplying the same generator polynomial to a considerably largermessage block. To make the extended code cyclic, the overallcode length is constrained to be an integer multiple of the basecode length. The captured syndrome uniquely identifies an errorpattern from the target list, but it can only point to a number ofpossible error-pattern positions. The final decision on the posi-tion is based on computing the maximum-likelihood (ML) posi-tion, given the error pattern and a number of possible positions.A biased form of soft metric is also proposed that can help re-duce the probability of miscorrection.

Digital Object Identifier 10.1109/TMAG.2006.878623

To demonstrate the viability of the proposed scheme, we com-pare its simulated bit-error rate (BER) performance with thatof post-Viterbi error correction in conjunction with a high-rateerror-detection code. The channel used for performance com-parison models jitter-dominant perpendicular recording.

II. HIGH-RATE ERROR-PATTERN-CONTROL CODES

A. Base Code Design

We first find a generator polynomial for a CC that producesdistinct syndrome sets for a given list of target error patterns. Let

’s, , be the targeted, dominant error patternsin the form of polynomials with binary coefficients. In a CC,a set (cycle) of syndromes corresponds to all possible cyclicshifts of a given error polynomial. To detect and correct a singleoccurrence of any error pattern from a given list of dominantpatterns, the syndrome sets corresponding to the targeted errorpatterns must all be different (which implies that the sets arealso nonoverlapping). Thus, our objective here is to construct agenerator polynomial in such a way that the syndrome sets forthe known dominant error patterns are all distinct. In this sense,our approach to designing ECC is markedly different from thetraditional approach of guaranteeing correction of up to anyerrors within the received data word. Given a CC witha generator polynomial of degree , the total number of distinctsyndromes is only , while there exist possible error events(including the all-zero error event). Our approach can be viewedas an effort to map this limited number of distinct syndromepatterns to the known set of dominant error patterns, as opposedto low-weight error events, as is done in traditional ECC design.

Our starting point in designing such a code is to establisha sufficient condition for having distinct sets of syndromesamong the targeted error polynomials. Let be the greatestcommon divisor (GCD) between the generator polynomialand . It can be shown that if ’s are all different, thenthe syndrome sets are guaranteed to be distinct among different

’s [1]. Let ’s, , be the irreduciblefactors making up all ’s, and the maximum power inwhich appears in any . It is theneasy to see that a generator polynomial of the form

(1)

0018-9464/$20.00 © 2006 IEEE

2574 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 10, OCTOBER 2006

TABLE IDOMINANT ERROR PATTERNS FOR VITERBI/PDNP DETECTORS

gives rise to distinct GCDs for all ’s.It turns out, however, that there may exist lower degree ’s

that can yield distinct syndrome sets for the targeted error poly-nomials. We search for such a generator polynomial by startingfrom the general form (i.e., the least common multiple (LCM) ofall ’s) given by (1), but with each power increased fromzero. The procedure is basically to find a lowest degree ofthe form (1) that gives either distinct ’s or else does not di-vide for any nonnegative integer , for which

(this last condition ensures that the syndromes ofand are different) [1].

As an example of a base code design for perpendicularrecording, a hyperbolic tangent transition response is assumed,with equalizer target response of at user density1.4 [2], which is defined as the ratio of the width over –50% to50% of the transition response’s saturation level to the user bitperiod. The noise mixture is assumed to be 10% additive whiteGaussian noise (AWGN) and 90% jitter noise. For the detector,we use a Viterbi detector with the branch metric modified toincorporate the pattern-dependent noise predictor (PDNP) [3],with four prediction taps for each branch. The predictor tapsoperate on the previous bits implied in the survivor path. TheViterbi detector based on the conventional Euclidean metric isalso considered for comparison.

Table I lists dominant error patterns that make up 98.78%and 98.13% of the observed error patterns at BERfor the conventional Viterbi detector and the Viterbi/PDNPdetector, respectively. While the error patterns and

are among the dominant patternswith the conventional Viterbi detector, they occur much lessfrequently at the output of the Viterbi/PDNP detector. Note thatas the equalizer target response changes, the error-pattern char-acteristics also change. Our code construction methodology,however, remains the same.

As discussed earlier, if the GCDs between all dominant errorpolynomials and the generator polynomial are different,then corresponding syndrome sets are also different. For the setof dominant error patterns for the Viterbi/PDNP detector, thereare three irreducible polynomial factors: ,and . A good choice for that canyield different GCDs for the dominant error patterns is

. With this , the GCDs for the five dominant errorpatterns are , and

, respectively. It turns out that as a bonus, this particularcan also produce different syndrome sets for five extra error

patterns outside the target set. This has period 12, and givesrise to a CC. Table II shows syndrome sets in decimalnumbers for five dominant error patterns, as well as five extra,nondominant (but not completely negligible) error patterns inpolynomial form. Since the syndrome sequence for each error

TABLE IISYNDROME SETS BY g(x) = 1 + x + x + x + x + x

pattern in the list eventually repeats itself as the feedback shiftregister content (that reflects the syndrome of the captured errorpattern) keeps shifting, only one syndrome in each syndromeset needs to be stored for recognizing the captured error pattern.This is assuming a relatively small latency can be tolerated (amaximum of 12 clock periods here, corresponding to the largestperiod of any syndrome set).

Note that also meets the designcriterion, but is less effective in terms of overall error-patterncorrection capability.

B. Base Code Extension

Once a degree- generator polynomial of period isdesigned, we have a base CC, and the base code canbe extended with any positive integer into a CC.With of period 12, we considera extended CC, the code rate of which is 0.98.

In this extended code, a given syndrome, while still pointingto a particular error pattern in the target list, now points to anumber of possible positions of that error pattern. Because ofthe cyclic property of the code, the syndrome cycles now repeatover the entire length of the code. For example, for the (300,294)CC, if the captured syndrome is 48 (in decimal expression), thenthe error pattern is identified as , from Table II. But nowthat the syndrome is repeated every 12 bits, possible starting po-sitions for this error pattern are . While cer-tain positions can be ruled out quickly by a simple bit-polaritycheck based on the already determined error pattern, the mostlikely position conditioned on a particular error pattern can befound using the metric proposed in [4], with a modification toincorporate PDNP. This would be equivalent to running a cor-relator-based post-Viterbi correction processor [5], but the dif-ference is that here only one correlator matched to the identi-fied error pattern is turned on per received block, and only theknown possible positions are scanned for the purpose of findingthe likely error position. We further show that introducing a biasterm in the metric of [4] can generate side information which canprovide additional error-correction capability and reduce prob-ability of miscorrection, as discussed next.

C. Biased Soft Metric

Let and ’s,, be the identified error pattern and

possible error pattern positions, respectively. Consider thesoft metric

(2)

PARK AND MOON: HIGH-RATE ERROR-CORRECTION CODES TARGETING DOMINANT ERROR PATTERNS 2575

Fig. 1. New error-pattern-correction scheme.

where is the Viterbi input sequence, is the convolution ofthe length- target response and the Viterbi output sequence,

is the convolution of and the corrected Viterbi output se-quence according to the known error pattern , and isthe convolution of and .

It can be shown that at each of the positions, the abovebiased soft metric is computed as

(3)

The noise-dependent terms are clearly eliminated by theadded term in (3). Note that the soft metric givesthe exact, noise-free value of the energy of the known outputerror sequence. As seen in Table II, an error pattern outside theoriginal target list may also map to the same syndrome as adominant error pattern. These nontargeted error patterns do notoccur frequently, but happen with nonnegligible probabilities.It is clear that based on the biased metric, we can easily dis-tinguish them from dominant error patterns and correct them,since the output energy is different in each case.

III. PERFORMANCE EVALUATION

Two schemes are considered in our simulation: “proposedECC I,” which attempts to identify the position of the capturederror pattern using the conditional ML metric of [4], with PDNPincorporated, and “proposed ECC II,” which further incorpo-rates the computation of the biased soft metric of (3) into theproposed ECC I scheme. The biased metric (3) enables correc-tion of a few additional error patterns, as well as reduction ofmiscorrection probability. The overall error-pattern-correctingscheme is shown in Fig. 1.

The BERs of the proposed schemes are compared in Fig. 2.For comparison, the BERs of the uncoded Viterbi detector, un-coded PDNP, and the post-Viterbi processor in conjunction witherror-detection coding are also shown, where a (300,296) cyclicredundancy check (CRC) code generated byis used as the error-detection code. The signal-to-noise ratio(SNR) has been defined as the energy of the first derivative ofthe transition response to the noise spectral density ,which signifies 90% jitter noise [6].

The proposed schemes outperform the post-Viterbi pro-cessor: the SNR gains of 0.61 dB for ECC I and 0.77 dB forECC II are observed at BER in Fig. 2.

Although not shown here due to the space constraint, thesector error-rate performance analysis, based on the block multi-nomial distribution of the probabilities for burst symbol errors,also exhibits similar SNR gains when the overall code rate withan outer Reed–Solomon code is fixed at 0.91.

Fig. 2. BER performance with target response of 5+6D�D for 10% AWGNand 90% jitter noise.

IV. CONCLUSION

An error-pattern control CC is developed that targets a setof known dominant error patterns. The generator polynomial istailored to the specific set of dominant error patterns to producedistinct sets of syndromes for the target error patterns. The typeand number of possible positions for the occurred error patternare easily determined based on its syndrome. A soft metric isused to find the most likely error position. A biased soft metricis also proposed that can improve the performance, by a smallmargin, by correcting additional error patterns, as well as re-ducing miscorrection probability.

ACKNOWLEDGMENT

This work was supported by the Samsung Advanced Instituteof Technology.

REFERENCES

[1] J. Park and J. Moon, “Cyclic codes tailored to a known set of error pat-terns,” in IEEE GLOBECOM, Nov. 2006, to be published.

[2] M. Madden, M. Öberg, Z. Wu, and R. He, “Read channel for perpen-dicular magnetic recording,” IEEE Trans. Magn., vol. 40, no. 1, pp.241–246, Jan. 2004.

[3] J. Moon and J. Park, “Pattern-dependent noise prediction in signal-de-pendent noise,” IEEE J. Sel. Areas Commun., vol. 19, no. 4, pp. 730–743,Apr. 2001.

[4] H. Sawaguchi, M. Izumita, and S. Mita, “Soft-output post-processingdetection for PRML channels in the presence of data-dependent medianoise,” in Proc. IEEE GLOBECOM, vol. 7, 2003, pp. 3913–3920.

[5] T. Conway, “A new target response with parity coding for high densitymagnetic recording channels,” IEEE Trans. Magn., vol. 34, no. 4, pp.2382–2386, Jul. 1998.

[6] J. Moon and J. Park, “Detection of prescribed error events: Applica-tion to perpendicular recording,” in Proc. IEEE ICC, vol. 3, 2005, pp.2057–2062.

Manuscript received March 13, 2006; revised May 15, 2006 (e-mail:jhpark@ece.umn.edu).