Burst or random error correction based on Fire and BCH codes

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1 Burst or random error correction based on Fire and BCH codes Wei Zhou, Shu Lin, Khaled Abdel-Ghaffar Department of Electrical and Computer Engineering University of California, Davis, CA 95616, USA {weizhou, shulin, Abstract A class of codes obtained by combining Fire codes, which are burst correcting codes, with BCH codes, which are random error correcting codes, is proposed. The proposed codes aresubcodesofbothfirecodesandbchcodes.lowerbounds on the burst error-correcting capabilities of the proposed codes arederived.thecodescanbeusedoveracompoundchannel that causes burst errors or random errors. I. INTRODUCTION In 1959, Fire[5] presented a construction of cyclic codes generatedbyaproductofabinomialandaprimitivepolynomial.thebinomialhastheform x c + 1forsomepositive odd integer c. The primitive polynomial generates a cyclic Hammingcodethatcancorrectasingleerror.Thelengthof thefirecodeistheleastcommonmultiple(lcm)of cand thelengthofthehammingcode.ifthedegreeoftheprimitive polynomial satisfies certain conditions, then the Fire code can correctasingleburstoflengthupto (c+1)/2.in1968,hsu, Kasami, and Chien[9] presented an interesting modification of Fire codes. They replaced the primitive polynomial generating the Hamming code with a polynomial generating a code that can correct more than one random error. Their motivation was to develop codes that can work over the compound burst/random error channel, i.e., a channel that causes either a burst of errors or random errors to the transmitted codeword andneithertheencodernorthedecoderknows,apriori,the typeoferrorscausedbythechannel.thecodeshsu,kasami, and Chien presented are not cyclic but rather shortened cyclic codes of length equal to the length of the random error correcting code used in the construction. Let v = (v 0, v 1,..., v n 1 )beabinarysequenceoflength ntransmittedoverachannelandlet r = (r 1, r 2,..., r n 1 )be thereceivedsequence.then e = r v = (e 0, e 1,...,e n 1 ) istheerrorsequencecausedbythechannel.iftheindices ofthe 1 sin eareconfinedwithin bconsecutiveintegers, i, i + 1,...,i + b 1,thenwesaythatthechannelcauseda burstoflengthupto b.if e i and e i+b 1 arebothequalto 1, thenwesaythatthebursthasstartingposition iandlength b. Iftheindicesofthe 1 sin e,reducedmodulo ntobeintheset {0, 1,..., n 1},areconfinedwithin bconsecutiveintegers, then eisacyclicburst.thus, e = ( )formsa cyclicburstoflengthsixstartingatposition12.suchaburst of errors, affecting the beginning and the end of a transmitted sequence,iscalledanend-aroundburst.acode Cissaidto have burst error-correcting capability of b if it can correct all bursts, not necessarily cyclic, of length up to b. Achannelmayalsocauserandomerrorsratherthanaburst oferrors.ifthenumberof 1 sin eis t,i.e., ehasweight t, thenwesaythatthechannelcaused trandomerrors.acode Cissaidtohaverandomerror-correctingcapabilityof tifit cancorrectallrandomerrorsofweight torless. Throughout this paper, we will only consider binary codes although generalizations to codes over larger finite fields are possible. For convenience, we represent a sequence e = (e 0, e 1,...,e n 1 )byapolynomial e(x) = n 1 e ix i.for brevity, we identify the sequence e with its polynomial e(x) andsaythat e(x)hasweight t,denotedby w(e(x)) = t,if thesequence eitselfisofweight t,i.e., e(x)hasexactly t nonzeroterms.wealsosaythat e(x)isaburstoflength bif thesequence eisaburstoflength b.inthiscase,wehave e(x) = x i (e i + e i+1 x + + e i+b 1 x b 1 ),i.e.,amonomial multipliedbyapolynomialofdegree b 1.Wesaythat e(x) is a cyclic burst if e is a cyclic burst. In this case, e(x) is congruentmodulo x n + 1 to a productof a monomial multipliedbyapolynomialofdegree b 1.If eisacodeword, thenwesaythat e(x)isacodepolynomial. Inthispaper,wefollowtheworkin[9]butrestrictourselves to cyclic codes. An advantage of cyclic codes over noncyclic codes, including shortened cyclic codes, is the ability of cyclic codes to correct end-around bursts. Suppose that...,v i 1,v i,v i+1,...isasequenceoftransmittedcodewords. Ifthechannelcausesaburstoferrorsaffectingboth v i 1 and v i andanotherburstoferrorsaffectingboth v i and v i+1,then thetwoburstsaffecting v i canbeconsideredasasingleendaroundburstwhichmaybecorrectablebyacodecapableof correcting such bursts. By modifying the construction in[9], we construct a class of cyclic codes that generalize Fire codes. However,weshowthatacodeinthisclassoflengthlarger than that of the random error-correcting code has poor random error-correcting capability regardless of that of the random error-correcting code. Hence, for the compound burst/random error channel, we consider cyclic codes in the proposed class of length equal to that of the random error-correcting code. Thepaperisorganizedasfollows.InSectionII,wepropose aclassofcycliccodesthatgeneralizefirecodes.theuseof the proposed codes over the compound channel is considered insectioniii.thepaperisconcludedinsectioniv.

2 II. GENERALIZATION OF FIRE CODES First, we present the construction of the well-known Fire codes[5]. Construction1. (FireCodes) Let g 0 (x)beanirreducible polynomialofdegree m 0 andperiod n 0 thatdoesnotdivide x c + 1,where cisapositiveoddinteger.then, g(x) = (x c + 1)g 0 (x)generatesafirecode,whichiscyclic,oflength n = LCM{c, n 0 }. Fire codes are capable of correcting bursts of errors as shown next. Theorem 1. The Fire code has burst error-correcting capabilityof min{m 0, (c + 1)/2}. Inparticular,if c = 2b 1and m 0 b,thenthecodecan correcteverysingleburstoflengthequaltoorlessthan b. Thepolynomial g 0 (x)intheconstructionoffirecodesis anirreduciblepolynomialofperiod n 0 and,hence,generates acodeoflength n 0 thatcancorrectasinglerandomerror.by replacing g 0 (x)withthegeneratorpolynomialofanyrandom error-correcting code, Hsu, Kasami, and Chien[9] presented the following construction of codes, which we call shortened cyclic HKC codes. Construction2. (ShortenedCyclicHKCCodes) Let g 0 (x) bethegeneratorpolynomialofacycliccode C 0 oflength n andminimumdistance 2t Let m 0 bethesumofthe degreesofthoseirreduciblefactorsof g 0 (x)whoseperiods areequalto n.then,thecode, C,obtainedbyshortening thecycliccodegeneratedby (x c + 1)g 0 (x)tolength nisa shortened cyclic HKC code. The following result shows that shortened cyclic HKC codes have burst error-correcting capability. Theorem 2. The shortened cyclic HKC code has burst errorcorrectingcapabilityof min{m 0, (2c + 3t 0 2)/4, c 1}. WenoticethattheclassofshortenedcyclicHKCcodesisan extensionoftheclassoffirecodes.however,bysetting t 0 = 1,itisclearthatFirecodesisnotaspecialcaseofshortened cyclic HKC codes. First, Fire codes, which are cyclic, have lengthsequaltotheleastcommonmultipleof candtheperiod of g 0 (x)whilethelengthsoftheshortenedcyclichkccodes, whicharenotcyclicif cisnotafactorof n,areequaltothe periodof g 0 (x).second,withthechoice c = 2b 1and m 0 b,thebursterror-correctingcapabilityoftheshortened cyclichkccodesasstatedintheorem2is b 1whilethat offirecodesis basstatedintheorem1.however,bya simple modification of the construction of shortened cyclic HKC codes, we can obtain the following construction of cyclic codes, which we call cyclic HKC codes, that include Fire codes asaspecialcase. Construction3.(CyclicHKCCodes) Let g 0 (x)bethegeneratorpolynomialofacycliccode C 0 oflength n 0 andminimum Thestatementofthetheoremin[9]doesnotlist c 1intheargument of the minimum function although this is explicit in the proof. distance 2t 0 +1.Let m 0 bethesumofthedegreesofthose irreduciblefactorsof g 0 (x)whoseperiodsareequalto n 0. Then,thecode, C,generatedby g(x) =LCM{x c + 1, g 0 (x)} isacyclichkccodeoflength n =LCM{c, n 0 }. Similar to their closely related shortened cyclic HKC codes, the cyclic HKC codes have also burst error-correcting capabilityasstatednext.theproofofthetheorem,whichverymuch followsthatoftheorem2in[9],isplacedintheappendix. Theorem 3. The cyclic HKC code has burst error-correcting capability of min{m 0, (c + t 0 )/2, c 1} for 1 t 0 2, min{m 0, (2c + 3t 0 2)/4, c 1} for t 0 3. Noticethatsetting t 0 = 1intheconstructionofcyclic HKCandTheorem3givestheconstructionofFirecodesand Theorem 1, respectively. Thefollowingresultshowsthatif n > n 0,thennomatter howlarge m 0 and t 0 are,thebursterror-correctingcapability ofthecyclichkccode CinConstruction3islessthan cand its random error-correcting capability is one. Theorem 4. If n > n 0,thenthecyclicHKCcode C has burst error-correcting capability less than c and random errorcorrecting capability of one. Proof: Let v(x) = x i + x n0 x i (mod x n + 1). Then, v(x)isanonzerocodepolynomial,whichisthesum oftwoburstsoflength c.wehave x i + x n0 x i + 1)(x c + 1) = (xn0, x + 1 whichisdivisiblebyboth x c + 1and g 0 (x),and,hence,by LCM{(x c +1), g 0 (x)}.thus, v(x)isacodewordin C.Since v(x)isasumoftwoburstsoflengthatmost c,theburst error-correcting capability of C is less than c. We also notice that (x+1)v(x) = 1+x c +x n0 +x n0+c isacodewordin Cof weight four. Hence, C has random error-correcting capability of one. Because of Theorem 4, we only consider in the following cyclichkccodesoflength n = n 0,whichisthecaseifand onlyif n 0 isamultipleof c.inthiscase,thebursterrorcorrectingcapabilityofthecodeneednotbelessthan cas dictated by Theorem 4. Actually, for shortened codes, the burst error-correcting capability can exceed c and, consequently, the guaranteed burst error-correcting capability specified in Theorem 2. Notice that Theorems 2 and 3 indicate that increasingtheminimumdistanceofthecode C 0 mayincrease the burst error-correcting capability of C. However, this is not the best way to achieve a larger burst error-correcting capabilityasitismoreefficient,intermsofcoderate,to keeptheminimumdistanceof C 0 equaltothreeandincrease

3 the value of c. Therefore, what is interesting in the shortened cyclichkccodesandcyclichkccodesisnottheuseof the these codes over a channel causing bursts but rather over compound channels that either cause bursts or random errors as explained in the next section. Before proceeding to the next section, let us mention the following two well-known bounds on the burst error-correcting capabilitiesofcodes.let Cbealinearcodeoflength nand dimension n rwhichcancorrectburstsoflength borless. Then, bisupperboundedbythereigerbound[15]as b r/2.ifthecodeiscyclic,then bisalsoupperbounded bytheabramsonbound[2],[12] 2 r n2 b 1 + 1,i.e., b r log 2 (n + 1) + 1.CodessatisfyingtheAbramsonbound with equality were studied in[1]. Let C 0 beabinarybchcodeoflength n 0 anddistance 2t Let mbethemultiplicativeorderof2modulo n 0,i.e., misthesmallestpositiveintegerforwhich 2 m 1 isdivisibleby n 0.Then, C 0 hasdimensionatleast n 0 mt 0 [11]. We conclude that the burst error-correcting capability, b,ofthecyclichkccode Coflength n = n 0,whichthen hasdimensionatleast n c mt 0,isupperboundedbythe Reigerboundas b (c + mt 0 )/2 andbytheabramson boundas b c + mt 0 log 2 (n + 1) + 1,whichreducesto b c + m(t 0 1) + 1incase n + 1isapoweroftwo. III. CODING FOR THE COMPOUND BURST/RANDOM ERROR CHANNEL Acompoundchannelconsistsofasetofchannelswiththe same input and output alphabets. All transmissions use one of thechannelsintheset.bothencoderanddecoderknowthe setofchannelsbutdonotknowwhichchannelisusedfor transmission. Compound channels and their capacities were introduced and studied in[3],[4],[16]. Explicit constructions ofcodeswereproposedin[9],[6],[7],[8]inthecaseinwhich the compound channel consists of two channels: one causing bursts of errors and the other causing random errors. The codes constructed in these papers are shortened cyclic codes. Consider a compound channel consisting of a burst error channel and a random error channel. For this compound channel, we are interested in a code, C, of length n and dimension n r that can correctasingle burst of length upto b comp orupto t comp randomerrors.suchacodeissaid tobe (b comp, t comp )-compounderror-correcting.weassumethat b comp > t comp otherwisethecompoundchannelcanbereduced toasinglechannelcausingupto t comp randomerrors. Necessary and sufficient conditions for the linear code C to be (b comp, t comp )-compounderror-correctingarepresentedin[9]. These conditions can be stated as follows: 1) Nononzerocodewordin Cisasumoftwoburstsof lengthupto b comp ; 2) nononzerocodewordin Chasweightlessthan 2t comp +1; and 3) nononzerocodewordin Cisasumofaburstoflength upto b comp andavectorofweightupto t comp. Thefirstconditionisequivalenttosayingthatthecodecan correctaburstoflengthupto b comp ifthechannelcausesa burst of that length or less. The second condition is equivalent tosayingthatthecodecancorrectupto t comp randomerrors ifthechannelcausesuptothatmanyrandomerrors.the third condition is equivalent to saying that no two distinct codewords,onesuffersfromaburstoflengthupto b comp,and theothersuffersfromupto t comp randomerrors,canleadto thesameword.posner[14]proposedasimpletesttocheck whetherornotcondition3)ismetincase t comp = 1. Anecessaryandsufficientconditionfor1),2),and3)to holdisthatalldistinctburstsofweightupto b comp andvectors ofweightupto t comp areindifferentcosets.basedonthis, Hsu, Kasami, and Chien derived the following bound, which combines the Abramson and the Hamming bounds[11]: t comp ( ) n 2 r n2 bcomp 1 + i t comp n i=1 ( bcomp 1 i 1 Clearly, if C is b-burst error-correcting, t-random errorcorrecting,and (b comp, t comp )-compounderror-correcting,and b and t are the largestnumbersforwhich this is true,then b comp band t comp t.typically,weareinterestedinthe maximumpossible b comp forwhichthe code C is (b comp, t)- compound error-correcting. This is obtained by checking the largest value not greater than b for which condition 3) holds. Following[9],acyclicorashortenedcyclic (b comp, t comp )- compound error-correcting code can be decoded as shown in Fig. 1. The receiver, which does not know whether the channel causing bursts or the channel causing random errors was used, decodes the received polynomial r(x) using two decoders: one tocorrectaburstoflengthupto b,whichcanbedoneusing bursttrapping,andtheotherforcorrectingupto trandom errors. From condition 1) above, it follows that the first decoder produces an output, which is the transmitted codeword, if the channel used for transmission causes a burst of length upto b.fromcondition2),itfollowsthattheseconddecoder produces an output, which is the transmitted codeword, if the channel used for transmission causes up to t random errors. The third condition guarantees that in either case only one of the two decoders succeeds in producing an output or that both produce identical outputs. This precludes the case in which the two decoders produce different codewords as outputs. Fig. 1. Decoder for the compound channel Hsu,Kasami,andChienin[9]presentedandprovedthe following theorem for the shortened cyclic HKC codes as given in Construction 2. Theorem5.Let CbeashortenedcyclicHKCcode.If Cisa b-burst error correcting code and t-random error correcting, ).

4 thenitisa(b comp, t)-compounderrorcorrectingcodewhere b comp = min{b, c}. Based on Theorem 5, Hsu, Kasami, and Chien constructed a classofshortenedcyclichkccodes,denotedbyclassk,for thecompoundchannelwhere C 0 isabchcode.thecodes constructed are not cyclic but rather shortened cyclic codes. By checkingtheproofoftheorem5,whichappearsastheorem1 in[9],itfollowsthatthetheoremholdsalsoforthecyclichkc codes presented in Construction 3. In Table I, we providesome cyclichkc codesforthe compound channel. Each code, C, is given by Construction 3, where cisafactorof n 0 = nandthecode C 0 isabch code.inparticular, CisasubcodeofaFirecodeandaBCH codeofequallength n.eachrowinthetablecorresponds toacode C.Thefirstcolumngivesthelength nandthe dimension kof C.Thesecondcolumngivesthevalueof c and the third column gives roots of the generator polynomial g 0 (x)intermsofpowersofanelement βoforder n.since the roots are consecutive powers of β starting with β, the code C 0 generatedby g 0 (x)isanarrow-sensebchcode[11].the polynomial g 0 (x)istheproductoftheminimalpolynomials overgf(2)oftheserootsandisgiveninoctalnotationinthe fourth column. For example, 3525 represents the polynomial x 10 + x 9 + x 8 + x 6 + x 4 + x Thefifthcolumngives the burst error-correcting capability, b, of the code C and the sixthcolumngivesthebchlowerbound[11]ontherandom error-correcting capability, t, of the code C, i.e., the maximum numberofconsecutivepowersof βthatarerootsofthegeneratorlcm{x c +1, g 0 (x)}of Cis 2tor 2t+1.Although tis alowerboundonthenumberofrandomerrorscorrectableby the code, decoding algorithms, such as the Berlekamp-Massey algorithm and Euclid s algorithm, can efficiently decode up to thebchbound[10].theseventhcolumngivesthe (b comp, t) compound error-correcting capability of the code C. IV. CONCLUSION In this paper, we considered cyclic codes for the compound burst/random error channel. The codes are obtained by combining Fire and BCH codes. We presented results on the burst error-correcting capabilities of some of the constructed codes. It is worth mentioning that the actual maximum burst error-correcting capabilities of the codes exceed those given in Theorems 3 and 5. Therefore, the burst error-correcting capabilities of the cyclic HKC codes need to be studied further. APPENDIX ProofofTheorem3. Since g 0 (x)divides x n0 1andboth x n0 1and x c 1arefactorsof x n 1,itfollowsthat g(x) divides x n 1and Cisacodeoflength n.toprovetheburst error-correcting capability of C we assume that the code has anonzerocodewordwhichisthesumoftwoburstsoflength atmost b < c.sincethecodeiscyclic,wecanassumethat thestartingpositionofoneoftheburstsis 0,i.e., e(x)+e (x) isacodepolynomialwhere e(x) b 1 e ix i (mod x n 1) and e (x) i 0+b 1 i=i 0 e i xi (mod x n 1).Hence, which implies that and From(2),wehave where e(x) e (x) (mod g(x)), (1) e(x) e (x) (mod x c + 1) (2) e(x) e (x) (mod g 0 (x)). (3) b 1 e (x) e i x i+cli (mod x n + 1), (4) i 0 i + cl i < i 0 + b (5) for 0 i < b.as b < c,then l i = l 0 or l 0 1.If l i = l 0 for all i, 0 i < b,then e (x) = x cl0 e(x) (mod x n + 1).Since e(x)+e (x)isanonzerocodepolynomial, cl 0 isnotdivisible by nand,therefore,isnotdivisibleby n 0 either.hence,no irreduciblefactorof g 0 (x)ofperiod n 0 divides x cl0 +1.From (3), (x cl0 + 1)e(x) 0 (mod g 0 (x)). Theproductofallirreduciblefactorsof g 0 (x)ofperiod n 0 divides e(x).sincethisproductisofdegree m 0 while e(x)is ofdegreelessthan b,itfollowsthat b m Therefore, l 1,...,l b 1 arenotallequalto l 0 asatleastoneofthem equals l 0 1.Let j 0 bethelargestindex jforwhich e j = 1 and l j = l 0 and j 1 bethesmallestindex jforwhich e j = 1 and l j = l 0 1.From(5),wehave j 0 + cl 0 < i 0 + band i 0 j 1 + c(l 0 1).Fromthesetwoinequalities,weget j 1 j 0 c b + 1. (6) Therefore, e j = 0for j 0 < j < j 1 andtheweightof e(x)is atmost b (j 1 j 0 1) 2b c.from(4),theweightof e (x),whichisthesameasthatof e(x),isalsoatmost 2b c. As e(x) + e (x) (mod x n + 1)isacodepolynomialinthe codegeneratedby g(x) =LCM{x c + 1, g 0 (x)}and,hence, ofevenweightnotlessthan 2t 0 + 1,wehave 2(2b c) w(e(x))+w(e (x)) w(e(x)+e (x)) 2t 0 +2, which gives b c + t (7) 2 Thisprovesthetheoremincase t 0 = 1or 2.For t 0 3,the proofcontinuesasgivenfortheorem2in[9]where tisused todenote t 0. ACKNOWLEDGMENT ThisworkwassupportedinpartbytheNSFunderGrant CCF and in part by gift grants from Northrop Grumman Corporation, LSI Corporation and SanDisk.

5 TABLE I PARAMETERS OF CYCLIC HKC CODES FOR THE COMPOUND CHANNEL. (n, k) c rootsof g 0 (x) g 0 (x) b t (b comp, t) (15, 8) 3 β (3, 1) (15, 6) 5 β (4, 2) (15, 4) 3 β, β (5, 3) (39, 24) 3 β (6, 1) (39, 12) 3 β, β (13, 3) (45, 30) 3 β (6, 1) (45, 28) 5 β (7, 2) (45, 26) 3 β, β (8, 2) (45, 24) 5 β, β (8, 2) (45, 20) 3 β, β 3, β (12, 3) (45, 18) 5 β, β 3, β (13, 3) (51, 40) 3 β (4, 1) (51, 32) 3 β, β (7, 2) (51, 24) 3 β, β 3, β (11, 4) (63, 54) 3 β (3, 1) (63, 50) 7 β (5, 1) (63, 48) 9 β (6, 1) (63, 48) 3 β, β (5, 2) (63, 44) 7 β, β (6, 2) (63, 42) 9 β, β (9, 2) (63, 42) 3 β, β 3, β (7, 3) (63, 38) 7 β, β 3, β (9, 3) (63, 36) 3 β, β 3, β 5, β (9, 4) (63, 33) 3 β, β 3, β 5, β 7, β (10, 5) (63, 32) 7 β, β 3, β 5, β (12, 5) (255, 244) 3 β (3, 1) (255, 242) 5 β (3, 1) (255, 236) 3 β, β (3, 2) (255, 234) 5 β, β (7, 2) (255, 228) 3 β, β 3, β (8, 3) (255, 226) 5 β, β 3, β (7, 3) (255, 220) 3 β, β 3, β 5, β (7, 4) REFERENCES [1] K.A.S.Abdel-Ghaffar,R.J.McEliece,A.Odlyzko,andH.C.A.van Tilborg, On the existence of optimum cyclic burst-correcting codes, IEEE Trans. Inf. Theory, vol. IT-32, no. 5, pp , Nov [2] N. M. Abramson, Error-correcting codes from linear sequential circuits, presented at the fourth Sypm. Inform. Theory, London, England, Aug. 29 Sep. 2, [3] D.Blackwell,L.Breiman,andA.J.Thomasian, Thecapacityofaclass of channels, Ann. Math. Statist., vol. 30, pp , Dec [4] R. L. Dobrushin, Optimum information transmission through a channel with unknown parameters, Radio Eng. Electron., vol. 4, no. 12, pp. 1 8, [5] P. Fire, E, P.(1959). A class of multiple-error-correcting binary codes for non-independent errors, Sylvania Reconnaissance Systems Laboratory, Mountain View, CA, Rept. RSL-E-2, [6] H.T.Hsu, Aclassofbinaryshortenedcycliccodesforacompound channel, Inform. Contr., vol. 18, pp , [7] H.T.Hsu, Onthedecodingofaclassofshortenedcycliccodesfora compound channel, Inform. Contr., vol. 22, pp , [8] H.T.Hsu, Aclassofshortenedcycliccodesforacompoundchannel, IEEETrans.Inf.Theory,vol.20,no.1,Jan [9] H.T.Hsu,T.Kasami,andR.T.Chien, Error-correctingcodesfora compound channel, IEEE Trans. Inf. Theory, vol. IT-14, no. 1, Jan [10] S. Lin and D. J. Costello, Jr., Error Correcting Coding: Fundamentals and Applications, 2nd ed. Upper Saddle River, NJ: Prentice-Hall [11] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, [12] R. J. McEliece, The Theory of Information and Coding, 2nd Edition. Cambridge, UK: Cambridge University Press, [13] W. W. Peterson and E. J.Weldon, Jr., Error-Correcting Codes, 2nd Edition, Cambridge, MA, MIT Press, [14] E. C. Posner, Simultaneous error-correction and burst-error detection using binary linear cyclic codes, J. Soc. Indust. Appl. Math., vol. 13, no. 4, pp , Dec [15] S. H. Reiger, Codes for the correction of clustered errors, IRE Trans. Inf.Theory,vol.IT-6,no.1,pp.16 21,Mar [16] J. Wolfowitz, Simultaneous channels, Arch. Rat. Mech. Anal., vol. 4, pp , 1960.

Fire Codes Capable of Correcting Bursts Beyond Their Guaranteed Burst-Correcting Capabilities

Fire Codes Capable of Correcting Bursts Beyond Their Guaranteed Burst-Correcting Capabilities Wei Zhou, Shu Lin, and Khaled Abdel-Ghaffar University of California Davis, CA 95616, USA E-mail: {weizhou,shulin,ghaffar}@ucdavis.edu