In this paper, we study linear codes capable of detecting and correcting s-periodic errors. Lower and upper bounds on the number of parity check digits required for codes detecting such errors are obtained. Another bound on codes correcting such errors is also obtained. An example of a code detecting such errors is provided

Das, Pankaj Kumar

Tyagi, Vinod

English

Pankaj Kumar Das

Vinod Tyagi

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RATIO MATHEMATICA 22 (2012) 61-68 ISSN:1592-7415Codes on s-periodic errorsPankaj Kumar Das* and Vinod Tyagi***Department of Mathematics, Shivaji College (University of Delhi),Raja Garden, Delhi-110 027, India**Department of Mathematics, Shyam Lal College (Eve.)(University of Delhi),Shahdara, Delhi-110032, India*pankaj4thapril@yahoo.co.in, **vinodtyagi@hotmail.comAbstractIn this paper, we study linear codes capable of detecting and cor-recting s-periodic errors. Lower and upper bounds on the number ofparity check digits required for codes detecting such errors are ob-tained. Another bound on codes correcting such errors is also ob-tained. An example of a code detecting such errors is provided.Key words: parity check matrix, syndromes, standard array, pe-riodic error.2000 AMS subject classifications: 94B25, 94B60, 94-02.1 IntroductionInvestigations in coding theory have been made in several directions butone of the most important directions has been the detection and correctionof errors. It began with Hamming codes[9] for single errors, Golay codes[10,11] for double and triple random errors and thereafter BCH codes[12, 13, 14]were studied for multiple error correction. There is a long history towardsthe growth of the subject and many of the codes developed have found appli-cations in numerous areas of practical interest. One of the areas of practicalimportance in which a parallel growth of the subject took place is that ofburst error detecting and correcting codes. It has also been observed thatin many communication channels, burst errors occur more frequently thanrandom errors. A burst of length b may be defined as follows:61P. K. Das and V. TyagiDefinition 1.1. A burst of length b is a vector whose only non-zero com-ponents are among some b consecutive components, the first and the last ofwhich is non zero.Extending the work of Hamming[9], Abramson[1] developed codes whichdealt with the correction of single and double adjacent errors. The work dueto Fire[8] depicted a more general concept of burst errors.Stone[19], and Bridwell and Wolf[4] considered multiple bursts. It wasnoted by Chien and Tang[5] that in several channels errors do occur in theform of a burst but not near the end of the vector. Channels due to Alexan-der, Gryb and Nast[2] fall in this category. In this light, Chien and Tangproposed a modification in the definition of a burst, now known as CT burst,according to which a CT burst of length b is defined as follows:Definition 1.2. A CT burst of length b is a vector whose only non zerocomponents are confined to some b consecutive positions, the first of which isnon-zero.Recently a new kind of error, known as repeated burst, has been observedby Berardi, Dass and Verma[3]. For further work on this type of error, onemay refer to [6, 7, 18] and references therein.It is very clear that the nature of error differ from channel to channeldepending upon the behaviour of channels or the kind of errors which occurduring the process of transmission. There is a need to deal with many typesof error patterns and accordingly codes are to be constructed to combatsuch error patterns. Though the errors are generally classified mainly intwo categories - random errors and burst errors, it has also been observedthat the occurrence of errors may follow a pattern, different from randomand burst. In certain communication channel like Astrophotography[21],small mechanical error occurs periodically in the accuracy of the trackingin a motorized mount that results small movements of the target that canspoil long-exposure images, even if the mount is perfectly polar-aligned andappears to be tracking perfectly in short tests. It repeats at a regular interval- the interval being the amount of time it takes the mount’s drive gear tocomplete one revolution. This type of error pattern is termed as periodic oralternate pattern. It was in this spirit that the codes correcting s-alternateerrors were developed by Tyagi and Das [20]. An s-periodic error is definedas follows:62Codes on s-periodic errorsDefinition 1.3. An s-periodic error is an n- tuple whose non zero compo-nents are located at a gap of s positions where s = 1, 2, 3,....,(n− 1) and thenumber of its starting positions is among the first s+ 1 components.For s=1, the 1-periodic error vectors are the ones where error may occurin 1st, 3rd, 5th...positions or 2nd, 4th, 6th,... positions. For example, in avector of length 8, 1-periodic error vectors are of the type 10101000, 00101000,0010101, 10101010, 10001010, 01010101, 01000101, 00000101, 00000001 etc.For s=2, the 2-periodic error vectors are those where error may occurin 1st, 4th, 7th,... positions or 2nd, 5th, 8th,...positions or 3rd, 6th, 9th,...positions. The 2-periodic error vectors may look like 10010010, 10000010,00010010, 01000001, 01000000, 00001001, etc in a vector of length 8.For s=3, in a code length 8, the 3-periodic errors are 10001000, 01000100,00100010, 00010001, 10000000, 01000000 etc.In what follows a linear code will be considered as a subspace of the spaceof all n-tuples over GF(q). The distance between two vectors shall be con-sidered in the Hamming sense.The rest of the paper is organized as follows:In section 2, we study the linear codes that detect any s-periodic error. Weobtain lower and upper bounds on the parity check digits for codes detectingsuch errors. It is followed by an example of such a code. In second 3, we givea bound (based on Reiger’s bound[16]) on codes correcting such errors .2 Codes detecting s-periodic errorsWe consider the linear codes that are capable of detecting any s-periodicerror. Clearly, the patterns to be detected should not be code words. In otherwords we consider codes that have no s-periodic error as a code word. Firstly,we obtain a lower bound over the number of parity-check digits required forsuch a code. The proof is based on the technique used in theorem 4.13,Peterson and Weldon [15].Theorem 2.1. Any (n, k) linear code over GF(q) that detects any s-periodicerror must have at least⌈ns+ 1⌉parity-check digits.Proof. The result will be proved on the basis that no detectable errorvector can be a code word.63P. K. Das and V. TyagiLet V be an (n, k) linear code over GF(q). Consider a set X of all thosevectors such that the non-zero components are located at the first positionand thereafter a gap of s positions.We claim that no two vectors of the set X can belong to the same coset of thestandard array; else a code word shall be expressible as a sum or differenceof two error vectors.Assume on the contrary that there is a pair, say x1, x2 in X belonging to thesame coset of the standard array. Their difference viz. x1-x2 must be a codevector. But x1-x2 is a vector all of whose non-zero components are locatedat the 1st position or after a gap of s position and so is a member of X, i.e.,x1-x2 is an s-periodic error, which is a contradiction. Thus all the vectors inX must belong to distinct cosets of the standard array. The number of suchvectors over GF(q) is clearly qp, where p =⌈ns+ 1⌉.The theorem follows since there must be at least this number of cosets. 2In the following, an upper bound on the number of check digits requiredfor the construction of a linear code discussed in theorem 2.1 is provided.This bound assures the existence of such a linear code and has been obtainedby constructing a matrix under certain constraints. The proof is based onthe well known technique used in Varshomov-Gilbert Sacks bound (referSacks[17], also theorem 4.7 Peterson and Weldon [15]).Theorem 2.2. There exists an (n, k) linear code over GF(q) that has nos-periodic error as a code word provided thatn− k ≥⌈ns+ 1⌉.Proof. The existence of such a code will be shown by constructing anappropriate (n − k) × n parity-check matrix H. The requisite parity-checkmatrix H shall be constructed as follows.Select any non-zero (n− k)-tuples as the first j − 1 columns h1, h2,..., hj−1;the jth(j > s+ 1) column hj is added provided thathj ̸=∑pi=1 uihj−i(s+1)where ui ∈ GF (q) and p =⌈js+ 1⌉− 1.This condition ensures that no s-periodic error will be a code word. Thenumber of ways in which the coefficients ui can be selected is clearly qp.64Codes on s-periodic errorsAt worst, all these linear combinations might yield a distinct sum.Therefore a column hj can be added to H provided thatqn−k > qp.or,n− k ≥⌈js+ 1⌉.For a code of length n, replacing j by n gives the result. 2Remark: The above two theorems can be combined as follows:For detecting s-periodic errors in a linear code of length n,⌈ns+ 1⌉paritycheck symbols are necessary and sufficient.Example 2.1. Consider a (7, 4) binary code with parity check matrixH = 1 1 1 0 0 1 00 1 0 1 1 1 00 0 1 0 1 1 1This matrix has been constructed by the synthesis procedure, outlined in theproof of Theorem 2.2, by taking s = 2 and n =7. It can be seen from Table1 that the syndromes of the different 2-periodic errors are nonzero, showingthereby that the code that is the null space of this matrix can detect all2-periodic errors.Table 1Error patterns Syndromes1000000 1000001000 0100000001 0011001000 1101000001 1010001001 0111001001 1110100000 1100000100 0110100100 1010010000 1010000010 1110010010 01065P. K. Das and V. Tyagi3 Codes correcting s-periodic errorsThe following theorem gives a bound on the number of parity-check dig-its for a linear code that corrects s-periodic errors. The proof is based onthe technique used to establish Reiger’s bound[16] (also refer Theorem 4.15,Peterson and Weldon [15]) for correction of s-periodic errors.Theorem 3.1. An (n, k) linear code over GF(q) that corrects all t-periodicerrors, t = 2s+ 1 must have at least⌈ns+ 1⌉parity-check digits.Proof. Any vector that has the form of an s-periodic error can be ex-pressible as a sum or difference of two vectors, each of which is an t-periodicerror. These component vectors must belong to different cosets of the stan-dard array because both such errors are correctable errors. Accordingly, sucha vector viz. s-periodic error can not be a code vector. In view of Theorem2.1, such a code must have at least⌈ns+ 1⌉parity-check digits.AcknowledgementThe authors are very much thankful to Prof. B. K. Dass, Department ofMathematics, University of Delhi for his valuable suggestions, revising thecontents and bringing the paper to the current form.References[1] N. M. Abramson, A class of systematic codes for non-independent errors,IRE Trans. on Information Theory, IT-5, No. 4(1959), 150-157.[2] A. A. Alexander, R. M. Gryb and D. M. Nast, Capabilities of the tele-phone network for data transmission, Bell System Tech. J., Vol. 39, No.3(1960), 431-476.[3] L. Berardi, B. K. Dass and R. Verma, On 2-repeated burst error de-tecting codes, Journal of Statistical Theory and Practice, Vol. 3, No.2(2009), 381-391.[4] J. D. Bridwell and J. K. Wolf, Burst distance and multiple-burst correc-tion, Bell System Tech. J., Vol. 49(1970), 889-909.[5] R. T. Chien and D. T. Tang, On definitions of a burst, IBM Journal ofResearch and Development, Vol. 9, No. 4(1965), 292-293.66Codes on s-periodic errors[6] B. K. Dass and S. Madan, Repeated Burst Error Locating Linear Codes,Discrete Mathematics, Algorithms and Applications, Vol. 2, No. 2(2010),181-188.[7] B. K. Dass, and R. Verma, Repeated Burst Error Detecting Lin-ear Codes, Ratio Mathematica Journal of Applied Mathematics, Vol.19(2009), 25-30.[8] P. Fire, A class of multiple-error-correcting binary codes for non-independent errors, Sylvania Report RSL-E-2, Sylvania ReconnaissanceSystems Laboratory, Mountain View, Calif,(1959).[9] R. W. Hamming, Error-detecting and error-correcting codes, Bell Sys-tem Technical Journal, Vol. 29(1950), 147-160.[10] M. J. E. Golay, Notes on digital coding, Proc. IRE, Vol. 37(1949), 657.[11] M. J. E. Golay, Binary Coding, IRE Trans., PGIT-4(1954), 23-28.[12] R. C. Golay and D. K. Ray-Chaudhuri, On a class of Error CorrectingGroup Codes, Inf. and Control, Vol. 3(1960), 68-79.[13] R. C. Golay and D. K. Ray-Chaudhuri, Further Results on Error Cor-recting Binary Group Codes, Inf. and Control, Vol. 3(1960), 279-290.[14] A. Hocquenghem, Codes Corecteurs d’Erreurs, Chiffres, Vol. 2(1959),147-156.[15] W. W. Peterson and E. J. Weldon(Jr.), Error-Correcting Codes, 2ndedition, The MIT Press, Mass, 1972.[16] S. H. Reiger, Codes for the correction of clustered errors, IRE Trans.Inform. Theory, IT-6(1960), 16-21.[17] G. E. Sacks, Multiple error correction by means of parity-checks, IRETrans. Inform. Theory IT, 4(1958), 145 - 147.[18] B. D. Sharma, B. Rohtagi, Some Results on Weights of Vectors Having2-Repeated Bursts, Cybernetics and Information Technologies, Vol. 11,No. 1(2011), 36-44.[19] J. J. Stone, Multiple burst error correction, Information and Control,Vol. 4(1961), 324-331.67P. K. Das and V. Tyagi[20] V. Tyagi and P. K. Das, s-Alternate Error Correcting Linear Codes, J. ofCombinatorics, Information & System Sciences, Vol. 35, No. 1-2(2010),17-26.[21] www.themcdonalds.net/richard/index.php?title=Astrophotography−Mounts:−Periodic−Error−Correction.68

Codes on s-periodic errors

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Codes on s-periodic errors

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