ERROR LOCATING CODES AND EXTENDED HAMMING CODE. Pankaj Kumar Das. 1. Introduction and preliminaries

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1 MATEMATIČKI VESNIK MATEMATIQKI VESNIK 70, 1 (2018), March 2018 research paper originalni nauqni rad ERROR LOCATING CODES AND EXTENDED HAMMING CODE Pankaj Kumar Das Absrac. Error-locaing codes, firs proposed by J. K. Wolf and B. Elspas, are used in faul diagnosis in compuer sysems and reducion of he reransmission cos in communicaion sysems. This paper presens locaing codes obained from he famous Exended (8, 4) Hamming code capable of idenifying he sub-block ha conains solid burs errors of lengh 2 (or 3) or less. We also make a comparison of informaion rae beween he exended Hamming code and obained codes. Furher, comparisons in solid burs error deecion and locaion probabiliies of he codes over binary symmerical channel are also provided. 1. Inroducion and preliminaries Wolf and Elspas [14] inroduced a midway concep (known as error locaion coding) beween error deecion and error correcion. Error locaing (EL) codes have been found o be efficien in feedback communicaion sysems. In such sysems, he whole code lengh is divided ino some finie number of sub-blocks which are muually exclusive. Each sub-block of received digis is invesigaed for he presence of errors. If error is occurred wihin a sub-block, hen he code has he capaciy o locae he corruped sub-block and he receiver can reques he reransmission of he corruped sub-block insead of he whole block, and his process is repeaed for each incoming sub-block. In order o send large amoun of daa, long code lengh is desired o increase coding efficiency and which in urn resuls in a low informaion rae. The use of EL codes sofens his deficiency by dividing long code lengh ino smaller subblocks and mainain he sysem o keep he informaion rae up. Some of very recen works on error locaing codes may be found in [5 7]. A good amoun of work dealing wih deecing and locaing random/burs error can be found in [8] (specially Chaper 6 and Chaper 9). The ype of error occurred on communicaion channel depends on he behaviour of channel. Solid burs error is one ype of error commonly found in many memory 2010 Mahemaics Subjec Classificaion: 94B05, 94B20, 94B60 Keywords and phrases: Pariy check marix, solid burs, syndrome, EL-codes, informaion rae, error probabiliy. 89

2 90 Error locaing codes and exended Hamming code communicaion channels viz. semiconducor memory daa, supercompuer sorage sysem [1 3, 11]. A solid burs may be defined as follows. Definiion 1.1. A solid burs of lengh b is a vecor whose all he b-consecuive componens are nonzero and res are zero. In wha follows an (n, k) linear code is a proper subspace of n-uples over GF (q). The block of n digis, consising of k informaion digis and n k pariy check digis, is divided ino s muually exclusive sub-blocks. Each sub-block conains = n s digis. The informaion rae (daa rae) of an (n, k) linear code is k n. We consider (n, k) linear codes over GF (q) ha are capable of deecing and locaing all solid burss of lengh b or less wihin a single sub-block. Such an EL-code capable of idenifying a single corruped sub-block conaining solid burs of lengh b or less mus saisfy he following condiions: (a) The syndrome resuling from he occurrence of a solid burs of lengh b or less wihin any one sub-block mus be disinc from he all zero syndrome. (b) The syndrome resuling from he occurrence of any solid burs of lengh b or less wihin a single sub-block mus be disinc from he syndrome resuling likewise from such errors wihin any oher sub-block. The paper [4] sudied codes ha deec and locae all solid burss of lengh b or less. The bounds on pariy check digis for he exisence of codes are obained. This paper presens linear codes ha are capable of deecing and locaing (i) all solid burss of lengh 2 or less (ii) all solid burss of lengh 3 or less. The codes are obained from he famous Exended (8, 4) Hamming code (refer [10], also [13, pp ]). The sudy of his paper is moivaed by he work done by Kai [12] where rearrangemen of he columns of he pariy marix of a sysemaic (16, 8) code (refer Gulliver and Bhargava [9]) gives rise o a code wih beer error deecion and correcion. We also give a comparison of informaion raes beween he Exended Hamming code and our second ype of codes. Furher, we also provide comparisons of solid burs error deecion and locaion probabiliies among hese codes over a binary symmerical channel. 2. Code Consrucion We sar wih he binary (8, 4) exended Hamming code ha can correc all single errors and deec all double errors. The pariy check marix H of he code is given by H = This code can no locae double errors. Now we consruc a linear codes ha is divided ino wo sub-blocks and is capable of locaing solid burs of lengh 2 or less

3 P. K. Das 91 occurring wihin a sub-block. We rearrange he columns of H as follows and rename i H 1 : H 1 = The null space of he marix H 1 is a binary (4 + 4, 4) linear code and is capable of deecing and locaing all solid burss of lengh 2 or less wihin a sub-block. This is because he condiions (a) and (b) are saisfied, i.e. he syndromes of all solid burss of lengh 2 or less are nonzero and disinc wihin one sub-block, furher he syndromes of such errors wihin one sub-block are disinc from he syndrome resuling likewise from any such errors wihin he oher sub-block. I can be easily verified from he error paern-syndrome able (as done in [4]). For locaion of errors, we proceed as follows. If he syndrome of any solid burs of lengh 3 or less is any one of he following: 0001, 0010, 0110, 1100, 0011, 0101, 1001, hen he error can be locaed in he firs sub-block. Again if he syndrome of such error is any one of he uples 1011, 1111, 0111, 1101, 0100, 1000, 1010, hen he locaion of he error is he second sub-block. We now again give anoher consrucion from H and which gives rise o a class of linear codes ha are capable of locaing solid burss of lengh 3 or less. Rearrange he columns of he marix H as [h 1 h 2 h 3 h 5 h 4 h 7 h 6 h 8 ] and hen repea he firs wo columns h 1, h 2 alernaively imes as h 1 h 2 h 1 h 2 h 1 h 2... and consider as he firs sub-block, again repea he nex wo columns h 3, h 5 alernaively imes for he second sub-block and so on for oher wo pair of columns (h 4, h 7 ), (h 6, h 8 ) for he hird and fourh sub-blocks. Then he resuling marix will give rise o a class of binary (4, 4 4) ( 3) linear codes. The new marix H 2 is given as follows: H 2 = [ h 1 h 2 h 1 h 2 h 3 h 5 h 3 h 5... h 4 h 7 h 4 h or, H 2 = h 6 h 8 h 6 h 8... ] The null space of he marix H 2 will deec and locae all solid burss of lengh 3 or less. This claim is also rue as we can verify ha he syndromes of all solid burss of lengh 3 or less are being nonzero and disinc wihin one sub-block, furher he syndromes of such errors wihin one sub-block are disinc from he syndrome resuling likewise from any such errors wihin any oher sub-block. For locaion of errors, we proceed as follows. If he syndrome of any solid burs of lengh 3 or less is any one of he following: 0001, 0011, 0010, hen he code will locae he error in he firs sub-block. Again if he syndrome of such error is any one of he uples

4 92 Error locaing codes and exended Hamming code 0101, 1001, 1100, hen he locaion of he error is he second sub-block. In he same way, if he syndrome is 0111 or 1101 or 1010, he fauly sub-block is he hird subblock. If syndrome is any of 1011, 1111, 0100, he sub-block wih errors is he fourh one. 3. Comparisons of codes wih respec o informaion rae and error probabiliy An error deecing code can only deec he presence of errors in he received vecor, whereas an error locaing code can also indicae he posiion of error and furhermore an error correcing code can correc he errors presen in he received vecor. As he purpose of he hree ypes of codes o handle errors is varying, so differen ypes of codes are o be consruced accordingly, bu in he consrucion of codes, one has also o keep in mind he informaion rae. The more is he informaion rae, he more is he speed of he sysem which ransmis he daa. Furher, as no all errors can be deeced/locaed, so i is always imporan o know he probabiliy of errors going undeeced/unlocaed despie he use of error deecion/locaion scheme. In his secion, we esablish a comparison of informaion raes among he exended (8, 4) Hamming code, he (4+4, 4) code and he new class of (4, 4 4) codes. Then, a comparaive sudy of solid burs error undeecion and unlocaion probabiliy of hese codes is followed. In Table 1 below, we pu he informaion raes of he (4, 4 4) codes for differen values of. As he informaion rae of he exended (8, 4) Hamming code as well as he (4 + 4, 4) code is 0.5, we can conclude ha he new class of (4, 4 4) codes has beer informaion rae han he exended (8, 4) Hamming code or (4 + 4, 4) code. For comparison of solid burs error undeecing/unlocaing probabiliy of he codes, le us consider a binary symmerical channel (BSC) wih error probabiliy p. For he (8, 4) exended Hamming code, we see ha he code can deec any solid burs of lengh 3 or less and probabiliy ha solid burs goes undeeced is 5p 4 (1 p) 4 + 4p 5 (1 p) 3 + 3p 6 (1 p) 2 + 2p 7 (1 p) 1 + p 8 = 5p 4 (1 p) 4, we can ignore oher erms for small value of p. Furher, his code can locae only single errors wihin a sub-block of lengh 4, so he probabiliy ha solid burs of lengh 2, 3, 4 can no be locaed wihin a sub-block is 6p 2 (1 p) 8 + 4p 3 (1 p) 5 + 2p 4 (1 p) 4 = 6p 2 (1 p) 8. For he (4 + 4, 4) code, he code can deec solid burss of lengh 3 or less, so he probabiliy ha solid burs goes undeeced is same as ha of he (8, 4) exended Hamming code i.e. = 5p 4 (1 p) 4. Bu he code can locae solid burss of lengh 2 or less occurring wihin a sub-block of lengh 4, he probabiliy ha solid burs error of lengh 3 and 4 goes unlocaed is 4p 3 (1 p) 5 + 2p 4 (1 p) 4 = 4p 3 (1 p) 5. For he (4, 4 4) code, i can also deec solid burss of lengh 3 or less, so he probabiliy ha solid burs goes undeeced is given by (4 3)p 4 (1 p) (4 4)p 5 (1 p) (4 5)p 6 (1 p) (p) 4 = (4 3)p 4 (1 p) 4 4. As solid burss of lengh 3 or less occurring wihin a sub-block of lengh can be locaed, so he probabiliy of no able o locae solid burs errors of lengh 4 upo by he code is

5 P. K. Das 93 4 { ( 3)p 4 (1 p) 4 4 +( 4)p 5 (1 p) 4 5 +( 5)p 6 (1 p) (p) (1 p) 3} = 4( 3)p 4 (1 p) 4 4. Le us assume he value of p is In Table 1 he probabiliies of solid burs error going undeeced and unlocaed by he (4, 4 4) codes for differen values of are lised. Table 1: The probabiliies of solid burs error going undeeced and unlocaed by he (4, 4 4) codes Informaion rae for he (4, 4 4) codes (appr. value) Solid burs error undeecing probabiliy for he (4, 4 4) codes (appr. value) for p = 0.01 Solid burs error unlocaing probabiliy for he (4, 4 4) codes (appr. value) for p = The probabiliy ha solid burs goes undeeced for he (8, 4) exended Hamming code or for he (4 + 4, 4) code for p = 0.01 is same i.e Bu he probabiliy ha solid burs goes unlocaed for he (8, 4) exended Hamming code is and he probabiliy ha solid burs goes unlocaed for he (4 + 4, 4) code is Thus, for locaion poin of view of solid burs, he (4 + 4, 4) code is a beer code han he (8, 4) exended Hamming code. Furher, from he able we can say ha (8, 4) exended Hamming code or he (4+4, 4) code has beer deecion rae of solid burs error han (4, 4 4) codes, bu (4, 4 4) codes has beer locaion rae of solid burs error han he (4 + 4, 4) code as well as (8, 4) exended Hamming code. Therefore, his new class of binary (4, 4 4) codes will be more useful if he purpose is o deec and locae solid burs error. Remark 3.1. Alhough he value solid burs error unlocaing probabiliy of (4, 4 4) codes is increasing which can be seen in Table 1, bu is solid burs error unlocaing probabiliy is always lesser han ha of he (4 + 4, 4) code. This is because of 4( 3)p 4 (1 p) 4 4 < 4p 3 (1 p) 5 i.e. ( 3)p(1 p) 4 9 < 1, for 3 and small value of p. We can verify his by Excel Sofware. 4. Conclusion This paper gives he consrucion of EL codes ha can deec and locae solid burs errors of 2 (or 3) or less. The obained class of (4, 4 4) codes is found o have beer informaion rae, locaing capabiliy, error locaion rae for solid burs error

6 94 Error locaing codes and exended Hamming code poin of view. One may work on o obain EL codes based on oher sandard codes ha can deec and locae solid burss of lengh b(> 3) or less. Acknowledgemen. The auhor is very much hankful o anonymous referee(s) for careful reading of he manuscrip and for valuable suggesions which improves he paper a lo. References [1] C. A. Argyrides, P. Reviriego, D. K. Pradhan, J. A. Maesro, Marix-based codes for adjacen error correcion, IEEE Transacions on Nuclear Science, 57 (4) (2010), [2] J. Arla, W. C. Carer, Implemenaion and evaluaion of a (b, k)-adjacen errorcorrecing/deecing scheme for supercompuer sysems, IBM J. Res. Develop. 28 (2) (1984), [3] D. C. Bossen, b-adjacen error correcion, IBM Journal of Research and Developmen, 14 (4) (1970), [4] P. K. Das, Codes deecing and locaing solid burs errors, Romanian Journal of Mahemaics and Compuer Science, 2 (2) (2012), [5] P. K. Das, L. K. Vashish, Error locaing codes by using blockwise-ensor produc of blockwise deecing / correcing codes, Khayyam J. Mah., 2 (1) (2016), [6] B. K. Dass, S. Madan, Repeaed low-densiy burs error locaing codes, Aca Universiais Apulensis, 33 (2013), [7] Ji-Hao Fan, Han-Wu Chen, A consrucion of Quanum Error-Locaing codes, Communicaions in Theoreical Physics, 67 (1) (2017), [8] F. Fujiwara, Code Design for Dependable Sysems, John Willey & Sons, [9] T. A. Gulliver, V. K. Bhargava, A sysemaic (16, 8) code for correcing double errors and deecing riple-adjacen errors, IEEE Trans. Compuers, 42 (1) (1993), [10] R. W. Hamming, Error-deecing and error-correcing codes, Bell Sysem Technical Journal, 29 (1950), [11] D. W. Jensen, Block code o efficienly correc adjacen daa and/or check bi errors, Paen number: US B1, Dae of Paen Aug 5, 2003, ( [12] R. S. Kai, Commens on a sysemaic (16, 8) code for correcing double errors and deecing riple-adjacen errors, IEEE Transacions on Compuers, 44 (12) (1995), [13] W. W. Peerson, E. J. Weldon(Jr.), Error-Correcing Codes, 2nd ediion, The MIT Press, Mass, [14] J. Wolf, B. Elspas, Error-locaing codes a new concep in error conrol, IEEE Transacions on Informaion Theory, 9 (2) (1963), (received ; in revised form ; available online ) Deparmen of Mahemaical Sciences, Tezpur Universiy, Napaam, Sonipur, Assam , India pankaj4hapril@yahoo.co.in, pankaj4@ezu.erne.in