On Probability of Undetected Error for Hamming Codes over Q-ary Symmetric Channel

Transcription

1 Joural of Coucato ad Coputer 8 ( O Probablty of Udetected Error for Hag Codes over Q-ary Syetrc Chael Mash Gupta, Jaskar Sgh Bhullar 2 ad O Parkash Vocha 3. D.A.V. College, Bathda 5, Ida 2. Malout Isttute of Maageet ad Techology, Malout 527, Ida 3. Ferozepur College of Egeerg ad Techology, Ferozepur 4252, Ida Receved: October 3, 2 / Accepted: Deceber, 2 / Publshed: Aprl 3, 2. Abstract: The probablty of udetected error Pu( ε for Hag codes of paraeter [,, 3] whe trastted over -ary syetrc chael s exaed. The perforace of a lear t-error correctg code over a -ary syetrc eoryless chael wth sybol error probablty Ε s characterzed by the probablty that a trassso error wll rea udetected. I past papers, t has bee show that probablty of udetected error Pu( ε for a bary (N = 2, K = N M, 3 Hag code (Q = 2 used for error detecto o bary syetrc chael satsfes the 2 -p boud, where P s the party check bts eual to N K, hece bary Hag codes are proper. I ths correspodece ths result s geeralzed ad t has bee show that ot oly bary but Hag Codes (for ay value of Q satsfy ths boud, so geeralzed Hag codes are proper. Key words: Hag code, udetected error probablty, BSC, cwlla detty, proper codes.. Itroducto I autoatc-repeat-reuest (ARQ error-cotrol syste, the udetected error probablty of a error-detectg code s oe of the ost portat perforace characterstcs. There are a uber of papers dedcated to exag the error detecto capablty for soe well kow classes of lear codes, such as Reed-Soloo codes, BCH codes ad Reed-Muller codes. For a geeral troducto to the theory of error detectg codes, we refer the readers to Klove [] ad ts refereces. A (, k lear block code, where s the block legth ad k s the uber of forato bts, wth u dstace d ca be used three ways for Jaskar Sgh Bhullar, Ph.D., assocate professor, research felds: forato theory, codg theory. Correspodg author: Mash Gupta, M.Sc., M. Phl, assstat professor, research felds: codg theory. E-al: ash_guptabt@yahoo.co. cotrollg trassso error a data coucato syste purely for error detecto, purely for error correcto or a cobato of error detecto ad correcto. If a code s solely for error detecto, ay cobato of d or fewer errors are guarateed t be detectable. If the code s used solely for error correcto, ay cobato of d or fewer 2 errors are guarateed to be correctable, where [x] s defed as greatest teger less tha or eual to x. I the last few years a uber of authors [2-6] have dscussed the udetected error probablty, P u (ε, of lear (, k block codes used solely for error detecto o a bary syetrc chael (BSC wth bt error rate ε. Most, but ot all, of the work reported the lterature regardg the udetected error probablty s restrcted to bary lear codes. Although research related to the udetected error probablty o the BSC s very portat fro the forato-theoretc

2 26 O Probablty of Udetected Error for Hag Codes over Q-ary Syetrc Chael vewpot, ts practcal value s restrcted by the fact that the BSC does ot always adeuately descrbe real coucato chaels [7]. The Glbert-Ellott ad Frtcha chael odels are exaples of foral odels used to aalyze the characterstcs of real chaels wth eory [8-]. The aalyss ad calculato of the udetected error probablty o such chaels becoe ore coplcated fro a aalytcal ad coputatoal pot of vew whe copared wth the sae proble for the BSC. ERROR detecto s used extesvely coucato ad coputer systes to cobat ose. Detecto s accoplshed by exag the receved word. If t s a codeword, the word s accepted as error-free. If t s ot a codeword, the word s rejected as beg erroeous. Further processg ay be eeded ths case to retreve the trastted forato. The udetected error occurs f a error-detectg schee fals to detect a error,.e., f the receved word s a codeword dfferet fro the trastted codeword. The probablty of udetected error s gve by Ref. [] ε Pu( ε = A ( ε, < ε < ( = where A s the uber of code words of weght code. For a code wth u dstace d A s zero for < < d. For =, A =. Aother forula to copute P u (ε s [2] ( k Pu( ε = B( Qε ( ε (2 = where Q = ad < ε < where B s the uber of code words of weght dual of code. Code C s called good f M Pu( ε Pu = (3 where M s uber of forato ad a code s proper f Pu ( ε s a creasg fucto for ε,. Proper codes are fe for error detecto. If ( k Pu ( ε for < ε <, the code s called ( k satsfyg boud. The code ot satsfyg ( k boud s ot fe for error detecto [4]. Upper boud o udetected error probablty for optal lear codes s also studed Refs. [4, 3-4]. It was beleved that ths upper boud holds for all codes sce t was assued that Pu ( ε s creasg for ε, ad Pu ( ε attas ts axu value at ε =. However, ths assupto was show to be wrog by soe codes that do ot obey the upper boud (see Refs. [2, 3, 5] soe classes of codes are kow to obey ths boud. To classfy codes as proper, o proper but good, or ot good, ofte turs out to be coplcated, ad such a classfcato has bee doe so far for relatvely few codes. It s worth etog that ay codes whch are kow to be optal or close to optal oe sese or other, tur out to be proper, such as Maxu Dstace Separable (MDS codes, the Hag codes, the Maxu Mu Dstace codes ad ther duals, etc.. For relevat forato ths regard we refer to the overvew [6]. Ths paper establshes that ot oly bary, but all Hag codes.e. for 2 ad 3 are proper whe trastted over ary syetrc chael wth ε as bt error probablty, where ε rages fro < ε <. So, Hag codes satsfy boud. ( k 2. Probablty of Udetected Error for Geeralzed Hag Codes Ths secto aalyses the properess of bary [, 3] splex code o bary syetrc chael wth sall bt error probablty ε by usg weght dstrbuto of splex code whch s kow.,

3 O Probablty of Udetected Error for Hag Codes over Q-ary Syetrc Chael 26 Theore: Paraeters of Hag codes are =, k = ad d = 3. Hag codes are proper for all values of. Proof: Weght Dstrbuto of Geeralzed Hag (, k code s gve by Ref. [7] ( + A x = + x + + x x ( + where = ( ( ( ( ( (, k = = = A x = Ax = + Ax (4 A( x = Ax Sce A = (5 = Probablty of Udetected Error of lear (, k code ca be expressed as ε P( ε = A ( ε ( = ε = ( ε A = ( ε ( ε = ( ε A ( ε ( P ( ε = ( ε +. ( + ε ( (. ( ( + + ε ( + ε ( ( ( ε ε ( ( + ( ( ε ( ε = ε ε + + ( +. ( + ε ε ( + ε ( ( + ε + ε = + ( ε ( + ( + {( + }( ε (6 Dfferetate both sdes w.r.t. ε + dp( ε ( + = ( ε. ( + + {( + }( ε ( + = ( ε ε (7 We wll ow prove that dp >, by provg ( ( > ε ε < ε ( ε ( ε > ε ( ( ( ε > ε ( Q > dp > for < ε < Ths cocludes the theore that probablty of udetected error creases the terval < ε <, ( k so Hag codes have the upper boud ad so are proper. The udetected error rate curves ad tables for dfferet values of ad also for = 2, 3, 4, 5, 6, 7 ad 8 are gve by followg tables ad graphs. 2. Case I for = 2 Table copares the probablty of udetected error for bary Hag codes wth 2 -p, whch also shows that bary hag codes satsfy 2 -p boud. 2.2 Case II for = 3 The udetected error curves for = 2, 3, 4, 5 ad 6 are show Fg., they do also satsfy 3-p boud eve though for large values of. The values of dp ad 3 -p for = 2, 3, 4, 5 ad 6 are lsted Table 2.

4 262 O Probablty of Udetected Error for Hag Codes over Q-ary Syetrc Chael Table Coparso of P ( ε ad 2 -p of Hag code for = 2. P ( ε 2 -p =3.7875E-.25E- = E E-2 =5 3.25E E-2 =6.5625E E-2 = E E-3 = E E-3 Udetected Error Probabty.7.63 = = =4.7 =2,=3 =364,= Bt Error Probablty Fg. Udetected error probablty for = 3. Table 2 Coparso of P ( ε ad 3 -p of Hag code for = 3. P ( ε 3 -p =2 6.25E-2.E E E-2 = E E-2 = E E-3 = E E Case III for = 4 The udetected error curves for = 2, 3, 4, 5 ad 6 for Hag codes are show Fg. 2, they do also satsfy 4-p boud eve though for large values of. The values of dp ad 3-p for = 2, 3, 4, 5 ad 6 are lsted Table Coclusos Probablty of udetected errors for Hag codes have bee exaed ths paper. Frstly we proved that Hag codes of ay legth whe trastted over -ary chael are proper codes. The t has bee Udetected Error Probablty = = =85.3 =34 = Bt Error Probablty Fg. 2 Udetected error probablty for = 4. Table 3 Coparso of P ( ε ad 4 -p of Hag code for = 4. P ( ε 4 -p = E E-2 = E E-2 = E E-3 = E E-4 = E E-4 show that Hag codes always obey -p boud o P ( ε for < ε <, however t ot essetally true for other cyclc codes. Referece [] T. Klove, V.I. Korzhk, Error Detectg Codes: Geeral Theory ad Ther Applcato Feedback Coucato Systes, Kluwer Acad. Press, Bosto, 995. [2] C. Leug, M.E. Hella, Cocerg a boud o udetected error probablty, IEEE Tras. Ifor. Theory 22 ( [3] C. Leug, E.R. Bares, D.U. Freda, O soe propertes of the udetected error probablty of lear codes, IEEE Trus. Ifor. Theory 25 ( [4] J.K. Wolf, A.M. Mchelso, A.H. Levesue, O the probablty of udetected error for lear block codes, IEEE Tras. Cou. 3 ( [5] T. Kasa, T. Klove, S. L, Lear block codes for error detecto, IEEE Tras. Ifor. Theory 29 ( [6] T. Kasa, S. L, O the probablty of udetected error for the axu dstace separable codes, IEEE Tras. Cou. 32 (

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