BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 9, No 3 Sofa 2009 O Error Detecto wth Block Codes Rostza Doduekova Chalmers Uversty of Techology ad the Uversty of Gotheburg, 41296 Gotheburg, Swede Abstract: I error detecto wth block codes over symmetrc memoryless chaels, the code performace s measured by the probablty of udetected error. Ths probablty depeds o code characterstcs ad o ε, the symbol error probablty of the chael. Whe the udetected error probablty behaves rregularly wth respect to ε, dffcultes arse fdg a code, approprate for error detecto over a chael wth ot exactly kow symbol error probablty (whch s most ofte the case). Good ad proper codes are to be preferred such cases. We preset a survey of kow methods ad techues for the study of block codes wth respect to properess ad goodess, together wth applcatos to famles of block codes, ad some ope problems. Keywords: Error detecto, block code, proper code, good code. 1. Itroducto: Brefly about chaels ad codes Codes are used to cotrol errors, whe formato s trasmtted over osy commucato systems. The system chael may be a telephoe le, a hgh freuecy rado lk, or a satellte commucato lk. The ose may be caused by huma errors, lghtgs, thermal fluctuatos, mperfecto eupmet, etc. I error cotrol, the orgal message s ecoded the begg of the chael by usg codewords. A ecoded message cotas redudat formato, used at the ed of the chael for better recoverg of the orgal message. The geeral model of a commucato system s sketched below. 3
Fg. 1. Basc model of a geeral commucato system The smple example, where the oly messages we wat to trasmt, are YES or NO, s llustrated below. YES s ecoded as 00000 ad NO as 11111. Suppose YES was set ad 01001 was obtaed. There are two basc methods of error cotrol used by the decoder, both agreemet wth the Maxmum Lkelhood Prcple: error detecto ad error correcto. I error detecto the decoder would ask for retrasmsso sce the vector obtaed s ot a codeword. I error correcto the decoder decodes the vector to the earest codeword, whch s 00000, YES. 4 Fg. 2. A smple example where the message source cossts of YES ad NO ad YES s trasmtted Further o we cosder the trasmsso over a -ary symmetrc memoryless chael (SMC). Such a chael has a alphabet wth symbols, each of them remag uchaged durg the trasmsso wth probablty 1 ε ad t may chage to ay of the other 1 symbols wth the same probablty ε/( 1). A atural assumpto for the chael s that t s more lkely for a symbol to rema uchaged durg the trasmsso, tha to be chaged to some other symbol,
whch results the restrcto 0 < ε < ( 1)/. The -ary symmetrc chael s memoryless, f the errors whch occur separate uses of the chael, are depedet. The ext pcture descrbes the mathematcal model of a terary SMC wth alphabet {0, 1, 2}. Fg. 3. A terary SMC wth symbol error probablty ε 2. Block codes A -ary block code s a set of seueces of the same legth (codewords) wth elemets from a fte set F wth elemets. It s well kow, that there exsts a feld GF() over F (the Galos Feld) f ad oly f s a prme power. Let F deote the -dmesoal vector space over GF(). The Hammg dstace betwee two vectors x ad y from F s the umber of o-zero elemets x y. The Hammg weght of a vector s ts dstace to the zero vector. The dstace dstrbuto of a block code C F s a collecto of umbers {A 0,, A }, where A euals the umber of pars of codewords C at dstace, dvded by the umber of all codewords. The smallest postve dstace betwee two codewords C s deoted by d ad called the code dstace. A lear code of dmeso k s a k-dmesoal subspace of F. I ths case d euals the mmum postve weght the code ad A euals the umber of codewords of weght, 0. The dual code of a lear code C s defed as the subspace C F orthogoal to C. Remark 1. Lear -ary codes are ot defed uless s a prme power. However, reasoable -ary codes ca be obtaed from lear codes dfferet ways, for example by omttg all codewords cotag a gve fxed symbol. Remark 2. The restrcto to lear codes s ot a sg of weakess. It turs out that codes that are optmal some way, very freuetly are lear. 5
3. Error detecto wth block codes Let C F be a block code wth M codewords. Ecodg wth C s carred out as follows. After data compresso, the tal source formato s preseted as a seres of symbols from F, whch s dvded to blocks of legth k <. Each block s a message, whch s ecoded a codeword from C. I ths way a message, wrtte k symbols, s after ecodg wrtte symbols, so that k symbols are redudat. Suppose the codeword x was set ad vector y was receved. I error detecto, the decoder accepts y as the codeword set, whe t s a codeword, or asks for a retrasmsso, whe t s ot. Thus trasmsso errors rema udetected oly f the codeword set chages durg the trasmsso to aother codeword. The probablty of udetected error of C s gve by [19, Ch. 2]: ε 1 (3.1) P ( C, ε) = A (1 ), 0. ue ε ε = 1 1 The formula s derved uder the assumpto of eually lkely messages. Ths assumpto s based o the Law of Large Numbers ad s basc C. Shao s fudametal paper Mathematcal theory of commucato from 1948. Aother expresso of the probablty of udetected error of C s MW (3.2) (, ) M 1 ε (1 ) P C ε = A ε ue C 1 where MW 1 1 z A ( z) = (1 ( 1) z) A C M 0 1 + ( 1) z 1 s the MacWllams trasformato [19, Ch. 2]. Whe C s lear (3.2) leads to ( k) ε 1 (3.3) P ( C, ε) = B 1 (1 ε), 0 ε. ue = 0 1 Here {B, 0 } s the weght dstrbuto of the dual code C. Suppose C s a lear code of dmeso k. Ay k matrx, the rows of whch form a bass C, s called a geerator matrx of C. I error detecto wth C, oe stadard way to check f the receved vector s a codeword or ot, s to compute ts scalar product wth a specfed geerator matrx of C. Ths product euals zero f ad oly f the vector s a codeword. Remark 3. I recet years the problem of makg a fast decso f a vector s a codeword or ot, has become stll more mportat coecto wth large data bases ecoded ad stored computers. For dfferet reasos we have to make a fast decso f the formato stored has ot bee substatally corrupted. To check for every sgle vector f t s a codeword or ot, mght be expesve. For ths reaso some stuatos we are cotet wth aswers of the type 75% of the formato s ot destroyed wth suffcetly hgh probablty of beg true, f oly 6
uck effcet algorthms exst for such aswers. These radomzed algorthms ad codes, for whch effcet radomzed algorthms exst, are called locally testable codes. Nowadays just a few codes are kow to be locally testable. Amog these are the shorteed frst-order Reed-Muller codes ad the Reed-Muller codes of costat order. 4. Good ad proper codes For a chael wth symbol error probablty ε, the most approprate for error detecto would be code C wth the smallest possble value of P ue (C, ε). It s dffcult, however, to fd such a code, sce o effcet method for such search exsts. Furthermore, the symbol error probablty ε of the chael s ofte ot kow exactly ad a code foud to be best for some ε may be completely approprate for the real chael. It s reasoable these stuatos to use codes, whch are good or proper. C s good for error detecto f 1 k 1 (4.1) P ( C, ε) P C, ( 1), 0 ε, ue ue = ad C s proper f P ue (C, ε) s a creasg fucto of ε [0, ( 1)/]. Thus a good code performs ay chael at least as well as t does the worst chael wth ε = ( 1)/, ad a proper code s just a good code wth the advatage that t performs better better chaels. I fact, the frst decades of the foudato of Codg Theory (4.1) ths was beleved to be true for ay lear code, but examples later dsproved ths. W o l f, M c h e l s o ad L e v e s u e [20] foud the average of P ue (C, ε) over all -ary lear codes of legth ad dmeso k. The result s a creasg fucto P ue (ε) = ( k) [1 (1 ε) k ]. Thus a hypothetcal average [, k] code would be proper, ad ths sese a proper code just mtates a average error detectg code. Ths s aother strog reaso to prefer a proper error detectg code to a o-proper oe stuatos, where t s mpossble to fd a optmal code (followg the rule to keep to the average, f othg better ca be doe). The cocepts of a good ad a proper code have bee troduced 1979 [18]. After ths, utl 1995, whe the frst moograph o error detectg codes appeared [17], just a few codes have bee studed regardg properess ad goodess. Amog them are the so called Maxmum Dstace Separable (MDS) codes, show to be proper 1984 [15]. 5. Study of block codes wth respect to goodess ad properess To fd out f a sgle code s proper, good, or o-good, we ca use computer graphs or umercal methods for the study of the polyomal represetg the udetected error probablty. Below the graph of the ormed probablty of code C 7
s show, whch s dual to [819, 12, 384] 2 Delsarte-Goethals cyclc code. The code s log ad has hgh dmeso, so oe ca expect ce propertes. Ideed, cosderg the pcture, ths code s proper. However, a proper scalg reveals aother pcture, see Fg. 2. I fact C s a member of a parametrc subclass of ogood Delsarte-Goethals cyclc codes, as show [14]: PC (, ε ) PC (, ε ) =.. 1 k 2 (2 1) Fg. 4. The ormed fucto P(C, ε) It becomes much more complcated whe we wat to vestgate parametrc famles of codes regardg properess ad goodess. Below we wll preset suffcet codtos for goodess or properess, whch have show to be effcet the study of parametrc famles of block codes, together wth some applcatos. Some codtos are expressed terms of basc code parameters ad may volve the so called exteded bomal momets of the code, others are aalytc. It should be metoed, however, that so far the aalytc study of the udetected error probablty fucto for famles of codes has show to be effcet oly a small umber of cases. The exteded bomal momets of a lear [, k, d] code C are syoymously related to ts weght dstrbuto {A 0,..., A } ad are defed as [2] A 0 * = 0, l l * () A = A, l l = 1, 2,,, = 1 () where j () deotes the -th factoral momet j(j 1) (j + 1) of j. I [2, 4, 5], the exteded bomal momets have bee used to study the udetected error probablty fucto, partcular, to obta dscrete suffcet codtos for properess ad goodess. 8
Fg. 5. A scaled pcture shows that C s ot good 6. Suffcet codtos for properess ad goodess ad terms of the weght dstrbuto ad basc code parameters As metoed Secto 4, the MDS codes have bee show to be proper [15]. A MDS code s a lear code whch s dstace optmal,.e., t has the largest code dstace amog the lear codes wth the same legth ad dmeso. Its dual code s MDS as well. Moreover, a lear code of legth s MDS, f ad oly f ts code dstace d ad the dual code dstace d satsfy d + d = + 2. Also, for ay o-mds code, we have d + d. The ext two theorems preset suffcet codtos for properess ad goodess of lear o MDS codes [4, 5, 2]. Theorem 6.1. Let C be a [, k, d] lear code wth d + d. The: () f the exteded bomal momets of C satsfy * * (6.1) A l A l 1, l = d + 1,, d + 1, the C s proper; () f the exteded bomal momets of the dual code satsfy * * (6.2) B l B l+1 k 1 ( 1), l = d + 1,, d + 1, the C s proper. The exteded bomal momets of a lear code are strctly creasg so that (6.1) s just a codto o the rate of crease. Though (6.1) ad (6.2) are euvalet, (6.2) s more effcet stuatos where the dual code dstace or the umber of o-zero weghts the dual code are small. Theorem 6.2. Let C be a [, k, d] code wth d + d. The: () f the exteded bomal momets of C satsfy 9
(6.3) A l * l ( k 1), l = d,, d, the C s good; () f the exteded bomal momets of the dual code satsfy (6.4) +l B * l k k+l, l = d,, d, the C s good. As above, the dual codtos (6.4) are more effcet stuatos, where the dual code dstace or the umber of o-zero weghts the dual code are small. Applcatos. The MDS codes are dstace optmal ad proper. Next optmalty are Near MDS (NMDS) codes, the the Maxmum Mmum Dstace (MMD) codes. The dual of a NMDS code s NMDS as well. May NMDS codes tur out to be proper, by Theorem 6.1; some are good by Theorem 6.2 [3], [8]. The MMD codes ad ther duals tur out to be proper [6], [1]. All uue optmal bary lear codes of dmeso at most seve ad ther dual codes are proper [11]. Also, may Cyclc Redudacy-Check codes (CRC) are proper or good, by the above theorems, but some stadardzed such codes are o-good [16]. 7. Suffcet codtos for properess terms of basc parameters Computato of the weght dstrbuto of a ler code s a NP hard problem. As a result, relatvely few codes are kow wth ther weght dstrbuto. For ths reaso, to have suffcet codtos for properess, ot volvg the code weght dstrbuto, would be very useful. Theorem 7.1. Suppose C s a -ary lear code of legth, code dstace d ad dual code dstace d. If max(dd ) [( 1) + 1]/, the both C ad ts dual code are proper [13]. Applcatos. Parametrc famles of Gresmer codes tur out to satsfy the above theorem [12, 13]. A Gresmer code s a lear code whch s legth optmal,.e., t has the smallest legth amog the lear codes wth the same dmeso ad code dstace. 8. Properess ad goodess tervals I our work we have ofte ecoutered codes, for whch the probablty of udetected error has extrema a relatvely small terval [0, a], ad the becomes a creasg fucto up to the edpot ε = ( 1)/. We call such a code proper [a, ( 1)/]. The followg results have bee show [13]. Theorem 8.1. Let C be a -ary lear code of legth ad dual code dstace d. If ( 1) + 2 ( 1) + 1 (8.1) d, + 1 10
the C s proper the terval (8.2) ( 1) d 1 1 +,. d 1 ( 1) d + 1+ 1 Corollary. Suppose the codto of Theorem (8.1) holds. If also ( 1) d + 1 d (8.3). d 1 ( 1) d + 1+ 1 the C s proper. Applcatos. The above Theorem 8.1 ad ts Corollary tur out to work well for parametrc famles of Gresmer codes, see [12, 13]. I all examples of terval the C s proper the terval properess cosdered these works the codes are also asymptotcally proper ad have small redudacy. Sce codes wth small redudacy are tesely used error detecto, such examples mght be of practcal terest. 9. Aalytc methods Ufortuately, at the preset tme we are ot aware of route aalytc methods for the study of parametrc polyomals represetg the udetected error probablty of parametrc famles of codes, ad developmet of such methods would of course be a challege. Below we preset two theorems, whch are obtaed by aalytc study of the udetected error probablty fucto. Theorem 9.1 [10]. A bary block code of legth ad code dstace d wth d ad symmetrc dstace dstrbuto (A = A ) s proper. 2 Theorem 9.2 [7]. Let C be a lear [, k, d] code ad assume that for some ε 0 (0, ( 1)/) we have ( k) (9.1) P ( C, ε ). ue 0 The C s ot good. Applcatos. Theorem 9.1 was used [10] ad to sow that some Kerdock codes ad the Preparata codes are proper. These are perhaps the frst examples of proper o-lear block codes. The Kerdock ad the Preparata codes seem to be of a permaet theoretcal appeal because of ther terestg algebrac-combatoral propertes. Also, t follows from Theorem 9.1, that bary self-complemetary block codes, lear ad o-lear, whch satsfy the so called Grey Rak boud are proper. 11
A code for whch (9.1) holds, s called ugly. Obvously, a ugly code s ogood. Accordg to theorem 9.2 whe a code s ugly, ts dual s o-good. We made use of ths result [7], where a full classfcato s gve wth respect to properess ad goodess of a parametrc class of -ary cyclc codes ad ther dual codes. 10. Lst of proper codes More detals regardg the lst below ca be foud [9]: All Perfect codes over fte felds; Some Reed-Muller codes; Some BCH codes; The MDS codes; The MMD codes ad ther duals; Some NMDS codes; Some CRC codes; Some Gresmer codes; The uue optmal bary codes of dmeso at most seve ad ther dual codes. 11. Ope problems There are may terestg uestos related to good ad proper codes. We wll meto two of them. Claude Shao proved that codes exst for relable trasmsso of formato at ay rate below the chael capacty, but dd ot provde a costructo of a optmal code. Eve today, t s ot kow what a optmal code looks lke. Istead, the efforts are devoted to the search for codes, for whch the performace error cotrol s effcet oe sese or aother. As a result, codes may be optmal may dfferet ways. Of greatest terest are the codes, whose parameters are some sese extremal, lke the MDS ad Gresmer codes. Our studes have show that may lear codes, whch are optmal some sese, or close to optmal, are also proper, ad most ofte ther dual codes are proper, too. It s atural to ask f properess ad optmalty are closely related propertes. If so, what kd of relatoshp would ths be? Aother terestg uesto s to compare the error detectg performace of a proper code wth the performace of a average code. I the case of bary lear codes our experece shows that a proper code s ever worse tha a average code. If ths was geeral, t would have a strog mpact o the theory ad practce commucatos. 12
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