Insertion/Deletion Correction with Spectral Nulls

Transcription

1 7 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO., MARCH 1997 Isertio/Deletio Correctio with Spectral Nulls H. C. Ferreira, Member, IEEE, W. A. Clarke, A. S. J. Helberg, K. A. S. Abdel-Ghaffar, ad A. J. Ha Vick Abstract Leveshtei proposed a class of sigle isertio/deletio correctig codes, based o the umber-theoretic costructio due to Varshamov ad Teegolt s. We preset several iterestig results o the biary structure of these codes, ad their relatio to costraied codes with ulls i the power spectral desity fuctio. Oe surprisig result is that the higher order spectral ull codes of Immik ad Beeker are subcodes of balaced Leveshtei codes. Other spectral ull subcodes with similar codig rates, may also be costructed. We furthermore preset some codig schemes ad spectral shapig markers which alleviate the fudametal restrictio o Leveshtei s codes that the boudaries of each codeword should be kow before isertio/deletio correctio ca be effected. Idex Terms Isertio/deletio correctio, spectral ulls, costraied codes, balaced codes. I. INTRODUCTION I 1965, Varshamov ad Teegolt s [1] proposed the followig code costructio to correct a sigle asymmetrical error o a chael where the probability of the symbol oe turig ito a zero is cosiderably less tha the probability of a zero turig ito a oe, or vice versa. Let x (x1;x; 111;x),xif0;1gdeote a biary codeword ad C the codebook. Varshamov ad Teegolt s required that ixi a(mod m) (1) x C () for some fixed itegers a ad m, where m +1. Note that i (1), the -dimesioal vector space is partitioed ito m differet codebooks, all havig the desirable error correctio capability ad we shall deote each of these codebooks with C(; m; a). We shall refer to the costructio i (1) ad other similar costructios as umber-theoretic code costructios. I 1966, Leveshtei [] described two classes of codes capable of correctig a sigle isertio or deletio error i a codeword. Such a error results i the deletio of a bit i a radom positio, or the isertio of a radom bit i a radom positio, ad it, respectively, chages the legth of the received word to 0 1 or +1. Briefly, Leveshtei oted that by settig m +1 i (1), a sigle isertio/deletio error ca be corrected uder the assumptio Mauscript received December 6, 1994; revised July 5, This work was supported i part by the South Africa Foudatio for Research Developmet, by the Deutscher Akademischer Austauschdiest, ad the Natioal Sciece Foudatio uder Grat NCR The material i this correspodece was preseted i part at the 6th Joit Swedish Russia Iteratioal workshop o Iformatio Theory, Mölle, Swede, August 3 7, 1993; at the 1994 IEEE Iteratioal Symposium o Iformatio Theory, Trodheim, Norway, Jue 7 July 1, 1994; ad at the 1996 Iteratioal symposium o Iformatio Theory ad Its Applicatios, Victoria, Caada, September 17 0, H. C. Ferreira, W. A. Clarke, ad A. S. J. Helberg are with the Departmet of Electrical ad Electroic Egieerig, Rad Afrikaas Uiversity, P.O. Box 54, Auckladpark, 006, South Africa. K. A. S. Abdel-Ghaffar is with the Departmet of Electrical ad Computer Egieerig, Uiversity of Califoria, Davis, CA USA. A. J. Ha Vick, is with Istitut für Experimetelle Mathematik, Uiversität GHS Esse, 4536, Esse, Germay. Publisher Item Idetifier S (97)0187-X. that the boudaries of each codeword, i.e., the locatio of bits x1 ad x, are kow. This sigle isertio/deletio correctio capability is deoted with the parameter s 1. Furthermore, it is show i [] that for m, either a sigle isertio/deletio error or a sigle reversal error (i.e., modulo additive error) ca be corrected, thus s 1or t 1, if parameter t deotes the reversal error-correctio capability. Leveshtei furthermore showed that the cardiality of the first class of codes is lower-bouded by jc(; m; a)j +1 : () Subsequetly, Gizberg [3] proved that the code cardiality ca be maximized by settig a 0ad miimized with a 1. The cocept of a subword obtaied whe deletig bits from a codeword also plays a importat role whe ivestigatig the correctio of deletios/isertios []. If a code corrects s isertios/deletios, the followig restrictio is imposed o the legth of the largest commo subword (x; y) y obtaied from codewords x ad y: j(x; y)j <0 s: (3) For the purpose of this correspodece, let us retur to (1). We ote here that first the series ixi, which sums all idices i where xi 1, represets the codeword momet, also more precisely referred to as the first-order momet [4]. By cosiderig the iteger codeword momet ixi, before it is reduced modulo m, as is doe i (1) ad i most ivestigatios, we preset some ew isight i the biary structure of Leveshtei s codes i Sectio II. For related work refer to [5] [9]. II. THE BINARY STRUCTURE OF LEVENSHTEIN S INSERTION/DELETION CORRECTING CODES I the rest of this correspodece we ivestigate Leveshtei s first class of codes i []. Uless otherwise idicated, we set m +1 ad a 0 to maximize the code cardiality. Usig the otatio itroduced i Sectio I, the results i this sectio thus maily pertai to the structure of Leveshtei s C(; +1; 0) class of codes, although some geeralizatios to C(; m; a), m 6 +1,a60, are also preseted. I later sectios, we show that costat-weight w subcodes of the C(; +1;0) codes, which we shall deote by C(; +1; 0; w), are also importat for other codig applicatios, such as the creatio of spectral ulls. We shall furthermore apply the results of this sectio i ew code costructios ad codig schemes. Let ad The ad sice (x1; x;111;x)c(; +1;a) (4) ixi : (5) ( +1) 0 (6) a mod ( +1) (7) belogs to a fiite set of at most bc +1itegers that differ by at least +1. Note that if m +1 ad a 0, the all-zeros codeword is icluded i the codebook. If, furthermore, is eve, the all-oes codeword is also icluded i C(; +1;0) /97$ IEEE

2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO., MARCH First, we are iterested i the Hammig weight structure of Leveshtei s first class of codes. Usig the theory of iteger partitios [10], a simple geeratig fuctio ca be set up to evaluate the umber of codewords of weight w. This follows from the observatio that each ozero codeword ca be associated with a partitio with o repeated parts, ad o part greater tha, of the iteger, which represets the ureduced codeword momet. Cosequetly, lettig t i represet the iteger part, or codeword idex i, ad u oe uit of weight, we ca set up the followig geeratig fuctio for determiig the weight spectrum: f (u; t) (1 + ut i ): (8) The umber of codewords N (w) of weight w ca be determied by summatio of the coefficiet(s) of u w t, 0 (+1), i (8). Several previous workers ivestigated the cardiality ad weight spectrum of Varshamov Teegolt s codes: see e.g., [3], [5], [7]. It is also iterestig to ote that Dickso [11, pp ] presets a formula which ca be used to compute the cardiality of C(; +1; a; w), attributed to R. D. vo Stereck (190). For a 0, it yields jc(; +1; 0;w)j (01)w (d)(01) bwdc ( +1)d bwdc dj+1 where is Euler s fuctio. For certai weights w, we are also able to obtai N (w) i explicit form, such as for w. Propositio 1: Let x C(; +1; 0; ). The two biary oes i each x occur i a symmetrically spaced pair at idices i ad 0i+1 ad N () bc. If there are two oes i ay biary word of legth bits, the momet is bouded by 3 ix i 01: Cosequetly, the momets of all codewords of C(; +1; 0; ) have to satisfy ix i +1: This is oly possible if the oes i a codeword occur i a symmetrically spaced pair at idices i ad 0 i +1. Cosequetly, there are bc codewords with w. We ext ivestigate whe the biary complemet of codeword x, i.e., x with x i x i +1(mod), is icluded i the codebook. This isight also eables us to further evaluate the weight spectrum. Propositio : Let x C(; +1;a)ad let be eve. The x C(; +1;0a). Thus ix i + ix i 0 ( +1) ix i 0mod(+1): ix i 0amod ( +1): Corollary 1: Let x C(; +1;0) ad let be eve. The x C(; +1; 0). The codewords thus occur i complemetary pairs. This leads to the followig corollary: (9) Corollary : For C(; +1; 0) with eve, the weight spectrum is symmetrical, i.e., N ( 0 i) N(i); for 0 i 0 1: Corollary 3: If is eve ad a 6 0mod(+1), the x ad its biary complemet x are i differet codebooks. The previous results ca be geeralized as follows: Propositio 3: If x C(; m; a), the x C(; m; e 0 a) where ( +1) e(mod m). Sice it follows that ix i + ( +1) ix i e(mod m): ix i a (mod m) ix i (e 0 a) (mod m): We ow ivestigate the Hammig distace structure. First, we preset some results which apply to all s 1isertio/deletio correctig codes. Propositio 4: Ay code correctig s 1isertios/deletios has d mi > 1. If codewords x ad y have Hammig distace d 1, a commo subword of legth j(x; y)j 0 1 ca be obtaied by deletig the distace buildig bit from each word. However, this cotradicts j(x; y)j <0 1, as required i (3). Sice the all-zero word is icluded i the C(; +1; 0) codebook, we have the followig corollary: Corollary 4: There ca be o codewords of weight w 1i the C(; +1;0) codebook, i.e., N (1) 0. The followig result agai applies to ay s 1isertio/deletio correctig code. Theorem 1: Ay code correctig s 1isertio/deletio has the followig property: two codewords with weight j ad j +1 have d 3. Two codewords with weights j ad j +1 have odd Hammig distace, d. Sice d > 1(Propositio 4), d 3. The followig result is kow to hold for the Varshamov Teegolt s costructio (see, e.g., [5], [9]): Theorem : Let C(; m ; a) be a Varshamov Teegolt s code. Two equal-weight codewords have d 4. Corollary 5: The Hammig distace properties i Theorems 1 ad also holds for Leveshtei s codes. We ow ivestigate the image ^x of codeword x. We defie the image of a codeword x to be the word ^x, foud by writig the bits i x i reverse order, i.e., ^x j x +10j; 1 j : The image word ^x will play a role i codig schemes preseted i later sectios. The proof of the followig propositio is straightforward: Propositio 5: Let x C(; +1;a). The it holds for the image ^x of x, that ^x C(; +1; 0a). Corollary 6: If a 0, it follows that x, ^x C(; +1; 0). Note that some codewords of legth, eve, may be their ow images, i.e., x j x +10j; 1 j : We shall call these codewords symmetrical.

3 74 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO., MARCH 1997 Cyclic shifts of codewords play a importat role i codig theory ad i some sychroizatio recovery schemes. Of particular iterest is if a cyclic shift of x C(; m; a) will also be i the codebook. The proof of the followig propositio is agai straightforward: Propositio 6: By cyclic shiftig a weight w codeword x C(; m; a) to the right (left), the resultig word x r (x l ) falls ito a ew codebook as follows: 1) x 0: x r C(; m; a + w) x 1 0:x l C(; m; a 0 w) ) x 1:x r C(; m; a + w 0 ) x 1 1:x l C(; m; a 0 w + ): Corollary 7: For x C(; ; 0; ), a codeword i the same codebook is obtaied after two cyclic shifts i the same directio. Corollary 8: For ay x C(; +1; 0;), shiftig x i the same directio, results i a codeword which belogs to oe of two other codebooks, as determied by x for right shifts ad x 1 for left shifts. III. SOME CONSTANT-WEIGHT SUBCODES OF LEVENSHTEIN S INSERTION/DELETION CORRECTING CODES I this sectio, we cosider costat-weight codes, icludig the balaced codes which have a spectral ull at dc. A. Bouds o the Cardiality of Costat-Weight Codes Costat-weight codes have bee ivestigated extesively i the past. The cardialities of these codes are usually deoted by A(; d; w), where is the legth of the biary block code with a miimum Hammig distace of d mi (deoted here with d d mi ) ad weight w. Upper bouds o the cardialities of these codes appear, e.g., i [6], [1], [13]. The Varshamov Teegolt s costructio has previously bee employed to costruct good costat-weight codes with d mi 4ad d mi 6(see, e.g., [5], [6].) The rate efficiecy of the costatweight Leveshtei codes per se ca be appreciated if the cardialities are compared with the latest results o A(; d; w) i [1]. From Corollary 5 it ca be see that the costat-weight Leveshtei subcodes will have miimum Hammig distace d mi 4. By evaluatig the geeratig fuctio i (8), it ca be see that the cardiality of the class of C(; +1;0;w)codes compares favorably to the upper bouds. I fact, the achievable code rates of iterest for implemetatio are ofte the same, i.e., R k blog jc(; +1; 0;w)jc blog A(; 4; w)c (10) or differ at the most with oe uit i the umerator. It is thus ofte possible to costruct a optimal rate R k, d mi 4 costat-weight code, ad also be able to correct isertios/deletios. B. Balaced Block Codes with 16,d mi 4 I [14], we preseted the costructio of a (; k) (16; 8) balaced or dc-free block code with d mi 4. Blaum improved o this result by costructig a (; k) (16; 9) balaced code i [15], while the bouds i, e.g., [1] idicates the existece of a (; k) (16; 10) code. Immik ad Beeker [4] proposed a 16dc -costraied code with cardiality 56 for this purpose. By evaluatig (8) for 16ad w 8, it ca be see that a (; k) (16; 9) balaced code with d mi 4ca be costructed usig C(16; 17; 0; 8). This code improves o the results i [14] ad [15], sice it ca correct either a sigle isertio/deletio i error or the sigle reversal error cosidered i [14] ad [15]. Furthermore, it represets a simpler costructio ad a simple decoder exists, as ca be see i Sectio V. I the ext sectio, we shall show that Immik ad Beeker s code is i fact oe possible subcode of this code. Usig a computer to trim selected code words from C(16; 17; 0; 8) which has cardiality 758, we ca reduce the maximum rulegth of the (; k) (16; 9) code, which eeds a codebook with 51 codewords, to a maximum rulegth of 5 bits (i compariso to 6 for Immik ad Beeker s code). The ew code has a ruig digital sum bouded by 05 z j 1 j x i 5; 1 j ; x i f01;+1g where we assume a mappig of x i f0;1goto x i f01;+1g. IV. SOME SPECTRAL NULL SUBCODES OF LEVENSHTEIN S INSERTION/DELETION CORRECTING CODES The balaced codes i the previous sectio have first-order spectral ulls at dc. I this sectio, we cosider block codes with either higher order spectral ulls, or spectral ulls at other frequecies. Note that oce we tur to the topic of spectral ulls, the polar represetatio of the biary symbols, i.e., x i f01;+1g must be used. I cotrast, whe codes with error-correctig capabilities are ivestigated, the x i f0;1grepresetatio is ofte used. By usig the straightforward mappig of f0; 1g oto f01; +1g, the results of the previous sectios, ca be applied to this sectio. A. Subcodes with Higher Order Spectral Zeros at Zero Frequecy Immik ad Beeker [4] described a subclass of balaced codes, sometimes called higher order dc-costraied codes, which suppresses low-frequecy compoets i the code s power spectral desity fuctio. Briefly, they ivestigated codewords x; x i f01; +1g, ofkth-order zero disparity, i.e., with the first K +1 codeword momets at dc all zero, or u k i k x i 0 kf0;1;111;kg: (11) This class of codes the has the property that the first K +1 derivatives of the power spectral desity vaish at dc or! 0, ad they show that codewords have miimum Hammig distace d mi (K +1): (1) Furthermore, it is show i [4] that should be divisible by 4, ad that x C, x +1 ix i 0 x 01 ix i ( +1)4: (13) If we assume a straightforward mappig of f01; +1g oto f0; 1g, set 4e, ad cosider the first-order spectral ull, it follows that for ay oe of the Kth-order zero disparity codes x C, ix i e( +1)0mod(+1): (14) Cosequetly, the momet of ay higher order dc-costraied code is e( +1)ad the code is a subcode of C(; +1;0;), which may also cotai codewords with other momets. We thus coclude that the class of higher order dc-costraied codes are capable of correctig a sigle isertio/deletio error, a property which seems to be hitherto ukow. We ow preset a result o the biary structure of higher order dc-costraied codes.

4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO., MARCH Fig. 1. The umber of iformatio symbols versus codeword legth of some Leveshtei subcodes. Propositio 7: Let x be dc -costraied. The x is also dc - costraied. ( +1) ix i + ix i ( +1) ix i (+1) 4 which idicates that x is also dc -costraied. 0 (+1) 4 B. Nyquist Null dc -Costraied Subcodes It is possible to further costrai the dc -costraied codes i the previous sectio, i order to obtai codes with a ull at the Nyquist frequecy f s. This class of codes is sometimes called miimumbadwidth codes. The Nyquist ull dc -costraied subcodes of C(; +1;0) were first described i [16]. Followig Kim [17], who ivestigated a zeroth-order ull at f s,we require i additio to (11) ad (13), that the ruig alterate sum for each codeword is zero, or RAS (01) i x i 0; x i f01;+1g: (15) By applyig the theory of iteger partitios, the structure of codewords ca be aalyzed ad a geeratig fuctio for eumeratig codewords ca be set up. Refer back to (13) ad cosider the idices p i where x i +1. Sice w, there are of these idices which partitio the iteger ( +1)4such that there are exactly parts, with o repeated parts ad o part larger tha. The same structure applies to the idices i where x i 01. A momet s reflectio reveals that (15) ca be writte i terms of these idices, as RAS N (p e ) 0 N (p o )+N( o )0N( e )0 (16) where the superscript e deotes eve ad o odd. Sice each codeword is balaced, we may also write RDS N (p e )+N(p o )0N( e )0N( o )0: (17) Furthermore, sice the codeword legth is eve, there are equal umbers of eve ad odd idices, thus N ( e )+N(p e )N( o )+N(p o ): (18) There are idices i total, or Solvig (16) (19) yields N (p e )+N(p o )+N( e )+N( o ): (19) N (p e )N(p o )N( e )N( o )4: (0) The Nyquist ull dc -costraied codes are thus defied by x C, 4 p o i + 4 p e i 4 e i + 4 o i ( +1)4: (1) Referrig back to (8), ad assumig a mappig of f01; +1g oto f0; 1g, the geeratig fuctio for this class of codes ca be set up. Let t accout for oe uit of weight, u i for idex i ad v for p o f (u; t; v) (1 + utv)(1 + u t)(1 + u 3 tv) 111(1 + u t) (1 + u i tv i0 i ): () Here the coefficiet of u (+1)4 t v 4 should be evaluated to eumerate this class of codes. The code rate is depicted i Fig. 1, from which it ca be see that the rate loss is ot sigificat whe further costraiig the dc - costraied codes. (For 1, there is o rate loss.) Note that every x C(8; 9; 0; 4) satisfies both the dc ad Nyquist ull properties. We ext preset some propositios o the biary structure of Nyquist ull dc -costraied codes. Propositio 8: If divides by 4, all symmetrical codewords x C(; +1;0;) with x i x +10i, 1 i, are icluded i the dc -code of legth.

5 76 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO., MARCH 1997 Let each codeword x costitute of two equal-legth subwords, (x 1; 111;x )ad (x +1 ; 111x ). Sice x is symmetrical, each subword cotais 4 oes. Thus there are 4 pairs of oes i x, which each cotribute i +(+10i)+1to the momet of x. Cosequetly the total momet is ix i ( +1)4 which is the momet of a dc codeword [4]. Hece, x is dc - costraied. I the previous propositio, we divided each codeword ito two equal-legth subwords. We ow ivestigate aother class of evelegth codewords where the secod subword is the iverse of the first subword, i.e., x +i x i : (3) Propositio 9: If divides by 4, all codewords x C(; + 1; 0; ) which have the iverse property, i.e., x +i x i, are icluded i the dc -code of legth. ix i ix i + ix i + j+1 i(x i + x i )+ i (+1) (+1) 4 jx j i x i x i which is the momet of a dc codeword. Hece x is dc -costraied. Propositio 10: All symmetrical balaced codewords x C(; +1; 0;), have a ull at the Nyquist frequecy. (01) i x i 0: (01) i x i 0 (01) i x i 0 i+1 j1 j1 j1 (01) i x i (01) +10j x +10j (01) j x +10j (01) j x j Propositio 11: If divides by 4, all codewords x C(; + 1; 0; ) which have the iverse property, i.e., x +i x i, 1 i, have a ull at the Nyquist frequecy. (01) i x i i+1 Substitute i j + i the last term. Thus (01) i x i 0: j1 j1 (01) i [x i + x +1 ] (01) i x i: (01) +j x +j (01) j x +j (01) i x +1 (01) i ; sice x i + x +i 1 Corollary 9: Both the symmetrical ad iverse subcodes of C(; + 1; 0;) are icluded i the Nyquist ull dc 0 - costraied code of legth. C. Spectral Null rf s N Subcodes As show i [18], spectral ulls at ratioal submultiples rn of the symbol frequecy f s ca be achieved if we restrict N to be a prime umber which divides, i.e., ad costrai every codeword to have where A i z01 0 Nz (4) A 1 A 111 A N (5) x N+1 ; i 1;111;N: (6) A spectral ull at frequecy f s N furthermore implies spectral ulls at frequecies rf sn gcd (r; N )1,. We ca ow set up a geeratig fuctio to eumerate subcodes of C(; +1; 0), which also coform with (5). Note that (5) ca oly be satisfied if the set of bits which cotribute to each A i, have the same weight h, where 0 h z. I the followig geeratig fuctio, x i represets bit x i i (6) ad u i represets a cotributio to A i, while t represets oe uit of weight as before f (u 1 ; 111;u k ;t;x) N01 (1 + tu 1x 1+Ni )(1 + tu x +Ni )111(1 + tu x N+Ni ): i0 (7) We thus eed to evaluate the coefficiets of u h 1 u h 111 uk h x, 0 h N, 0mod( +1), to obtai the cardiality of the spectral ull rf sn subcode of C(; +1; 0), or the coefficiets of u1 h u h 111 uk h t w x if we wat the cardiality of a weight w subcode of C(; +1;0). Theorem 3: A spectral ull A 1 A 111 A N subcode of C(; +1;0) has d mi mi f4; Ng.

6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO., MARCH Sice A 1 A 111 A N, it follows that each of the N subsets of code bits fx i;x i+n;111;x i+0ng,i 1;111;N, should have the same weight h, 0 h z. Cosequetly, ay codeword (x 1 ;x ;111;x )has weight w Nh. Thus ay two codewords of differet weights have d N. Furthermore, ay two codewords of the same weight have d 4. It follows that d mi mif4;ng. Code rates of spectral ull rf s3 subcodes are also depicted i Fig. 1. V. DECODING THE CONSTANT-WEIGHT AND SPECTRAL SHAPING SUBCODES The costat weight ad spectral shapig subcodes C(; +1; a) are highly structured ad the decodig procedures are thus simple. We propose a decodig process that cosists of three phases ad ow briefly discuss these three phases. Note that all the codes uder cosideratio here correct either s 1isertio/deletio error, or t 1reversal error i each codeword. A. Error Detectio The decoder fuctios uder the assumptio that the begiig ad ed, ad thus also the legth of each received word, is kow. This is the assumptio also made by Leveshtei whe the C(; m; a) class of codes was proposed, ad it may be achieved i practice by isertig periodic markers or sychroizatio words i the trasmitted sequece. Both the ature of the error ad the restored value f0;1gof the affected bit ca be determied uiquely from the legth 0 ad weight w 0 of the received word x 0 as follows: ad a) 0 w 0 w +1(w01) : reversal error 0(1)! 1(0) b) w 0 w(w 0 1) : deletio error 0(1)!^ c) 0 +1; w 0 w+1(w) : isertio error ^!1(0): Here ^ deotes the empty word ad the weight w is kow for costat-weight w codes, or for higher order spectral ull subcodes (w ), while for the spectral ull rf sn subcodes, we use w mi fjw 0 0 Nhjg; 0 h z: (8) B. Restoratio of the Trasmitted Codeword I order to determie the idex of the affected bit, we proceed as follows. For higher order spectral ull subcodes, the restored codeword ca have oly oe momet, ( +1)4, ad the followig sydrome fuctio ca always be used: S ( +1)40 ix 0 i : (9) For costat-weight subcodes, or for spectral ull rf sn subcodes, the momet of the restored codeword is determied by the ature of the error ad the restored value of the affected bit as follows: If a reversal error with 0, or a isertio is detected, select to miimize the sydrome fuctio S ix 0 i 0 ; S>0 (30) thus S ix 0 i (mod +1): (31) If a reversal error with 1, or a deletio is detected, select to miimize the sydrome fuctio thus S 0 S (+1)0 ix 0 i; S>0 (3) ix 0 i (mod +1): (33) The value of S i oe of (9) (33), as dictated by the class of codes ad, is the used to determie the idex of the corrected bit as follows: a) For a reversal error, ivert the bit at idex S, sice this is the oly way to restore the momet to. b) For isertios/deletios we may apply the decodig algorithm proposed by Leveshtei []. However, some simplificatio is possible if we make use of the fact that each codeword s complemet is also icluded i the codebook ad satisfies (1). (Refer back to Corollary 1.) For isertio/deletio of a zero, we delete/isert a zero to the right of the Sth oe from the left i (x 0 max; 111;x 0 1), this is agai the oly way to restore the momet to. For isertio/deletio of a oe, we first complemet x 0, the compute S ad proceed as above for isertio/deletio of a zero, ad agai complemet the restored word. C. Mappig of the Restored Codeword Oto Iformatio Bits From Fig. 1 ca be see that the code rates of all the balaced w subcodes which we ivestigated evetually exceeds rate R 1. Cosequetly, it is ot possible to always costruct a systematic code: for example the all zeros iformatio word will require w < ad the all oes iformatio word w >. However, for rates R 1, it may be possible to fid a systematic mappig by ispectio [16]. Alteratively, a lookup table should be used. VI. INSERTION/DELETION CORRECTING CODING SCHEMES The practical applicatio of the class of C(; m; a) Leveshtei codes has bee restricted by the requiremet that the boudaries of each received word be kow before isertio/deletio correctio ca be affected ad framig of words be maitaied. I this sectio we propose two codig schemes which determie the boudaries of each codeword as log as s 1isertio/deletio per codeword is ot exceeded, ad if every word with a error is followed by a errorfree word. Though the rate of the secod scheme is somewhat less tha the rate of the first scheme, it also offers correctio of t 1 reversal errors. A. Codig Scheme 1 I this codig scheme, we use a subcode of C(; m; a; w), 0 < w <, such that each codeword has a momet from a set fg, where j 1 0 j+1for ay distict 1, i fg. Note that it is possible to use a dc -costraied code, sice each codeword has w ad fixed l ( +1)4. We furthermore eed to isert two buffer bits, b 1 b 0, betwee every two codewords. I the followig descriptio, it is assumed that a dc - costraied code is used. The trasmitted sequece is thus costituted

7 78 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO., MARCH 1997 as follows: 111 x j01 b j01 1 b j01 x j xj b j 1 bj xj x j+1 b j+1 1 b j (34) where x j is the jth trasmitted codeword. For each codeword, the receiver establishes a frame of +1bits ad computes the momet +1 iy i (35) where (y 1; 111;y +1) is the received word correspodig to (x 1 ; 111;x ;b). Whe decodig x j, we assume that framig up to x j01 has proceeded correctly. Without ay isertio/deletios the decoder observes +1 iy i ( +1)4 (36) ad proceeds to x j+1. If a bit is deleted i x j, b j replaces b j 1 i (35) ad the decoder observes +1 iy i ( +1)4: (37) If a bit is iserted i x j, x j replaces b j 1 i (35) ad the receiver observes +1 iy i ( +1)4: (38) Note that equality i (37) ad (38) is oly achieved if there are oly zeros to the right of the iserted/deleted bit ad if the iserted/deleted bit is a zero. If this is the case, the decoder observes a word which resembles x j i the first bits of the frame ad eed ot take ay actio. However, this evet will the be reflected as a isertio/deletio at idex i 1i x j+1 whe the decoder icorrectly establishes the ext frame, ad it is corrected if x j+1 is received error-free. If iequality is observed i (37) ad (38), the decoder detects ad discrimiates betwee a isertio/deletio, ad uses the observed weight w 0 to determie the restored value of the affected bit, as i Sectio V. If oe of the buffer bits b j01 is deleted, this is perceived at the,is perceived as the isertio of a 0 at i 1i x j. Restoratio of the trasmitted codeword proceeds as i Sectio V ad the receiver adjusts the framig if ecessary. Note that the decoder is uable to discrimiate betwee isertios/deletios ad reversal errors, cosequetly, the capability to do reversal error correctio is lost. I case the set of momets fg cotais more tha oe iteger, we see that the decoder eeds to discrimiate betwee a deletio from a codeword with momet 1 or a isertio i a codeword with momet, where 1 >, hece i geeral, for C (; m; a; w) we require decoder as the deletio of x j 1, while a isertio precedig b j01 i j 1 0 j+1 (39) for ay distict 1, i fg. The codig rate of this scheme, whe implemeted with dc -costraied codes, is R k( +), where k ad ca be obtaied from Fig. 1 for the dc -costraied codes. The overall rate is also depicted i Fig. 1. For example, R 1is achieved whe usig the (; k) (16; 9) dc -costraied code. B. Codig Scheme Alteratively, we ca achieve isertio/deletio correctio, by makig use of a symmetrical subcode of C (; +1;0), which has x j x +10j, 1 j, ad which was ivestigated i Propositios 5, 8, ad 10. If we are furthermore iterested i correctig reversal errors, we require d mi 4, hece we eed a symmetrical costat weight subcode of C (; +1; 0), which the achieves the highest cardiality for w. From Propositios 8 ad 10 it thus follows that we shall make use of a subcode of the Nyquist ull, dc -costraied codes discussed i Sectio IV. No buffer bits eed to be trasmitted. The decoder i this codig scheme assumes that framig up to x (j01) has proceeded correctly, establishes a frame of bits, iputs x j, ad checks for the symmetry. If a deviatio from symmetry is observed, it iputs x (j+1). It performs reversal error correctio o x j if x (j+1) is symmetrical. If x (j+1) is ot symmetrical, it attempts to restore its symmetry by meas of oe bit left shift or oe bit right shift. If it is ecessary to perform a right shift o x (j+1), a deletio occurred i x j, while a left shift idicates a isertio. The decoder thus sets up the correct frame for x j, ad uses the observed weight w 0 withi this frame, to determie the restored value of the affected bit. Restoratio of the trasmitted codeword proceeds as i Sectio V. By careful cosideratio of cadidate symmetrical codewords, it ca be show that there are four codewords, which may achieve symmetry i x (j+1) agai, after two like shifts, which will the lead to a icorrect decisio about the ature of the deletio/isertio i x j. These are the all-zeros ad all-oes codewords, ad the two complemetary codewords which each costitute oly of rus of legth like symbols, except for x 1 ad x, which each costitute a ru of legth 1. These words have to be omitted from the codebook. The cardiality of the codes i this scheme ca be determied by otig that oce the first subword (x 1; 111 x ) of x is specified, the secod subword is fixed. Furthermore, ay pair of oes (x j ;x +10j ) cotributes ( +1)to the codeword momet, hece ix i 0(mod+1) ad for ay symmetrical word x, x C(; +1;0). For oly isertio/deletio correctio, we allow the first subword to have ay weight ad the omit the four words discussed jc 1 j 0 4 : (40) To afford reversal correctio i additio we eed a costat-weight w subcode ad allow the first subword to cotai w 4 oes. Thus the all-zeros ad all-oes codewords are excluded, ad jc j 0 : (41) 4 For both these classes of codes, the asymptotic code rate is R 1 1: (4) Code rates for fiite are depicted i Fig. 1. Although these rates appear low, the miimum-badwidth property of the secod class of codes, together with the rate, should be cosidered to evaluate the efficiecy o badwidth-limited chaels. VII. SPECTRAL NULL MARKERS FOR THE DETECTION OF DELETION/INSERTION ERRORS A. Prelimiaries I this sectio the use of a marker (or sychroizatio sequece) is proposed to separate each codeword i the chael. Markers

8 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO., MARCH are predetermied sequeces used for the aligmet of trasmitted sequeces. We preset markers for use with either ucostraied or spectral ull C(; +1;a)codes. Sellers [19] itroduced the idea of a marker for deletio/isertio error cotrol. No data is usually mapped oto a marker ad therefore the overall rate is reduced. Sellers method eeded to wait for two markers before ay detectio could be doe. I this sectio, a mergig of Sellers method ad the Leveshtei codes is proposed. The goal of the markers itroduced i this sectio is to afford spectral ulls whe ecessary, as well as to detect errors. The correctio of errors must be doe by the decodig procedures for the isertio/deletio error correctig code. The idea of a marker is better uderstood by first itroducig the cocept of a sychroizatio error correctig sequece. A sychroizatio error correctig sequece cosists of a isertio/deletio correctig codeword (e.g., a Leveshtei codeword) x (x; 111;x 1)of legth followed by a marker b bm 111b b 1 of legth Ms. The isertio/deletio correctig codeword x will be called the iformatio segmet because this is the part of the sequece carryig the iformatio. The total legth of the sychroizatio error-correctig sequece is thus + Ms : (43) Sellers [19] showed that the miimum legth of a marker b must be Ms s +1; s 1 (44) where s is the isertio/deletio correctig ability of the code. For a s 1code the miimum legth of the marker must be 3. Before costructig the markers, it is ecessary to ivestigate the fuctio of a marker. Similarly as for the codig schemes i the previous sectio, the assumptio is made that ay sychroizatio error-correctig sequece with error should be followed by oe without error. A marker must eable the receiver to detect the occurrece of a error, be that a isertio/deletio or a reversal error. The order of detectio is also importat. A isertio/deletio shifts the rest of the symbols i a codeword up or dow, depedig o the error. Up to the poit of the error, the symbols will be correct. From the isertio/deletio error positio i owards all the symbols may be wrog. If the receiver first checks for reversal errors, the icorrectly framed received word x 0 may appear to cotai a maximum of 0 i +1 reversal errors. If the reversal error-correctig ability of the code is t ad t<0i+1, the the receiver will be uable to correct or eve detect the errors. The word may eve look like aother valid codeword, oly with t or fewer additive errors ad the receiver may the map it oto a wrog codeword. If, however, the receiver first checks for isertio/deletio errors by evaluatig a marker, it will first detect ad correct the isertio/deletio error if it exists. The ext codeword will the agai be correctly framed. B. Rules for the Costructio of Markers It is assumed that whe codewords are trasmitted, the x 1 is set first ad x last. (Refer to (34).) Let B be a marker codebook ad ^b B be a marker of legth Ms, Ms 3, ad b b 1b 111bM (45) with bi f0;1gfor a biary marker. The resultig marker after a sigle deletio i the previous codeword is b d b b 3 111bM x 1 (46) where x 1 is the first symbol from the ext codeword. The first symbol of b, b 1 is shifted out ad becomes the last symbol of the previous iformatio segmet. The last symbol of b d ow cotais the first symbol of the followig iformatio segmet. Note that x 1 may assume ay biary value. The word b d, which will be called a deletio idicator, is the resultig marker b due to a deletio i the precedig iformatio segmet. Similarly, the resultig marker after a isertio i the previous iformatio segmet is b i xb 1b 111bM 01 (47) where x is the last symbol from the precedig iformatio segmet. The word b i will be called a isertio idicator. For a sequece b to be cosidered as a marker, the followig rules must be valid: 1) b i 6 b d ) b i 6 b; b d 6 b. Rule 1) esures that the decoder ca differetiate betwee isertio ad deletio errors. Rule ) eables the receiver to detect the occurrece of a isertio/deletio. If more tha oe marker is to be used together i the same trasmissio, the detectio ad differetiatio rules must still be valid. Let b 1 ad b be two markers of equal legth. For b 1 ad b to be valid markers i the same marker codebook B, they must first comply with Rules 1) ad ) ad with the followig additioal rules, where bj i is the isertio idicator for marker j, bj d is the deletio idicator of marker j ad b fb 1;b g: 3) b i 1 6 b;b d d 1 6 b i 4) b i 1 6 b; b d 1 6 b; b i 6 b; b d 6 b: Agai, Rule 3) esures that the receiver will be able to differetiate betwee isertio ad deletio errors. Rule 4) eables the receiver to detect sychroizatio errors. For a 3-bit marker code book, oly four codewords comply with both Rules 1) ad ), i.e., 001; 100; 011; ad 110. The markers 001 ad 100 are the oes which Sellers [19] proposed i his isertio/deletio correctig codes. The marker 011 was the oe Ullma [0] used i his codes. Noe of these valid markers however ca be used together i a marker codebook because ay two markers violate Rules 3) ad 4). Util ow, valid markers oly eabled the decoder to detect ad differetiate isertio/deletio errors i the previous iformatio segmet. It is also possible for a isertio/deletio or reversal error to occur i the marker itself. To combat this situatio, it is first ecessary for the marker codebook to have a miimum Hammig distace of d mi. This will guaratee that a reversal error i a marker codeword will be detected. To combat isertio/deletio errors, the marker codebook must furthermore have isertio/deletio-detectig capabilities. It is proposed to use markers from a Leveshtei code. Note that these markers will the also have miimum Hammig distace of d mi. C. Example of Some 4-Bit Marker Codebooks Table I presets eight 4-bit marker codebooks. These marker codebooks comply with the marker costructio rules (1) (4) give i the previous sectio. The markers may be used to detect as well as idetify the type of sychroizatio error i the previous iformatio segmet exclusively. Table II lists all the valid 4-bit marker codebooks which furthermore has a s 1 isertio/deletio correctig ability (ad thus also d mi ). Note whe used with the ucostraied Leveshtei code, the rate of the codig scheme with marker ca be icreased by mappig oe iformatio bit oto a set of two marker words chose from Table I or II. D. Decodig Procedure The decodig of the sychroizatio error-correctig sequece cosists of two steps. A lookup table is ecessary for the marker

9 730 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO., MARCH 1997 TABLE I VALID BINARY 4-BIT MARKER CODEBOOKS TABLE III DECODING SYNCHRONIZATION ERROR CORRESPONDING SEQUENCES TABLE II VALID 4-BIT MARKER CODEBOOKS Fig.. Power spectral desity of a 9-bit Leveshtei subcodes with spectral zeros at multiples of fs3. codebook ad its error idicators. Whe a sychroizatio errorcorrectig sequece is received, the decoder separately checks if there are ay errors i the iformatio segmet or i the marker. By applyig (1) to the iformatio segmet, the receiver ca detect whether a error has occurred or ot. If the received iformatio segmet does ot comply with (1), the receiver kows that a error has occurred i that iformatio segmet. Let r i be the output of the iformatio segmet check ad r m the output of the marker check ad r i ;r m f0;1g. The values or r i ad r m are zero whe there are o errors ad oe if there are ay errors. Table III lists the result of the variables r i ad r m ad also the meaigs ad remedies. It assumed that the iformatio segmet is represeted by a codeword from C(; m ; a), or from C(; +1; 0;w), hece reversal error correctio ca be achieved. The deletio/isertio error correctio for the iformatio segmet may be doe by the procedures give by Leveshtei []. If a marker idicates that a deletio occurred, the bits of the iformatio segmet precedig the begiig of the marker are take to be the iformatio segmet i error. For the correctio, however, the last symbol of the iformatio segmet must be left out, because this will be the first symbol from the origial marker. The total legth of the iformatio sequece passed o to the correctig procedure will therefore be 0 1. The framig must the be adjusted accordigly. For a isertio the same procedure is followed. The symbols of the iformatio segmet precedig the begiig of the marker, together with the first symbol of the marker, are take as the received word to be corrected by the isertio correctig procedure. The first symbol of the marker must be icluded because it is the last symbol of the origial iformatio segmet. The total legth of the word to be corrected is thus +1symbols. The decoder framig must be adjusted to correct the framig of subsequet codewords. If the error is a reversal error i the iformatio segmet, the symbols precedig the begiig of the marker are take to be the word to be corrected by the reversal error-correctig procedure. Sometimes it may happe that a reversal or isertio/deletio error i the marker appears to be the same as a isertio or deletio idicator. I these cases, the decoder simply igores the idicator ad cotiues with the ext codeword. If the error was a isertio/deletio, the ext codeword will be corrupted because of this isertio/deletio. If the ext sychroizatio error-correctig sequece is received error-free, the marker i this sequece will detect the error ad the decoder will be able to correct the error. If more tha oe isertio/deletio occur per sychroizatio errorcorrectig sequece, the decoder will be uable to correct these errors ad the framig will be lost. If more tha oe reversal error occur per sequece, or if two cosecutive sequeces with errors are received, the decoder will be uable to correct the errors, but word framig will still be itact. E. Spectral Shapig Parameters of Markers The questio may arise why a marker codebook is eeded, whe traditioally oly oe marker was used. If oe uses a arbitrary sigle marker, the spectral properties of the chael sequece may be altered. The marker, or set of markers, must therefore be chose carefully to match the spectral properties of the isertio/deletio error correctig code. Table IV lists the values of the ruig digital sum, secod-order momet at dc, ruig alterate sum ad secod-

10 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO., MARCH Fig. 3. Some spectra of dc-free Leveshtei subcodes. TABLE IV SPECTRAL PROPERTIES OF MARKERS Fig. 4. Spectra of miimum-badwidth Leveshtei subcodes. order momet at the Nyquist frequecy for every cadidate marker. Note that the biary symbols 0 ad 1 are mapped oto 01 ad 1, respectively. The markers for which these spectral shapig parameters have value zero, are of particular iterest, for usig as a sigle marker i cojuctio with the subcodes i this paper. VIII. CONCLUSIONS I this correspodece we have preseted some ew isight ito the structure of a class of Varshamov Teegolt s codes, as well as several importat subclasses. We have show that there is a relatio betwee Leveshtei s codes for correctig deletios/isertios ad some classes of codes with ulls i the power spectral desity fuctio. Some of these power spectral desity fuctios are depicted i Figs. 4. From Fig. 1 we see that the rates of the Nyquist-ull subcodes are ideed higher tha those of the dc-free subcodes. Aother observatio is that the ull otch width decreases as the codeword legth icreases i both the dc-free ad Nyquist-ull Leveshtei subcodes. This ca be explaied by the fact that the digital sum variatio icreases. Although we have preseted some results to this effect, future work ca focus o improvig the rates of codig schemes which alleviate the restrictio o Leveshtei s codes that the boudaries of each codeword should be kow. REFERENCES [1] R. R. Varshamov ad G. M. Teegolt s, Correctio code for sigle asymmetrical errors, Avtom. Telemekh., vol. 6, o., pp. 88 9, Feb [] V. I. Leveshtei, Biary codes capable of correctig deletios, isertios ad reversals, Sov. Phys. Dokl., vol. 10, o. 8, pp , Feb [3] B. D. Gizburg, A umber-theoretic fuctio with a applicatio i the theory of codig, Probl. Cyber., vol. 19, pp. 49 5, 1967, i Russia. [4] K. A. S. Immik ad G. F. M. Beeker, Biary trasmissio codes with higher order spectral zeros at zero frequecy, IEEE Tras. Iform. Theory, vol. IT-33, o. 3, pp , May 1987.

11 73 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO., MARCH 1997 [5] P. H. Delsarte ad P. H. Piret, Spectral eumerators for certai additiveerror-correctig codes over iteger alphabets, Iform. Cotr., vol. 48, pp , [6] R. L. Graham ad N. J. A. Sloae, Lower bouds for costat weight codes, IEEE Tras. Iform. Theory, vol. IT-6, o. 1, pp , Ja [7] T. Helleseth ad T. Kløve, O group-theoretic codes for asymmetric chaels, Iform. Cotr., vol. 49, pp. 1 9, [8] L. E. Mazur, Correctig codes for asymmetric errors, Probl. Pered. Iform., vol. 10, o. 4, pp , Oct. Dec [9] W. H. Kautz ad B. Elspas, Sigle-error-correctig codes for costatweight data words, IEEE Tras. Iform. Theory, vol. IT-11, o. 1, pp , Ja [10] J. Riorda, A Itroductio to Combiatorial Aalysis. Priceto, NJ: Priceto Uiv. Press, [11] L. E. Dickso, History of the Theory of Numbers, vol.. New York: Chelsea, 195. [1] A. E. Brouwer, J. B. Shearer, N. J. A. Sloae, ad W. D. Smith, A ew table of costat weight codes, IEEE Tras. Iform. Theory, vol. 36, o. 6, pp , Nov [13] F. J. MacWilliams ad N. J. A. Sloae, The Theory of Error-Correctig Codes. New York: North-Hollad, [14] H. C. Ferreira, Lower bouds o the miimum Hammig distace achievable with rulegth costraied or dc-free block codes ad the sythesis of a (16; 8) dmi 4 dc-free block code, IEEE Tras. Mag., vol. MAG-0, o. 5, pp , Sept [15] M. Blaum, A (16; 9; 6; 5; 4) error-correctig dc free block code, IEEE Tras. Iform. Theory, vol. 34, o. 1, pp , Ja [16] A. S. J. Helberg, W. A. Clarke, H. C. Ferreira, ad A. S. J. Vick, A class of dc free sychroizatio error correctig codes, IEEE Tras. Mag., vol. 9, o. 6, pp , [17] D. J. Kim ad J. Kim, A coditio for stable miimum badwidth lie codes, IEEE Tras. Commu., vol. COM-33, o., pp , [18] K. A. S. Immik, Codig Techiques for Digital Recorders. Eglewood Cliffs, NJ: Pretice-Hall, [19] F. F. Sellers, Jr, Bit loss ad gai correctio code, IRE Tras. Iform. Theory, vol. IT-8, o. 1, pp , Ja [0] J. D. Ullma, Near-optimal, sigle-sychroizatio-error-correctig code, IEEE Tras. Iform. Theory, vol. IT-1, o. 4, pp , Oct A Note o the -ary Image of a -ary Repeated-Root Cyclic Code Li-zhog Tag, Cheog Boo Soh, ad Erry Guawa Abstract For (; q) p s, where p ch(fq); s1; V a q m -ary repeated-root cyclic code of legth with geerator polyomial g(x), we give a partial aswer about whether the q-ary image of V is cyclic or ot with respect to a certai basis for Fq over Fq. Idex Terms Cyclic code, q-ary image, ideal, rigs, repeated-root cyclic code. I. INTRODUCTION Let m ad be two positive itegers, ad let F q be a q-ary fiite field of characteristic p. The we kow that q is a power of p. Now let ( 0 ; 1 ;111; m01) be a basis (ordered) for F q over F q Mauscript received February 16, 1996; revised August 5, The authors are with the School of Electrical ad Electroic Egieerig, Nayag Techological Uiversity, Sigapore Publisher Item Idetifier S (97) ad defie the mappig as follows: where ad d (;m;) : F q [z](z 0 1) 0! F q [z ](z m 0 1) m01 01 d (;m;) (a(z)) a i;jz mj+i a(z) a j 01 j0 m01 i0 a jz j i0 j0 a i;j i ; a i;j F q : The d (;m;) has the followig properties: i) It is a bijective map (ijective ad surjective). ii) It is q-ary liear (i.e., liear over F q). iii) d (;m;) (g(z)a(z)) g(z m )d (;m;) (a(z)) for ay g(z) F q [z] ad a(z) F q [z]. If V is a q m -ary [; k] cyclic code, the its q-ary image with respect to the basis is d (;m;) (V ) where d (;m;) (V )fd (;m;) (a(z)) j a(z) V g: It follows that d (;m;) (V ) is a q-ary [m; km] liear code ivariat uder multiplicatio by z m ; hece d (;m;) (V ) is a q-ary quasicyclic code [1], []. The the geeral problem is as follows: For which pair (;V), where V a q m -ary cyclic code ad a basis for F q over F q,isd (;m;) (V ) a cyclic code? Several authors attacked this problem [3] [11]. Especially, i [11], uder the oly restrictio (; q) 1,Ségui gave a very simple characterizatio of all the cyclic codes V for which there exists a basis such that d (;m;) (V ) is cyclic. His mai result is quoted as follows: Lemma 1.1 ([11, Theorem 10]): Let (; q) 1ad let V be a q m -ary cyclic code of legth with geerator polyomial g(z). The there exists a basis for F q over F q for which d (;m;) (V ) is cyclic if ad oly if: i) g(z) F q [z], i which case d (;m;) (V ) is cyclic for every basis ad the geerator polyomial of d (;m;) (V ) is g(y m );or ii) g(z) g 0 (z)(z 0 0q );g 0 (z) F q [z];f q 6 F q F q() Fq ;v Z k, ad! m 0 has a divisor over F q of degree e mk. I this case, d (;m;) (V ) is cyclic if ad oly if m01; m0;111; m0e are F q -idepedet ad ad j e a q i j+i ; 0 j<m0e a(!)! e 0a 1! e a e F q [!] divides! m 0. Moreover, the geerator polyomial of d (;m;) (V ) is a01(y)g 0 (y m ), where a01(y) is the reciprocal of a(y); or /97$ IEEE