The Coset Distribution of Triple-Error-Correcting Binary Primitive BCH Codes

Transcription

1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., APRIL The Coset Distribtion of iple-error-correcting Binary Primitive BCH Codes Pascale Charpin, Member, IEEE, TorHelleseth, Fellow, IEEE, VictorA. Zinoviev Abstract Binary primitive triple-error-correcting Bose Chadhri Hocqenghem (BCH)codes of length 1 have been the object of intensive stdies for several decades. In the 1970s, their covering radis was determined in a series of papers to be 5. However, one problem for these codes that has been open p to now is to find their coset distribtion. In this paper this problem is solved the nmber of cosets of each weight in any binary primitive triple-error-correcting BCH code is determined. As a conseqence this also gives the coset distribtion of the extended codes of length with minimm distance 8. Index Terms Bose Chadhri Hocqenghem (BCH)codes, coset distribtion, covering radis. I. INTRODUCTION Let GF( m ) denote the finite field with m elements. A binary linear [n; k]-code C is a k-dimensional sbspace of GF() n. The Hamming distance between two vectors is the nmber of components in which they differ. The minimm distance of the code is the smallest Hamming distance between any two distinct codewords in the code. A coset of the code C is the set of vectors a + C forsome a GF() n. The coset leaderfora coset of a linearcode is a vectorof smallest weight in the coset. The weight of a coset is the weight of a coset leader. For an optimal complete decoding algorithm for a linear code, where all codewords are sent eqally often, the errors which are corrected are exactly the coset leaders. An important problem in determining the performance of a linear code is therefore to determine the distribtion of the cosets of the code (i.e., to determine the nmber of cosets of each weight). The family of triple-error-correcting binary primitive Bose Chadhri Hocqenghem (BCH) codes of length n m 0 1 has been thoroghly stdied since the 190s. The weight distribtion of these codes was determined by Kasami [9] for odd m. Foreven m the method of Kasami did not work. Berlekamp [, Table 1.5], [] [] gave the weight distribtion of the extended codes for even vales of m. The covering radis of a code is the maximm weight of a coset leader. In the 1970s, the covering radis of these codes were shown in a series of papers by Assms Mattson [1], van der Horst Berger [8] Helleseth [7] to be 5. However, one problem for the triple-error-correcting binary primitive BCH codes that has been open p to now is to find the coset distribtion of these codes. We will assme throghot the rest of this paper that m 5 since the nmber of cosets of the binary triple-error-correcting code in this case Manscript received Janary 19, 005; revised November 1, 005. This work was spported by INRIA-Rocqencort, by the Norwegian Research Concil, by the Rssian fnd of fndamental research nder Project Nmber The material in this correspondence was presented in part at the 005 IEEE International Symposim on Information Theory, Adelaide, Astralia, September 005. P. Charpin is with INRIA, Codes, Domaine de Volcea-Rocqencort, Le Chesnay 7815, France ( pascale.charpin@inria.fr). T. Helleseth is with The Selmer Center, Department of Informatics, University of Bergen, Bergen N-500, Norway ( tor.helleseth@ii.ib.no). V. A. Zinoviev is with the Institte for Problems of Information ansmission, Rssian Academy of Sciences, Moscow 1017, Rssia ( zinov@iitp.r). Commnicated by A. E. Ashikhmin, Associate EditorforCoding Theory. Digital Object Identifier /TIT is m (cf. MacWilliams Sloane [10, p. ]), while the case m is degenerated in the sense that it only contains 5m cosets, since 5 has a minimm polynomial of degree m. Let K i denote the nmberof cosets of weight i. Since the code is triple-error-correcting the vales of K 0, K 1, K K are immediately given by K 0 1; K 1 n 1 ; K n ; K n : Since the covering radis of the code is known to be 5, it is therefore sfficient to determine K K 5. Some partial reslts were given by van derhorst Berger[8] who came p with the following bonds for K K 5 : K 1 n(5n +10n 0 ) K 5 n(n +) they conjectred that eqality holds for m 8. They were able to prove this conjectre for 8 m 1 sing a comptersearch. They frther verified that the conjectre does not hold for m 5, 7. The main reslt of this paper is to show that their 0-year-old conjectre holds therefore to prove the following theorem. Theorem 1: Let K i denote the nmberof cosets with a coset leader of weight i in a triple-error-correcting binary primitive BCH code of length n m 0 1 where m 8. Then the distribtion of the cosets is given by K 0 1 K 1 n 1 K n K n K 1 n(5n +10n 0 ) K 5 n(n +): One shold observe that an even harder problem is the fll problem to find the weight distribtion of the vectors in any coset of the binary primitive triple-error-correcting codes (or their extended codes). This problem has been stdied in the papers by Charpin Zinoviev [] Charpin, Helleseth, Zinoviev [5]. This problem remains open foreven m while it was solved (i.e., expressed in terms of some exponential sms) forodd m in these papers. II. COSETS OF BCH CODES OF LENGTH n m 0 1 The proof of or main reslt in Theorem 1 will be rather self-contained. We will apply simplify some of the techniqes developed in van derhorst Berger[8] in combination with some new ideas needed to prove their conjectre. The focs of the very long technical paper by van der Horst Berger [8] was to constrct a complete decoding algorithm for the triple-error-correcting BCH code. Their stdies of the complete decoding algorithm was partially motivated by the desire to se the triple-error-correcting BCH code in sorce coding. We will mainly focs on the distribtion of the weights of the cosets in order to prove their conjectre. The binary triple-error-correcting primitive BCH code has paritycheck matrix H defined by H 1... n (n01) (n01) /$ IEEE

2 178 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., APRIL 00 where is an element of order n m 0 1 in GF( m ), i.e., the code consists of all binary codewords c (c 0 ;c 1 ;...;c n01) GF() n sch that ch tr 0. The syndrome s of a received vector r is s rh tr ( ;S ;S 5 ) where r (r 0 ;r 1 ;...;r n01) H tr denotes the transpose of the matrix H. The decoding starts by compting the syndrome of the received vector then finding a coset leaderin the coset with the same syndrome s (;S ;S 5). The vectors in a coset all have the same syndrome or problem is to find a coset leader, i.e., a vector of smallest possible weight with this syndrome. Therefore we need to find nonzero distinct elements X 1;X ;...;X w (the error-locations) in GF( m ) with the smallest possible w sch that w S i Xj i (1) j1 for i 1; ; 5. Eqivalently, we need to determine a locator polynomial (X) w (X + X i)x w + 1X w w of minimal weight w, sch that (1) holds with nonzero distinct X i s in GF( m ). The Newton identities give a relation between the coefficients of the locatorpolynomial (X) the error-locations via S S + 1 S S i + 1 S i01 + S i i i 0 for i w. Fori>wthen 111 S i + 1S i w01s i0w+1 + ws i0w 0: We se the convention that i 0when i>w. The Newton identities imply 1; S 1 S ; S 5 1 S 1 + S + S : Throghot this correspondence, we define the trace fnction from GF( m ) to GF() by (x) A. The Cosets of Weight w 1 m01 i0 x : The locatorpolynomial of a coset leaderof weight 1 is (X) X + since the syndrome ( ;S ;S 5 ) is given by S j X j 1 for j 1; ; 5. The cosets of weight 1 are therefore the n cosets with syndromes of the form ;S 1;S 5 1 where 0. B. The Cosets of Weight w The locatorpolynomial of a coset leaderof weight corresponding to the error-locations X 1 X leads to a syndrome given by S j X j 1 + Xj for j 1; ; 5. Note that 0since the error-locations are distinct. We obtain by simple direct calclations from the Newton identities that S S 1 + S 5 S S. Comparing the two expressions for,wehave (S + S 1 ) S 5 + S 5 1 S : Since 1, the locatorpolynomial in the case of a coset leader of weight is given by (X) X + X + S + S 1 : () This polynomial has two distinct nonzero zeros in GF( m ) when 0, S S 1 S + S 1 S 1 0. Note that the two expressions for lead to the following relations between the syndrome components M S + S 1S + S 5 + S 1 0: Observe that there are n choices of 0 (n 0 1) choices of S sch that S + 0. Therefore the cosets with syndromes ( ;S ;S 5 ) where 0, S, S + 0 S 5 is sch that the condition M 0 holds are all the n cosets of weight. C. The Cosets of Weight w In this case, the Newton identities lead to 1 the following two expressions for : S + S 1 + S 1 S 5 + S S : Note that if S the first expression sbstitted into the second one implies S 5, 5 contradicting that w. Therefore, S comparing the expressions above leads to S 5 + S 1S S + S 1 sbstitting for in the expression for gives where S + S 1 + S + S 1 + S 5 + S 1S S + S 1 M S + S 1 M S + S 1S + S 5 + S 1: Hence, the locatorpolynomial in the case w is (X) X + X + S5 + S 1S X M S + + : () S + Note that M 0since X 1 X X 0(all X i are nonzero elements) for the case of an error corresponding to a coset leader of weight. Even thogh we do not need this fact in this paperit is sefl forthe decoding algorithm of these codes to observe that it follows from well

3 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., APRIL known properties of cbic polynomials that a necessary condition for three distinct zeros of the locator polynomial is that S 5 + S 5 1 (S : )5 D. The Cosets of Weight w 5 We will stdy the conditions forcosets of weight w 5with a locator polynomial with nonzero distinct zeros X 1, X, X, X X 5. The following theorem characterizes some particlar cosets of weight 5 in the triple-error-correcting codes with a syndrome (;S ;S 5) where 0. These will laterbe shown to be all cosets of weight w 5 with 0. Theorem : Let D be a coset with syndrome ( ;S ;S 5 ) sch that 0, M S + S + S 5 + 0; S + 1. Then D is a coset of weight 5. Proof: Since the covering radis of the triple-error-correcting BCH code is known to be 5, it is sfficient to show that the weight w of the coset D is at least 5. Since 0 the coset has weight at least 1. By the trace condition in the theorem it follows that S therefore the coset has weight at least. Frther, since S + 1 it follows that w since the locator polynomial () for w does not have its zeros in m GF( ) nless S + 0. The coset cannot have weight either, since the locatorpolynomial () fora vectorof weight has constant term 0 when M 0 therefore cannot have three distinct nonzero zeros. It follows that the coset has weight at least. We therefore assme that the coset has weight we will show that this is impossible. From the Newton identities, we obtain S + + S5 + S5 1 + S + S 1 : Sbstitting for in the expression for gives S 5 + S S 1 + (S + S 1 ) : Hence, the locatorpolynomial is (X) X + X + X + S + S 1 + X + S5 + SS 1 + S + : () Since this polynomial has fordistinct zeros in m GF( ) it can be factored as (X) (X + ax + b)(x +(a + )X + d): (5) Comparing coefficients, we obtain d + b + a(a + ) ad + b(a + ) bd: From the first identity, we obtain from the second, we get Hence, d + b + a(a + ) ad + b(a + ) a + ab + a (a + )+ab + b a + a (a + )+b: b + a + a (a + ) sbstitting the vale for b in the expression for d above gives +(a + ) + a(a + ) d : The third identity now gives +a +a (a + ) +(a+) +a(a+) S 1: Sbstitting S + S 1 + S5 + SS 1 + S + S 1 into this expression, we obtain (a + ) + S + S 1 + a (a + ) a + S + S 1 + a(a + ) S 5 + S S 1 + S + S 1 : Collecting the coefficients of, the constant term dividing the reslting expression by a(a + )S 1 gives where + S 1 k S + S 1 + S 5 + S S 1 S 1 a(a + ) + k 0 () + S + S 1 S 1 M S a(a + ) a (a + ) + a (a + ) S 1 MS 1 S x (x +1) (7) x a. The trace of k is given by (k) M + S + + x + x M S + + : (8) Since M 0, the trace condition in the theorem gives S + (k) 1: This contradicts () which only has a soltion in m GF( ) when (k) 0, therefore a coset weight of w for D is impossible. Since the code has covering radis 5, we conclde that the coset D has weight 5. Remark1: Note that the nmberof cosets of weight w 5of this form is n(n +1). This follows since there are n choices of 0 foreach nonzero, there are (n +1) choices of S sch that S + S 1 S 1 1. Observe that S 5 is niqely determined by S since M 0. The following reslt can be fond in [8]. We inclde a simpler more direct proof for the sake of completeness. Lemma 1: There are (5n +1n) cosets of weight 5 with 0. Proof: Note that, for m >, there are (n +1) cosets with 0. Frther, observe that all nonzero cosets with 0mst have odd weight. Indeed, as we saw above, it is impossible forcosets of weight, since X 1 X. That cosets of weight with 0 are impossible can be seen from the following simple argments. Assme that some coset D has syndrome ( 0;S ;S 5 ) assme that X 1, X, X, X are error-locations of some vector of D. Recall, that X 1, X, X, X are mtally distinct nonzero elements in GF( m ). Now we see from (1) that for any h GF( m ) a -tple (X 1 + h; X + h; X + h; X + h) forms the error-locations for a

4 170 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., APRIL 00 vectorwith the same syndrome ( 0;S ;S 5). This can easily be verified by checking the syndrome of this -tple sing that similarly Xi X i X i X i S 1 S 1: Now take h X 1, which is not forbidden. Then we obtain a -tple (X + X 1;X + X 1;X + X 1) which has this syndrome. Bt this means that D is a coset of weight. The nmberof nonzero cosets of weight with 0 are the nmber of (nordered) ways to select nonzero distinct X 1, X X sch that X 1 +X +X 0 is therefore eqal to n(n 0 1). This means that the nmberof cosets of weight 5 with 0therefore is ((n +1) 0 1) 0 n(n 0 1) (since we cannot choose S S 5 both eqal to zero), which eqals (5n +1n). Remark: Note that adding the nmberof cosets in Lemma 1 with 0to the nmberof cosets with 0in Remark 1 obtained from Theorem we obtain the following sefl bond n(n +1) K 5 n(n +): E. The Cosets of Weight w + 5n +1n To prove the conjectre by van der Horst Berger [8] it is sfficient to show that all cosets with 0, except forthe ones in Theorem, have weight at most. In order to prove Theorem, we need the following crcial lemma which is proved by Moreno Moreno [11]. Lemma : Let F be the algebraic closre of F GF( m ). Let f (x);g(x) F [x] where deg f < r deg g g(x) is a polynomial with t distinct zeros in F. Let L denote the set of zeros of g(x) in F.If f(x) g(x) h(x) + h(x) forany rational fnction h(x) F (x), then xf nl p (01) (t + r 0 ) m +1: Based on this lemma we are able to prove the following theorem. This is a key reslt in order to complete the determination of the coset distribtion. Lemma : Let 0, F GF( m ).Form 10 for given i 0or1 there exists an element x F nf0; 1g sch that i x (x +1) 0: Proof: Let T be the nmberof elements in F 0 F nf0; 1g, obeying the conditions in the theorem. Frther, let e(f) (01) (f). Then stard methods give 8T xf 1+e 1+ 1+e x (x +1) xf xf xf xf 1+ e xf e + x (x +1) + xf x (x +1) + xf e xf (x + x +1) x (x +1) : x (x +1) (x +1) x In order to apply the bond of Lemma, we have to check that all fnctions a(x)b(x) which appearin the sms above satisfy the condition of the lemma. We take, forexample, the fnction x(x +1). We have to show that there is no rational h(x) F (x) sch that x (x +1) h(x) + h(x): (9) Sppose, on the contrary, that there is a rational fnction h(x) a(x)b(x) which satisfies (9). We can assme withot loss of generality that gcd (a(x);b(x)) 1. We obtain xb(x) (x +1) a(x)(a(x)+b(x)): This is a polynomial eqality. So, the left-h side of this eqality is divisible by (x +1). This means that b(x) is divisible at least by (x +1). Bt then, the right h side of this eqality is divisible by (x +1), which is impossible, since a(x) cannot be divisible by x +1 when b(x) is divisible by x+1. We therefore conclde that the eqality (9) is impossible. Hence we can apply Lemma to this sm. The other fnctions in all the sms above are treated in the same way. Using the bonds forthe exponential sms given in the lemma above implies that all the last seven sms are pper bonded by c p m + 1, where c takes the vales ; ; ; ; ; ;, respectively, starting from p the first sm after 1. It follows therefore that 8T m 0 0 m 0 7 so T is strictly positive for m 10. Theorem : Let m 10 let D be a coset sch that 0 which does not obey both the conditions M S + S 1S + S 5 + S 1 0 S + S 1 S 1 1. Then D is a coset of weight at most. Proof: We will show that we can find a polynomial (X) of degree with for zeros (not necessarily distinct) in GF( m ) with the reqired syndrome. We will show that we can select the coefficient of X in the polynomial (X) in () sch that it has all zeros in GF( m ). This implies that (1) holds therefore that the coset has weight at most. If some of the zeros of (X) are 0 orrepeated this leads to a coset of weight less than. Set x to be an element of GF( m ) nf0; 1g, sch that x (x +1) S + S 1 S 1 0: This can be done by Lemma. Set to be a soltion in GF( m ) of the qadratic eqation where + S 1 + k 0 k MS 1 + S + + x (x +1):

5 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., APRIL This can be done since (k) 0, by the choice of x in the previos step. Then set 1 S + S 1 + S5 + SS 1 + (S + S 1 ) a x b + a + a (a + ) d +(a + ) + a(a + ) (X) (X + ax + b)(x +(a + )X + d): Note that since x f0; 1g 0, we cannot have a 0nor a. Frther, observe that (ba ) Since a x, then we obtain (ba ) 0: Similarly we obtain d (a + ) + a + a (a + ) a S + S 1 +(a + ) + a (a + ) a S + S 1 a + a + a + a +1 S + a + a + a a + +1 : S + + S 1 + x + x +1 (S + ) + k x + x +1 x (x +1) + x +1+x +1 x (x +1) +(a + ) + a(a + ) (a + ) S + S 1 + a + a(a + ) (a + ) : Note that this is the same formla as for (ba ) with a+ replaced by a oreqivalently x replaced by x +1. Therefore we have d (a + ) 0: Then note that (X) factors into linear factors (X) X + X +(a(a + )+b + d)x rotine calclations show that +(ad +(a + )b)x + bd (X) X + 1 X + X + X + : Foreach i let T i be the sm of the ith powers of the roots of (X) (repeating roots according to their mltiplicity in sch sms). Then it remains to show that T i Si for i 1; ; 5. The Newton identities give s T 1 1 T 1 T 1 + T 1 + T 5 1T 1 + T + T 1 + T 1 since 1,wehaveT 1. Then T 1 S S by the definitions of 1. Frther, T 5 1 S 1 + S + S 1 + S 5 by the definitions of 1,. We conclde that the zeros of (X) obey the syndrome eqations in (1). Note that if some of the zeros of (X) are 0 oreqal, it means that the coset has a coset leaderof weight less than. In any case, the coset D has weight at most. Proof of Theorem 1: The theorem was shown to be tre for 8 m 1 by a compter search de to van der Horst Berger [8]. From Remark 1 following Theorem we know that the nmber of cosets of weight w 5 with 0 is at least n(n +1). Frther, from Remark following Lemma 1 we know that the nmber of cosets of weight w 5with 0eqals (5n +1n), which implies that K 5 n(n +). According to Theorem all cosets with 0otherthan those in Theorem have weight at most. It follows therefore that K 5 n(n +) therefore, since the total nmber of cosets is (n +1), that K 1 n(5n +10n0 ). III. THE COSET DISTRIBUTION OF EXTENDED BCH CODES OF LENGTH n m WITH MINIMUM DISTANCE d 8 In this section we determine the coset distribtion of the extended codes of the triple-error-correcting BCH codes. Denote by B the extended binary primitive BCH code of length N m with minimm distance 8. The parity-check matrix H ext of B is defined by H ext n (n01) (n01) where is an element of order n m 0 1 in GF( m ). Note that the syndrome in this case is a -tple (S 0 ; ;S ;S 5 ). Denote by 0 i the nmberof cosets of B of weight i. Since B is an extended binary [N m ;k N 0 m 0 1]-code with covering radis at most, we have immediately that 0 i N i0 i0 0 i 0 i01: Using that that clearly 0 i N i for i 1; ;, we dedce that 0 +0 N N 0 5 N N N N (N 0 1)(5N +8): 1 Hence, to find the complete coset distribtion we have to find the nmber 0 or 0. From the previos sections we know the coset

6 17 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., APRIL 00 TABLE I COSET DISTRIBUTION OF BCH CODES TABLE II COSET DISTRIBUTION OF EXTENDED BCH CODES distribtion for the code of length n m 0 1. Frther, we observe that 0 K 5 : This can be seen since any coset of weight of the code B of length N m is redced to a coset of weight 5 of the corresponding code of length n m 01, vice versa, when we do overall parity checking. In other words, more precisely, a coset with syndrome ( ;S ;S 5 ) has weight 5 in the binary triple-error-correcting BCH code of length n m 0 1 if only if (S 0 0;;S ;S 5) is the syndrome of a coset of weight in the extended code. Therefore, 0 K 5 the coset distribtion of the extended code follows directly from the coset distribtion of the binary primitive triple-error-correcting BCH code this gives the final reslt below. Theorem : Let 0i denote the nmberof cosets with a coset leader of weight i in the extended triple-error-correcting binary primitive BCH code of length N m where m 8. Then the distribtion of the cosets is given by N 1 REFERENCES [1] E. F. Assms Jr H. F. Mattson Jr, Some -error-correcting BCH codes have covering radis 5, IEEE ans. Inf. Theory, vol. IT-, pp. 8 9, May 197. [] E. R. Berlekamp, Algebraic Coding Theory. New York: McGraw-Hill, 198. [], The weight enmerators for certain sbcodes of the second-order Reed-Mllercodes, Inf. Contr., vol. 17, pp , [], Weight enmeration theorems, in Proc. Sixth Allerton Conf. Circit Systems Theory, Urbana, IL, 198, pp [5] P. Charpin, T. Helleseth, V. A. Zinoviev, On cosets of weight of binary BCH codes with minimal distance 8 exponential sms, Prob. Inf. ans., vol. 1, no., pp. 01 0, 005. [] P. Charpin V. A. Zinoviev, On coset weight distribtions of the -error-correcting BCH-codes, SIAM J. Discr. Math., vol. 10, no. 1, pp , Feb [7] T. Helleseth, All binary -error-correcting BCH codes of length 1 have covering radis 5, IEEE ans. Inf. Theory, vol. IT-, pp , Mar [8] J. A. van der Horst T. Berger, Complete decoding of triple-errorcorrecting binary BCH codes, IEEE ans. Inf. Theory, vol. IT-, pp , Mar [9] T. Kasami, Weight distribtions of Bose-Chadhri-Hocqenghen codes, in Combinatorial Mathematics Its Applications, R. Bose T. Dowling, Eds. Chapel Hill, NC: University of North Carolina Press, 199, ch. 0. [10] F. J. MacWilliams N. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherls: North-Holl, 199. [11] C. Moreno O. Moreno, Exponential sms Goppa codes: I, in Proc. Amer. Math. Soc., vol. 111, 1991, pp On Qasi-Cyclic Interleavers for Parallel bo Codes Joseph J. Botros, Member, IEEE Gilles Zémor, Member, IEEE 0 N 0 N 0 1 (N 0 1)(N 0 5N 0 ) N (N 0 1)(5N +8) 0 (N 0 1)(N +1): Forcompleteness sake, the nmberof cosets with a coset leaderof weight i in the triple-error-correcting binary primitive BCH code of length n m 0 1 in the extended code are given in the Tables I II when m 5; ; 7. IV. CONCLUSION We have determined the distribtion of cosets in the binary primitive triple-error-correcting BCH codes of length n m 0 1. This solves the conjectre from 197 by van der Horst Berger. As an easy conseqence this also gives the coset distribtion of the extended codes of length N m minimm distance d 8. ACKNOWLEDGMENT The athors wold like to thank the anynomos reviewers for their very detailed carefl comments that considerably improved the presentation of this paper. Abstract In this correspondence, we present an interleaving scheme that yields qasi-cyclic trbo codes. We prove that romly chosen members of this family yield with probability almost 1 trbo codes with asymptotically optimm minimm distance, i.e., growing as a logarithm of the interleaver size. These interleavers are also very practical in terms of memory reqirements their decoding error probabilities for small block lengths compare favorably with previos interleaving schemes. Index Terms Convoltional codes, iterative decoding, minimm distance, qasi-cyclic codes, trbo codes. I. INTRODUCTION It is now well known that the behavior of trbo codes, althogh very powerfl nder high noise, exhibits an error floor phenomenon that can be explained by poor minimm distance properties. More specifically, it can be shown that for romly chosen interleavers, the expected minimm distance of a classical two-level trbo code remains constant [1], [19], i.e., does not grow with block length. Can the error floor behavior of trbo codes be improved by designing the interleaver in a way that differs from pre rom choice? Manscript received Janary 1, 005; revised December 0, 005. The material in this correspondence was presented in part at the 00 IEEE International Symposim on Information Theory, Chicago, IL, Jne/Jly 00. The athors are with École Nationale Spériere des Télécommnications, 75 Paris 1, France ( botros@enst.fr; zemor@enst.fr). Commnicated by M. P. Fossorier, Associate Editor for Coding Techniqes. Digital Object Identifier /TIT /$ IEEE