on the minimal interpolation problem and decoding rs codes

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 4, JULY 2000 1573

On the Minimal Interpolation Problem and Decoding RSCodes

Xiao Ma and Xin-mei Wang, Member, IEEE

Abstract—Some properties of the minimal interpolation problemare investigated, from which a simple proof of the validity of theWelch–Berlekamp algorithm is presented. A new key equation is derived,which is closely related to the classical key equation of syndrome-decodingalgorithm and can be solved by the Welch–Berlekamp algorithm.

Index Terms—Decoding Reed–Solomon codes, Euclidean algorithm,key equation, minimal interpolation problem, Reed–Solomon codes,Welch–Berlekamp algorithm.

I. INTRODUCTION

Reed–Solomon (RS) codes have found increasing application insuch areas as telecommunications, data storage and transmission,video systems, etc. They have been adopted for several internationalstandards and continue to be attractive for new applications. There-fore, investigating the efficient decoding algorithm for RS codes isvery important. The following three algorithms are well known [5]:Peterson algorithm, Berlekamp–Massey algorithm, and Euclideanalgorithm. All of these algorithms require calculating the syndromes.We called them syndrome-decoding algorithms. In the early 1980’s,L. Welch and E. R. Berlekamp [1] proposed a key equation fordecoding RS codes, which does not require the prior computation ofthe syndromes and can be solved by the Welch–Berlekamp algorithm(WB algorithm). The Welch–Berlekamp key equation depends onlyon the remainder polynomial and the generator polynomial. So theWelch–Berlekamp decoding algorithm is referred to as the remainderdecoding algorithm. In 1992, M. Morii and M. Kasahara [2] deriveda generalized key equation from which the Welch–Berlekamp keyequation can be derived. Another Welch–Berlekamp key equation canbe found in [3] or [8]. These key equations show that the problem ofdecoding RS codes is related to the minimal interpolation problem,which has been deeply studied in system theory where real or complexfield is concerned, see, for example, [6]. The algorithms for solvingthe WB-type key equation have been studied by several authors, see[3], [4], [7]–[10], etc.

The minimal interpolation problem studied in this correspondenceis specified by Berlekamp [3], which is a special case of the weak gen-eralized rational interpolation problem introduced by Blackburn [7].The problem may also be equivalently described in the form of poly-nomial congruence [8]. We derive some properties based on the equiv-alent description. The derivation is independent of any algorithm forsolving the problem, and thus different from that in references, wheresome derivation is based on some recursive algorithms, and hence byinduction. Using these properties, a simple proof of the validity of theWB algorithm is presented. The equivalent problem also shows that theminimal interpolation problem can be solved not only by the WB algo-rithm but also by the Euclidean algorithm, a fact similar to the problemof decoding RS codes. Based on a new key equation, we prove that theproblem of decoding RS codes can be transformed into the minimalinterpolation problem.

Manuscript received July 2, 1998; revised November 17, 1999.The authors are with National Key Lab of ISN, XiDian University, Xi’an,

710071, China.Communicated by I. F. Blake, Associate Editor for Coding Theory.Publisher Item Identifier S 0018-9448(00)05022-7.

II. THE RATIONAL INTERPOLATION PROBLEM

A. Descriptions of the Problem

Let FFF be a fixed field. Supposexi 2 FFF , yi 2 FFF , 1 � i � j, andyi’s (1 � i � j) arej distinct points. Consider the following problem.

Problem 1: Find a pair of polynomialsN(z),W (z) satisfying

N(yi) =W (yi)xi; 1 � i � j:

Remark 1: Clearly, Problem 1 has a trivial solution,N(z) =W (z) = 0. Here we are only interested in the nontrivial solutions.

As in [3], we letz denote the indeterminate, and writeN for N(z),W for W (z). We refer to a solution as “N=W ” even though this refersto the pair of polynomials, not to their quotient, which in this corre-spondence we denote by the horizontal bar,N

W. SupposeN=W is a

solution of Problem 1,f is a common factor of them, i.e., there existsome pair of polynomialsn, w satisfying

N = nf W = wf:

If n=w is also a solution, we refer to thatN=W can be reduced ton=w. In this case, we callf removable if the degree off(denoted bydeg f) < 0. N=W andn=w are considered as the same solution ifdeg f = 0 (deg f = 0 iff f 2 FFF , f 6= 0, and we assume thatdeg f = �1 if f = 0). The reader should notice that, for example, acommon factor of the form(z� yi) might or might not be removable,and only common factors which might not be removable are of thatform. We call a solution which has no removable common factors irre-ducible. An irreducible solution may have one or more common factorsof the form(z � yi). Define the rank of a solutionN=W as

Rank (N=W ) = maxf2degW; 1 + 2degNg:

Berlekamp [3] proved that there can never be more than one irre-ducible solution of rank� j. The WB algorithm can always find sucha solution, thereby proving that the irreducible solution of rank� j isnot only unique, but that it exists. We will give another description ofProblem 1, then prove the same conclusion.

Remark 2: It appears that Problem 1 is to find a rational functionN(z)W (z) passing through the points(yi; xi) for 1 � i � j. So we callProblem 1 a rational interpolation problem, which is a special case ofthe weak generalized rational interpolation problem introduced by S.R. Blackburn [7]. Also note that the definition of the rank of a solutionis equivalent to that of the complexity in [7] in the sense that a solutionhas a minimal rank if and only if it has a minimal complexity.

LetP (z) be any interpolating polynomial such thatP (yi) = xi for1 � i � j. For instance,P (z) is taken to be the Lagrange interpolatingpolynomial

P (z) =

j

i=1

xiPi(z)

Pi(yi)

where

Pi(z) =

k=j

k=1k 6=1

(z � yk):

Let�(z) = j

i=1(z� yi), it can be shown that Problem 1 is equiv-alent to Problem 2 below (note that a similar polynomial congruenceis also found in [8]).

0018–9448/00$10.00 © 2000 IEEE

1574 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 4, JULY 2000


N(z) =W (z)P (z)mod�(z):

B. Some Properties of the Solutions of the Rational InterpolationProblem

In this subsection, we will prove some lemmas leading to the WBalgorithm.

Lemma 1: There exists at least one irreducible solution of Problem 2with rank� j.

Proof: Consider the set

S = f(N;W )jmaxf2degW; 1 + 2degNg � jg:

For any(N;W ) 2 S, (M; V ) 2 S, f 2 FFF , define

(N;W ) + (M;V ) = (N +M;W + V )

f(N;W ) = (fN; fW ): (1)

It can be verified thatS is a linear space of dimensionj +1. By theEuclidean division algorithm, there exists one unique pair of polyno-mialsQ, R such that

N �WP = Q�+R; degR < j:

Define the mapping

EEE: S ! ff 2 FFF [z]jdeg f < jg

EEE((N;W )) = R: (2)

Clearly,EEE is a linear mapping from the linear space of dimensionj+1into the linear space of dimensionj. Therefore, the dimension of itskernel space is greater than zero, which implies that there exists at leastone solution of Problem 2 with rank� j. Hence there exists at least oneirreducible solution of Problem 2 with rank� j.

It should be pointed out that Lemma 1 is clearly implied by the resultsin the related references, for example, see [3], [7], [8], since some avail-able algorithms do find the solution with rank� j. The proof above isindependent of any algorithms.

Lemma 2 [3, Lemma 1], or [9, Theorem 1]:If N=W is an irre-ducible solution of Problem 2 andM=V is another solution, such that

Rank (N=W ) + Rank (M=V ) � 2j: (3)

ThenM=V can be reduced toN=W .Proof: For any two solutions, in particular forN=W andM=V ,

there exists two polynomialsQ1, Q2, such that

N �WP = Q1� M � V P = Q2�: (4)

Then

(NV �MW )P = (MQ1 �NQ2)�: (5)

and

GCD(�; P )jN GCD(�; P )jM (6)

whereGCD(�; P ) represents the greatest common factor of the twopolynomials�, P . From (5) and (6),

(NV �MW )P

GCD(�; P )=MQ1 �NQ2

GCD(�; P )� (7)

which implies

�j(NV �MW ): (8)

From the definition of the rank and (3)

degNV =degN+degV �Rank (N=W )�1

2+Rank (M=V )

2<j

degMW =degM+degW �Rank (M=V )�1

2+Rank(N=W )

2<j

we havedeg (NV �MW ) < j. From (8), we have

NV �MW = 0: (9)

Let d = GCD(W;V ). Then there exist two polynomialsw, v whichare relatively prime such that

W = dw V = dv: (10)

Substituting (10) into (9), we haveNdv = Mdw, wjN , vjM . LetN

w= M

v= h, so

N = hw M = hv: (11)

Substituting (10) and (11) into (4), we have

hw � dwP = Q1� hv � dvP = Q2�: (12)

SinceGCD(w;v) = 1, there exist two polynomialss, t such thatsw + tv = 1. Combining with (12), we obtain

h� dP = (sQ1 + tQ2)�: (13)

Equation (13) shows thath=d is also a solution. From (10) and (11),bothN=W andM=V can be reduced toh=d. By the irreducibility ofN=W , we havedeg w = 0. This completes the proof.

Suppose thatN=W ,M=V are two solutions of Problem 2, such that

Rank (N=W ) + Rank (M=V ) = 2j + 1 (14)

and

NV �MW = f� (15)

wheredeg f = 0, we callN=W andM=V complementary. The def-inition is simpler than that given by Berlekamp [3], however, the fol-lowing lemmas show that they are equivalent.

Lemma 3: If N=W andM=V are two solutions of Problem 2 andthey are complementary, then both of them are irreducible solutions,and one of them has rank not greater thanj.

Proof: From (14), the second conclusion is clearly correct. Sup-pose that the first conclusion is not correct. Then there exist two irre-ducible solutions,n=w andm=v, and two polynomialsf , g such thatN = fn, W = fw, M = gm, V = gv, anddeg f + deg g > 0.From (14)

Rank(n=w) + Rank(m=v) = 2j + 1� 2(deg f + deg g) � 2j:

By Lemma 2,n=w andm=v at most differ by a constant commonfactor. Hence the left-hand side of the preceding equation is even, acontradiction.

Lemma 4: LetN=W be the irreducible solution of Problem 2 withrank� j. Then there exists at least one solution which is a complementof N=W .


Proof: The proof is similar to that of Lemma 1. Consider the set

S = f(M;V )jmaxf2deg V; 1 + 2degMg

� 2j + 1� Rank(N=W )g:

It can be verified thatS is a linear space of dimension2j + 2 �Rank (N=W ) under the operations defined by (1). LetK be the kernelspace of the linear mapping defined by (2). It can be proved that the di-mension ofK is greater thanj+1�Rank(N=W ). We can prove theset

T = f(gN; gW )jg is a polynomial such that

Rank(gN=gW ) � 2j + 1� Rank (N=W )g

is a subspace ofK, whose dimension is not greater thanj + 1 �Rank (N=W ). In fact, from

Rank (gN=gW ) � 2j + 1� Rank (N=W )

and

Rank (gN=gW ) = 2deg g + Rank (N=W )

we have

deg g � f2j + 1� 2Rank (N=W )g=2 � j � Rank (N=W ):

Hence there exists at least one solution satisfyingM=V 62 T and

Rank (N=W ) + Rank (M=V ) � 2j + 1:

By Lemma 2, we have

Rank (N=W ) + Rank (M=V ) = 2j + 1: (16)

Therefore, one ofRank (N=W ) andRank (M=V ) is even, and theother is odd. We have either

Rank (N=W ) + Rank (M=V )

= (1 + 2degN) + 2degV > 2degW + (1 + 2degM)

or

Rank (N=W ) + Rank (M=V )

= 2degW + (1 + 2degM) > (1 + 2degN) + 2degV

which impliesdeg (NV �MW ) = j. Since any two solutions satisfy(8), we have

MW �NV = f�; degf = 0: (17)

Equations (16) and (17) show thatM=V is a complement ofN=W .

Lemma 5: If N=W is the irreducible solution of Problem 2 andM=V is one of its complements, then for anym, n 2 FFF , n 6= 0,(nM �mN)=(nV �mW ) is also one of its complements.

Proof: Clearly,(nM �mN)=(nV �mW ) is a solution satis-fying (15).

From

Rank((nM �mN)=(nV �mW ))

� maxfRank(M=V );Rank(N=W )g = Rank(M=V )

and noticing that(nM � mN)=(nV � mW ) cannot be reduced toN=W , we obtain (14) by Lemma 2.

Remark 3: By these lemmas, we conclude that there exists one andonly one irreducible solution of Problem 2 with rank� j. Furthermore,this unique solution has at least one complement. The reader should

notice that the definition of complementary here (essentially borrowedfrom Berlekamp [3]) is closely related to the basis of theFFF [zzz]-moduleintroduced in [8]. In fact, any two complementary solutions define abasis matrix [8, Definition 1 and Lemma 3].

C. The Minimal Interpolation Problem and the Euclidean Algorithm

The minimal interpolation problem related to Problem 1 is describedas


N(yi) =W (yi)xi; 1 � i � j

minimizeRank (N=W )

A problem equivalent to Problem 3 is


N(z) =W (z)P (z)mod�(z)

minimizeRank (N=W )

By Lemmas 1 and 2, the solution of Problem 3 (equivalently,Problem 4) is just the irreducible solution of Problem 1 (equivalently,Problem 2) with rank� j. Problem 4 can be solved by the Euclideanalgorithm.

Theorem 1 (Euclidean Algorithm) :Let s(0)(z)=�(z), t(0)(z)=P (z), and

AAA(0)(z)=1 0

0 1:

Calculate the following recursive equations:

Q(r)(z) =s(r)(z)

t(r)(z):

(Denote the quotient ofs(r)(z) divided by t(r)(z) according to theEuclidean division algorithm)

AAA(r+1)(z) =0 1

1 �Q(r)(z)AAA(r)(z)

s(r+1)(z)

t(r+1)(z)=

0 1

1 �Q(r)(z)

s(r)(z)

t(r)(z):

If 1 + 2degP (z) > j, there exists someR such that

1 + 2deg t(R�1)(z) > j

1 + 2deg t(R)(z) � j:

In this case,t(R)(z)=A(R)22 (z) is the solution of Problem 4, hence also

the solution of Problem 3, whereA(R)22 (z) is the right-low element of

the matrixAAA(R)(z). If 1 + 2degP (z) � j, letR = 0.Proof: The proof is analogous to that of[5, Theorem 7.7.3].

D. The Welch–Berlekamp Algorithm

Sometimes it is not convenient to solve Problem 3 using the Eu-clidean algorithm because it requires calculating the interpolating poly-nomial. However, the WB algorithm can solve Problem 3 directly. Inthis subsection, we will give a simple proof of the validity of the WBalgorithm.

Given1 � k < j. Consider the following problems.

Problem 1(k): Find a pair of polynomialsN(z),W (z) satisfying

N(yi) =W (yi)xi; 1 � i � k:


Problem 3 (k): Find a pair of polynomialsN(z), W (z) satisfying

N(yi) = W (yi)xi; 1 � i � k

minimizeRank (N=W ):

Suppose thatN (k)=W (k) andM (k)=V (k) are two complementarysolutions of Problem 1(k). By Lemmas 1–4, one of them with lowerrank is the solution of Problem 3(k). Without loss of generality, supposethatN (k)=W (k) has lower rank. Let

nk = N (k)(yk+1)� xk+1W(k)(yk+1)

mk = M (k)(yk+1)� xk+1V(k)(yk+1):

We have

Theorem 2 (Welch–Berlekamp Algorithm):If nk = 0 thenN (k)=W (k) and(z � yk+1)M

(k)=(z � yk+1)V(k) are two comple-

mentary solutions of Problem 1(k + 1). Furthermore,N (k)=W (k) isthe solution of Problem 3(k + 1).

If nk 6= 0 then

(z � yk+1)N(k)=(z � yk+1)W

(k)

(nkM(k) �mkN

(k))=(nkV(k) �mkW

(k))

are two complementary solutions of Problem 1(k + 1). One of themwith lower rank is the solution of Problem 3(k + 1).

Proof: SinceN (k)=W (k) andM (k)=V (k) are complementary,

Rank (N (k)=W (k)) + Rank (M (k)=V (k)) = 2k + 1 (18)

and

N (k)V (k) �M (k)W (k) = f

k

i=1

(z � yi) (19)

wheredeg f = 0:nk = 0. It is easily verified thatN (k)=W (k) and

(z � yk+1)M(k)=(z � yk+1)V

(k)

are two solutions of Problem 1(k + 1) satisfying

Rank(N (k)=W (k)) + Rank((z � yk+1)M(k)=(z � yk+1)V

(k))

= 2k + 3 (20)

and

(z�yk+1)N(k)V (k)�(z�yk+1)M

(k)W (k) = f

k+1

i=1

(z�yi) (21)

which implies thatN (k)=W (k) and

(z � yk+1)M(k)=(z � yk+1)V

(k)

are complementary. FurthermoreN (k)=W (k) has lower rank.nk 6= 0. By Lemma 5,N (k)=W (k) and

(nkM(k) �mkN

(k))=(nkV(k) �mkW

(k))

are two complementary solutions of Problem 1(k+1), hence they sat-isfy (18) and (19). So, clearly

(z � yk+1)N(k)=(z � yk+1)W

(k)

and

(nkM(k) �mkN

(k))=(nkV(k) �mkW

(k))

are two solutions of Problem 1 (k+ 1) satisfying (20) and (21), whichimplies they are complementary.

By Theorem 2, in thekth iteration, the WB algorithm can find thesolution of Problem 3(k) and one of its complements from the solutionof Problem 3(k � 1) and one of its complements. According to thefollowing flowchart (see Fig. 1), we will obtain the solutionN (j)=W (j)

of Problem 3 afterj iterations.

III. D ECODING REED–SOLOMON CODES

A. Reed–Solomon Codes

Let � be a primitive element of the finite field GF(q), whereq is aprime power. The RS code over GF(q) with length ofn = q � 1 ischaracterized by the generator polynomial

g(x) = (x� �b) (x� �b+1) � � � (x� �b+d�2)

where b is a nonnegative integer less thanq � 1. A vectorccc = (c0; c1; � � � ; cn�1) is a codeword if and only if the correspondingpolynomial

c(x) =

n�1

i=0

cixi

is a multiple ofg(x). The RS code defined byg(x) has parameters[n = q�1; k = n�d+1; d], which is a maximum distance separablecode. So we can select anyk locations as message locations. Here wewill selectLm = f�k; d � 1 � k � n � 1g as message locationsandLc = f�k; 0 � k � d� 2g as check locations. As an anonymousreferee suggested, it should be explicitly said that “location”�k corre-sponds tock.

Let ccc = (c0; c1; � � � ; cn�1) be the transmitted codeword, andvvv = (v0; v1; � � � ; vn�1) be the received vector. Then the error patterneee = vvv � ccc. Suppose that there are onlyt errors, the error locationssetLe = fXl; 1 � l � tg, and the error value which occurs atXl isYl, i.e.,

ei =0; �i 62 Le;

Yl(6= 0); �i = Xl 2 Le;0 � i � n� 1:

It is known that the error patterneee can be evaluated from the receivedvectorvvv as long ast � (d � 1)=2.

B. Some Known Key Equations of Remainder Decoding Algorithm

The remainder polynomial

r(x) =

d�2

i=0

rixi

is obtained by

r(x) = v(x)mod g(x):

Define the error locator polynomial (note that it is different from thatused in syndrome decoding algorithm, see next subsection)

W (x) =

t

l=1

(x�Xl):

We can writeW (x) as

W (x) = Wm(x)Wc(x)

where

Wm(x) =

c

l=1

(x�Xl); Wc(x) =

t

l=e+1

(x�Xl)


Fig. 1. The flowchart of the Welch–Berlekamp algorithm.

and

Xl 2 Lm; for 1 � l � e

Xl 2 Lc; for eee+ 1 � l � t:

Considering the case ofe = 1 then utilizing the linearity, Welchand Berlekamp [1] proved that there exists a polynomialNm(x)whichsatisfies

pk�kNm(�

k) = rkWm(�k); for �k 2 Lc�fXl; e+1 � l � tg

and

degNm(x) < degWm(x)

where

d�2

i=0

pixi =

d�2

k=1

(x� �b+k):

Let N(x) = Nm(x)Wc(x).Clearly,

pk�kN(�k) = rkW (�k); for �k 2 Lc

and

degN(x) < degW (x) � (d� 1)=2:


This is Welch–Berlekamp key equation. AfterW (x) andN(x) arecalculated, the error values are given by

Yl = f(Xl)N(Xl)

W 0(Xl); if W (Xl) = 0 andXl 2 Lm

where

f(x) = x�b

d�2

i=0

pi�i(b+1)

�i � x

andW 0(x) is the formal derivative ofW (x).In 1992, Masakatu Morii and Masao Kasahara [2] presented the fol-

lowing generalized key equation:

N(�k) = rk kW (�k); for �k 2 Lc

and

degN(x) < degW (x) � (d� 1)=2

where

k = �bkd�2

i=0i6=k

(�i � �k):

The error values are given by

Yl = �(Xl)N(Xl)

W 0(Xl); if W (Xl) = 0 andXl 2 Lm

where

�(x) =1

xbd�2

i=0

(�i � x)

:

From this key equation, both Welch–Berlekamp key equation andLiu’s key equation [2], [11] can be derived.

In [3], Berlekamp gave another key equation

N(�k) = rkF0(�k)W (�k); for �k 2 Lc

and

degN(x) < degW (x) � (d� 1)=2;

where

F (x) =

d�2

i=0

(x� �i):

This key equation is also derived by Dariush Dabiri and Ian F. Blake[8]. The error polynomial is given by

Yl =�N(Xl)

W 0(Xl)F (Xl); if W (Xl) = 0 andXl 2 Lm:

C. A New Key Equation from the Key Equation of Syndrome Decoding

Define syndromesSj = v(�b+j) for 0 � j � d� 2, where

v(x) =

n�1

i=0

vixi:

Define the error locator polynomial

�(x) =

j

l=1

(1� xXl)

and denote

S(x) =

d�2

j=0

Sjxj :

It is known that there exists some polynomial(x) (the so-called error-evaluator polynomial) such that

�(x)S(x) = (x) (modxd�1) (22a)

and

deg(x) < deg�(x) � (d� 1)=2: (22b)

The problem of decoding RS codes may be transformed into

Problem 5: Find a pair of polynomials�(x),(x) satisfying

�(x)S(x) = (x) (modxd�1)

deg(x) < deg �(x) � (d� 1)=2:

minimize deg�(x):

Problem 5 can be solved by Berlekamp–Massey algorithm or by Eu-clidean algorithm. The error values are given by

Yl = �X�b+1(X�1

l )

�0(X�1l )

; for �(X�1l ) = 0: (23)

Define

Vj =

n�1

i=0

�ijvi; for j = 0; � � � ; n� 1:

Then [5, Theorem 8.1.2]

vi = �

n�1

j=0

��ijVj ; for i = 0; � � � ; n� 1:

This pair of vectors,

vvv = (v0; v1; � � � ; vn�1)$ VVV = (V0; V1; � � � ; Vn�1)

is referred to a Fourier transform pair.Let

(r0; � � � ; rd�2; 0; � � � ; 0)$ (R0; � � � ; Rn�1)

be the Fourier transform pair with respect to the remainder polynomial.DenoteSj = R((b+j)) for j = 0; � � � ; n� 1, where the double paren-theses denote modulo-n arithmetic on indices. It is clear that

(s0; � � � ; sd�2; 0; � � � ; 0)$ (S0; � � � ; Sn�1)

is a Fourier transform pair such thatsi = �biri, for i = 0; � � � ; n � 1[5, Theorem 8.1.4].

DenoteS(x) =n�1

j=0

Sjxj . We have

S(��k) = ��bkrk; for k = 0; � � � ; d� 2 (24a)

S(��k) = 0; for k = d� 1; � � � ; n� 1: (24b)

Notice thatS(x), which is slightly different from that defined in thebeginning of this subsection (reminding of that the former isS(x) =

d�2j=0 Sjx

j , so they are equal in the sense ofmod xd�1), satisfies thekey equation (22a)

�(x)S(x) = (x) (modxd�1): (25)

Then there exists some polynomialQ(x) such that

�(x)S(x) = (x) + xd�1Q(x)

= (x) (1� xn) + xd�1(Q(x) + xn�d+1(x)): (26)


Let

Fm(x) =

n�1

k=d�1

(1� x�k) Fc(x) =

d�2

k=0

(1� x�k):

From (24b),Fm(x)jS(x), we have

�(x)S(x)

Fm(x)= (x)Fc(x) + xd�1N(x) (27)

where

N(x) =Q(x) + xn�d+1(x)

Fm(x):

From (27), we obtain

�(x)S(x)

Fm(x)= xd�1N(x)modFc(x): (28)

Conversely, we can obtain (25) from (28).Now, let us consider the relationship between the degrees of some

polynomials in the derivation above.Since in (26),deg �(x)S(x) � n + t � 1, deg (x) < t, so

deg (x) (1 � xn) < n + t and

deg (Q(x) + xn�d+1(x)) < n� d+ 1 + t:

From (27), we havedeg N(x) < t.Therefore, we have proved that Problem 5 is equivalent to Problem

6 given below.

Problem 6: Find a pair of polynomials�(x),N(x) satisfying

�(x)S(x)

Fm(x)= xd=1N (x)modFc(x)

degN(x) < deg �(x) � (d� 1)=2:

minimize deg �(x):

SincedegN(x) < deg�(x) � (d� 1)=2 impliesRank(N=�) =2deg�(x) � d � 1 and the solution of Problem 7 given below isunique, we assert that the solution of Problem 6 must be the solutionof Problem 7.


�(x)S(x)

Fm(x)= xd�1N(x)modFc(x)

minimizeRank(N(x)=�(x)):

Settingx = ��k and substituting (24a) into Problem 7, we get anequivalent problem of Problem 7


N(��k) =��(b+d�1)k

Fm(��k)rk�(�

�k) for k = 0; � � � ; d� 2

minimizeRank(N(x)=�(x)):

This is the new key equation, which is derived from the classical keyequation of syndrome decoding, and only requires the remainder poly-nomial. From the discussions above, we have proved that the problemof decoding RS codes (Problem 5) can be transformed into the minimalrational interpolation problem (Problem 8). The later can be solved byWB algorithm.

From (27)

(x) =�(x)

Fc(x)�S(x)

Fm(x)�

xd�1

Fc(x)N(x)

we have

(X�1l

) = �X�d+1

Fc(X�1l

)N(X�1

l);

for �(X�1l

) = 0 andXl 2 Lm: (29)

Substituting (29) into (23), we get the error values at message locations

Yl = �X�b+1l

(X�1l

)

�0(X�1l

)=

X�(b+d�2)l

Fc(X�1l

)�N(X�1

l)

�0(X�1l

);

for �(X�1l

) = 0 andXl 2 Lm: (30)

Remark 4: By definition, the constant term of the error locatorpolynomial is always equal to1. However, because�(0) can be can-celled from the numerator and the denominator of (30), any solutionof Problem 8 can be used to correct the errors occurring in informationlocations by (30) as long as it satisfies thatdegN(x) < deg�(x) and�(x) hasdeg�(x) distinct nonzero roots in GF(q).

D. An Example of Decoding RS Codes

In this subsection, we give an example ofb = 0:The sequence

a(k) =�(d�1)k

Fm(��k); for k = 0; � � � ; d� 2

can be calculated by recursive equation

a(0) =1

Fm(1)= Fc(1)

a(k + 1) =�d�1(1 + �n�k+1)

1 + �d�k�2a(k);

for 0 � k � d� 3:

These values can also be precomputed for storage. The sequence

b(k) =��(d�2)k

Fc(��k); for k = d� 1; � � � ; n� 1

can be treated with in a similar manner.Let GF(24) = f0; 1; �; �2; � � � ; �14g, where1+� = �4. We have

x 0 � �2 �3 �5 �6 �7 �11

1 + x 1 �4 �8 �14 �10 �13 �9 �12

Let

g(x) = (x+ 1) (x+ �) � � � (x+ �5)

= x6 + �9x5 + �12x4 + �x3 + �2x2 + �4x+ 1:

Theng(x) generates a[15; 9; 7] RS code over GF(24), which can cor-rect three or less errors. Letf�k; 6 � k � 14g be the informationlocations. The values ofa(k) are listed in the following table:

k 0 1 2 3 4 5

a(k) �7 �6 �2 �5 1 �7

Suppose that the received word contains three errors at the field lo-cationsX1, X2, andX3 exhibiting erroneous valuesY1, Y2, andY3distributed as follows:

l 1 2 3

Xl �4 �6 �7

Yl 1 �2 �4

Fromr(x) = v(x) = c(x) + e(x) = e(x)mod g(x), we have

k 0 1 2 3 4 5

rk �14 �7 �10 �13 �14 �10

Hence the inputs of Problem 3 are (note thatxk = a(k) � rk)

yk 1 ��1 ��2 ��3 ��4 ��5

xk �6 �13 �12 �3 �14 �2:


By the WB algorithm given in Fig. 1, we obtain the table at the top ofthis page. The solution is

�(z) = �12(�7z2 + �14z + �4) (z + �11)=

N(z) = �12(�z + �5) (z + �11):

The polynomial�(z) has three distinct roots�11, �9, �8. Hence theerror locations areX1 = �4,X2 = �6,X3 = �7. By (30), we get theerror valuesY2 = �2, Y3 = �4.

IV. CONCLUSION

We have proved that there exists one and only one solution of theminimal interpolation problem (Problem 3). Furthermore, the solutionhas at least one complement. The simple proof is independent of any al-gorithm. The Welch–Berlekamp algorithm is justified, which can findthe solution and one of its complements of the minimal interpolationproblem withk interpolating points from such pair solutions of the cor-responding problem withk � 1 points. We also derived a new keyequation. Although the new key equation does not offer more prac-tical advantages over the other previously known key equations, it is oftheoretical interest. The derivations show that the new key equation isclosely related to the classical key equation of syndrome decoding.

ACKNOWLEDGMENT

The authors wish to thank Dr. Bai Junfeng of Tsinghua Universityfor his providing [4], and Prof. van Lint for providing some useful ma-terials. The authors also wish to thank Prof. Ian F Blake for his usefulsuggestions on this version, and the two anonymous reviewers for theirvaluable comments.

REFERENCES

[1] L. Welch and E. R. Berlekamp, “Error correction for algebraic blockcodes,” U.S. Patent 4 633 470, Sept. 1983.

[2] M. Morii and M. Kasahara, “Generalized key-equation of remainderdecoding algorithm for Reed-Solomon codes,”IEEE Trans. Inform.Theory, vol. 38, pp. 1801–1807, Nov. 1992.

[3] E. R. Berlekamp, “Bounded distance+1 soft-decision Reed-Solomondecoding,” IEEE Trans. Inform. Theory, vol. 42, pp. 704–720, May1996.

[4] X. Dingjia, “Homogeneous interpolation problem and key equation fordecoding Reed-Solomon codes,”Sci. in China (Ser. A), vol. 37, pp.1387–1398, 1994.

[5] R. E. Blahut,Theory and Practice of Error Control Codes. Reading,MA: Addison-Wesley, 1983.

[6] A. C. Antoulaset al., “On the solution of the minimal rational interpo-lation problem,”Liner Algebra and Its Applications, vol. 137/138, pp.511–573, 1990.

[7] S. R. Blackburn, “Fast rational interpolation. Reed-Solomon decoding,and the linear complexity profiles of sequences,”IEEE Trans. Inform.Theory, vol. 43, pp. 537–548, Mar. 1997.

[8] D. Dabiri and I. F. Blake, “Fast parallel algorithms for decoding Reed-Solomon codes based on remainder polynomials,”IEEE Trans. Inform.Theory, vol. 41, pp. 873–885, July 1995.

[9] W. G. Chamberset al., “Algorithm for solving the Welch-Berlekampkey-equation, with a simplified proof,”Electron. Lett., vol. 29, no. 18,pp. 1620–1621, 1993.

[10] W. G. Chambers, “Solution of Welch-Berlekamp key-equation by Eu-clidean algorithm,”Electron. Lett., vol. 29, no. 11, p. 1031, 1993.

[11] T. H. Liu, “A new decoding algorithm for Reed-Solomon codes,” Ph.D.dissertation, Univ. Southern Calif., Los Angeles, CA, 1984.

On Memory Redundancy in the BCJR Algorithm forNonrecursive Shift Register Processes

Michael Schmidt, Student Member, IEEE,andGerhard P. Fettweis, Senior Member, IEEE

Abstract—For computation of a posterioriprobabilities (APP’s), the well-known BCJR algorithm is often applied. If the underlying model is a non-recursive shift register process, it will be shown that this algorithm obtainsa general memory redundancy if the computation is performed over a com-mutative semiring with an absorbing zero element. The result is of partic-ular interest for the BCJR algorithm carried out in the max-log domain.

Index Terms—APP decoding, BCJR algorithm, log domain, max-log do-main, semiring.

I. INTRODUCTION

Symbol-by-symbola posteriori probability (APP) decoding (alsoknown as maximuma posteriori (MAP) decoding1 ) has become animportant tool, due to the implicit generation of soft information. Themost prominent APP decoding algorithm is often derived from the al-gorithm by Bahl, Cocke, Jelinek, and Raviv (BCJR algorithm), whichrecursively computes certain state and transition probabilities [1]. Apossible application of the BCJR algorithm is APP decoding of binaryconvolutional codes with nonrecursive encoders. In this context, Pe-terson reported that the BCJR algorithm can be simplified, meaning

Manuscript received July 14, 1998; revised October 25, 1999.The authors are with the Mannesmann Mobilfunk Chair, Dresden

University of Technology, Chair for Mobile Communications Systems,D-01062 Dresden, Germany (e-mail: [email protected]; [email protected]).

Communicated by F. R. Kschischang, Associate Editor for Coding Theory.Publisher Item Identifier S 0018-9448(00)04643-5.

1The term APP decoding is more appropriate since computing all APP’s in-stead of a maximum APP is usually desired.

0018–9448/00$10.00 © 2000 IEEE

on the minimal interpolation problem and decoding rs codes

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