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The BCH-Goppa decoding as a moment problem and a tau function over finite fields
__ -&If3
a
25 November
1996
PHYSICS
ELX3’IER
LETTERS
A
Physics Letters A 223 (1996) 75-81
The BCH-Goppa decoding as a moment problem and a tau function over finite fields Yoshimasa Nakamura ’ Department
cfl Electronics,
Received 21 November
Doshisha
Universi~,
Tunabe. Kyoto 610-03. Japarl
1995; accepted for publication Communicated by A.P. Fordy
18 September
1996
Abstract A tau function of the finite nonperiodic Toda equation over finite fields is introduced. A new decoding algorithm of the BCH-Goppa codes is presented being based on a moment problem over finite fields. A relationship of the zeros of orthogonal polynomials to Goppa polynomials is also discussed.
1. Introduction
The finite nonperiodic Toda equation [ 1] is one of the fundamental equations in nonlinear sciences. It is well known that the Toda equation can be regarded as a continuous limit of the QR algorithm for computing eigenvalues of a Jacobi matrix [ 21. An expression of the Toda equation was already found by Rutishauser [ 31 in 19.54 as a continuous limit of the qd algorithm. Recently the application of the qd algorithm to coding theory was for the first time discussed in Ref. [ 41. The reduced form of the Toda equation called Moser’s dynamical system [ I] also appears in many areas. In Ref. [ 51, a hierarchy of Moser’s system is introduced as a deformation equation of discrete and continuous Stieltjes measures. An integrable deformation of Stieltjes measures can be described explicitly. The key is that the tau function for the Toda equation (cf. Ref. [ 61) is essentially a positive Hankel determinant ’ Present address: Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka 560, Japan.
of moments. The positivity therefore guarantees the Toeplitz condition for the existence of a solution to the moment problem. Here the moment problem is the inverse problem of constructing a Stieltjes measure from a given set of moments [ 71. In this paper we consider an analogue of the moment problem over finite fields GF( 4). The aim is to investigate the intrinsic structure of the BCH-Goppa decoding in coding theory [ 81 in terms of the tau function of the finite nonperiodic Toda equation. In Section 2, we review the tau function and a moment problem. As an application a new decoding algorithm for the BCH-Goppa codes is designed in Section 3. Here moments and Stieltjes measures correspond to coefficients of a syndrome polynomial and error location polynomials in BCH-Goppa decoding, respectively. It is proved that the algorithm solves any BCH-Goppa decoding problem. Two decoding examples are given. In Section 4, it is shown that any Goppa polynomial should be coprime with an orthogonal polynomial defined by a Jacobi matrix. This is a theoretical contribution of our algorithm.
0375-9601/96/$12.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SO375-960 1(96)007 18-9
Y. Nukumura / Physic> Letters A 223 (1996175-81
76
2. The finite Toda equation
and a moment
problem
ho h,
hl h2
...
..
H,= First of all, we note a relationship among four sets of variables to the finite nonperiodic Todaequation which will be the starting point of this paper. The equation of motion for n particles has the Lax representation dL dt:
L - L+T,
= rL+
h
al
..
0
.
’ a,_~
h, .
I
.
(6)
h2p2
In our case H,, is a positive definite matrix [ 51. Let us set & = detHk,
‘.. ‘.
h,_, h,
k = 1,2,.
,
3
0
al b2
L=
L]
i h,_l
.
(1)
ho hl . hl 112. . & = det .. . hk_, hk . r
d, = h,,
a,-] 6,
hk_2 hk_1
hk hk+,
h2k_3 h2k-, (7)
k=2,3,....
where a; > 0, [A,B] = AB - BA, and L+ indicates the strictly upper triangular part of the Jacobi matrix L. Moser [ I] introduced the rational function of degree N
Now we proceed to a tau function. Define new variablesg(t) >Oandgk(r),k=l,2,...,through
f(r:)=E,T(Zf-L)-lE,,
Let us introduce an infinite sequence of k x k Hankel determinants rk ( t) ,
E,=
(
1,o )...(
; EC.
0 T, > (2)
The numerator and the denominator have no common zeros. Since L is symmetric and aj > 0, the poles of f( z ), denoted by Aj, are real and mutually distinct. Moreover, every residue of f( z ) at z = Aj is positive. Hence f( z ) admits the partial fraction expansion
f(z)=-fpL ,j=I
ZpAj’
-5j2=1.
Let us introduce the Markov parameters tional function f(z) by
f(Z)
=g-$.
(3)
j=I
hk of the ra-
(4)
I;=0
$ iogg
= -2h,,
/
Tk(t)
=det
@ = hk. g
r!?
I
gl
. . .
.!?I g2 .
. ..
.
gk-I
gk
(8)
gk-l
\
gk
(9)
. . &k-z . I
wherek= 1,2,.... Since gk come from the Markov parameters hk of f( z ) , Tk = gkAk. Hence 7k ( t) > 0 for k = 1,2,. . , II. In Ref. [ IO] it is shown that the sequence rk( t) can be regarded as a tau function for the finite Toda equation. We can construct a solution of the Toda equation ( 1) by the direct formula
,
k=1,2
,...,
n-1,
Tk
The residues r12 and the poles Al directly determine the Markov parameters hk by
(5)
ho = 1, j=l
It is known (cf. Ref. [ 91) that the wherek= 1,2,.... condition deg f( i ) = n is equivalent to the regularity of the II x II Hankel matrix
If We substitute dk = gdkTk and dk = gek?k into ( 10) we obtain
Y. Nakumura/Physics
LII,=
xl%zG
,
k= 1,2 )...,
rz-
k= I,2 ,...I
II,
(12)
where we set A0 = 1 and & = 0. This is an alternative proof of the classical identity [ 111 in orthogonal polynomials. We here call Ak and & the tau function for the finite Toda equation. In conclusion we have the diagram {ak.
bk}
(2)
A
lhd’
(7)
Four sets of variables to the Toda equation are related to each other in a purely algebraic manner. Though each arrow in ( 13) can be found in the literature (cf. Refs. [ 1,lO,l 1 ] ) , the cyclic relationship helps us to design a new algorithm for computing {T,~, Aj} from {hk} in this paper. Next we discuss a moment problem. First we consider the following expression of f(z),
f(z)=
Dci@u(v) J
d/*(V) = 2 rj26( v - hi)dv, ,j=l
(14)
where 6( z ) is the delta function. Let (z k, be moments defined by the discrete Stieltjesmeasure p(z), namely (:“) =
d&z)
= $,‘S(z
- A.i)dz
(17)
j=l
The condition dk > 0, k = I, 2,. is known as the necessary and sufficient condition by Toeplitz for the existence of solution to moment problems [ 71. Since Ak(t) > 0, k= l,...,n, themomentproblem (16) associated with the finite nonperiodic Toda equation is always solvable. Here we have established a method for solving the moment problem ( 16) explicitly.
3. Tau function over finite fields and BCH-Goppa decoding algorithm
z-v
-32
Remark 2.1. Let {hk; k = 1, . ,2n - 1} be a given set of moments. If Al > 0, . , A,, > 0, then p(z) defined by
is a Stieltjes measure which gives rise to {hk; k = 1,2,. . .} as a set of moments, namely, hk = k dp( z). Conversely, the Stieltjes measure J-“, ,’ (17) always gives a sequence of positive Hankel determinants Al,. . . , A, through (3) and (14).
{r,*,Aj}
{Adk} -
77
Our idea is to find a way through {dk, &} and {ak, bk} to { rj*, Aj) by using the diagram ( 13). As was shown in this section we have
I,
Ak _
_
Letrers A 223 (1996) 75-81
J--3o 2ER. m zkd&),
Let us briefly review the BCH-Goppadecoding [ 81. Let k be a positive odd number and let a be a root of unity such that ak = I. Let GF(2”‘) be a simple extension field of GF(2) having the primitive element (Y.We write a generator polynomial of a BCH code I as G(X). Let ffl’, a’+’ ,...,a Tfr-’ be roots of I. Then the check matrix H can be expressed as the Vandermonde matrix
(15)
Noting (z’) = cy=, hikrj2, k = 0, 1,. . . and (5), we see that the moments (z”) are nothing but the Markov hk of f(z).Let us consider a moment problem for finding the Stieltjes measure ,u( z) from a givensetofmoments(zk)=hkfork=1,2,...,2n-
paKm%XS
H=[;??;
(f;f;,j.
(18)
1, {rj”*Aj;j=l....,?Z} T
a moment problem
{hl; k=0,...,2n-
1)
(16)
Suppose that one sent the code word c(x) a(x)G(x) andtheotherreceived b(x) = c(x)+e(s), where e(x) is the unknown error. We set s(x)
E b(x)
(modG(x)).
=
(19)
78
Y. Nu!umuru/Physics
Letters A 223 (1996175-81
The decoding problem is the problem of finding e(x) from the syndrome s(x). In vector notation, s = bHT = eHT. Define the syndrome polynomial
S(Z)
Sj=S((Cl’)‘+‘).
=CSjZ",
(20)
.i=o
’
The basic idea of the BCH-Goppa decoding is as follows (cf. Ref. [ 81). Let 1J be Gauss’s symbol. If one can find polynomials a(z) and w(z) such that
w(z)
ff(z)
E S(z)
(modz’),
degg(z)
where P is a permutation of k-words. (ii) Introduce the Jacobi matrix over GF( 2”)
< [i-J, L=
degw(z)
< lki
- 1,
where the elements are defined by
ak
=
JAk- I Ak+I ,
k=1,2
,...,
n-1,
&
W(C) E c U(f)
5
(22)
(modz’),
jCJ
then we find the error e(.r) = cjEJ xj. It should be noted that the decoding problem is very similar to the moment problem (16) of finding {rj*,Aj;j = l,... , n} from given {hk; k = 0,. ,2n - l}. See diagram (13). In this section we present a new algorithm for the BCH-Goppa decoding problem by extending Remark 2.1 over the finite field GF(2”‘). Our decoding algorithm is as follows. (i)For a set of given syndrome {Se, St , . &n-1 }, where Sk E GF( 2”‘)) calculate the determinants
, Sk-t
k=2,3
(25)
(21)
one can know the error location d by factoring a( z ) into p(z) = njEJ( z - cd), where J = {jle, $ 0). Namely, if
.&=s,,
(24)
,...,
rz,
& . . . &k-2
&=det[‘,
,...,
k=1,2
;;;;
;;si,j,
n.
Here we define the determinant
(23) over GF(2”‘) by
I
_
b+
Ak- I -1 &I
k= 1,2 ,...,
?I,
(26)
where A0 = 1 and & = 0. (iii) Compute the rational function
w(z)
= EIT(d - L)-‘E,, c(z) El = (I, 0,. . , O)T.
(27)
(iv) Factor the polynomials a(z) and w( z ) and calculate the partial fraction expansion of w( z) /a( z ) -o(z) u(z)
E CL jEJ
z - ,i
(modz’).
(28)
One obtains the error polynomial e(x) = C,EJ XJ and then an n-error decoding of BCH-Goppa code has been finished. RemarkJ.1. Since L is symmetric, ak always appear in (T( z ) and w (z ) as the quadratic form ~2. Hence the square root in the definition of ak (26) does not cause any problem. It is known [ 121 that Ak $0 if k is equal to II, the number of errors that actually occurred, as the central theorem of Peterson’s decoding algorithm. Moreover, dk = 0 if k is greater than II. To know the
Y. Nukumura / Physics Lerters A 223 (1996) 75-81
79
number of errors II we need computing the tau function &, k = K, K - 1,. . . , where K is a large number. If
S(Z) = &+SIZ modz* .Usingcr’= 1 anda =a+l, we have S( z ) = cr4 + cyz. Consequently, we get
dl io,
Al = a4,
....
d”#O,
(29)
then our algorithm solves any n-error decoding problem. When Ak $ 0 and Ak+l = 0 for some k E {I.. . , II - 1}, we set n = k and consider the k-error decoding problem. Remark 3.2. We can generalize this algorithm to a suitable extension of any finite fields GF(p”‘), where p is a prime number. Moreover, we can present a Goppa decoding algorithm when we replace cuj by mutually distinct elements LY.,E GF(p”‘) and replace Z” by a Goppa polynomial M( z ) of degree r satisfying MC&j)
i0
[
131.
Finally we give two decoding examples. Example 3. I ( 1-error decoding). Let G(x) = x3 + x + 1 be a generator polynomial and let z2 be a Goppa polynomial. We here consider the extension field GF( 2’) = (0, 1, a,cu+l,a2,Cu2+1,(Y*+~,*2+ (US- I}, where cr’ = 1. Since the corresponding BCHGoppa code I : c = (CO,cl,. . . , cf,) is a cyclic Hamming code, c satisfies
j=O
F c.i = 0 2 - aJ
(1, l,O, quence and b = sequence. Problem (O,O, O,O, 1, 0,O) the corresponding Let
c
=
Since 61 = dl/Al = a -3 = cr4, the Jacobi matrix L and the rational function w( z ) /CT(z 1 are simply L =
cx4
w(z) _
(a4),
a(i)
-
(mod:*).
Z -cX4
This indicates e = (O.O,O, 0, l,O, 0). Example 3.2 (3-error decoding). Let G(x) = xi0 + x8 + x5 + x4 + x2 + x + 1 be a generator polynomial and let z6 be a Goppa polynomial. We here consider the extension field GF( 2”)) where CX” = 1. Let S(Z) = l+~+a’0~2+~“+cu’0~J+~s~s
The sum of the numbers of time of products and additions of this algorithm for 2-error decoding is less than that of the Euclidean algorithm [ 141 which is known as a useful BCH-Goppa decoding algorithm. However, with respect to the computational complexity, our algorithm is not superior than the BerlekampMassey algorithm [ 131 and the Euclidean algorithm for multi-error decoding. The merit is a theoretical contribution on Goppa polynomials discussed in the next section.
e
d, = a.
(modz6)
be a given syndrome polynomial. Note that cuJ+cu+ 1 = 0 and cy” + cy5 + 1 = 0. The determinants are then Al=l,
A2=a5,
A3=1,
f&=1,
&=a5,
&=l.
The Jacobi matrix L takes the following
form,
L=(+?Jj).
This gives a solution of the finite Toda equation over the finite field GF( 24). The rational function defined by L is -w(z) r(z) _
z2 +a5
=
23 + 22 + cy5
a3 Z -CXs
I
cl5 + Z -ff5
ff’2
12 Z-Lu
(modt6).
In
conclusion we obtain the error polynomial = x3 + x5 + x1*, or equivalently, e = e(x) (0,0,0,1,0,1,0,0,0,0,0,0,1,0,0).
(modz*).
l,O,O, 0) be a transmitted se(1, l,O,l,l,O,O) be a received in to know the error vector e = and c from b. We first derive syndrome polynomial S(Z). Set
4. Goppa polynomial polynomials
and zeros of orthogonal
One of the most challenging problems in coding theory [ 81 is to find an asymptotically good Goppa polynomial M(z). In Section 3 we set simply M(z) = zr.
80
Y. Ndumura
/ Physics Lertrrs A 223 (1996) 75-81
Namely, we formulate our decoding algorithm only for the BCH-Goppa code. As was mentioned in Remark 3.2, there is no essential difficulty extending the algorithm to other Goppa polynomials, however, the choice of good Goppa polynomials is still an open problem. Here let us consider how to test a candidate for the Goppa polynomial. This is a theoretical contribution of our algorithm. The characteristic polynomial pn ( z ) of the fz x II Jacobi matrix L(“) = L in ( 1) over R is an orthogonal polynomial of degree n (cf. Refs. [ 5.1 I ] ) Here the eigenvalues a, (‘I) of L’“), or equivalently, zeros of the orthogonal polynomial pn (Z ) are mutually distinct < a!“’ < real numbers. We set . . < a;!‘, < al”) Jfl . . Then there is the following property of orthogonal polynomials [ 1 I]: . . . < ai”, < a;“-‘) < cxjn’ < &-I) < dfl) < , . . , where crj(‘-‘) are eigenvalues
A=
(31)
ck =
dk-
d,=$
I bi+ I
An2
,
k=
I,2 ,...,
A.!- I A: dk = p+ AkA;-,
as
I1 - 1, AkA;_,
Ak- I A;_,
k = 2,3, . . . , n,
A; =O,
AL = det
/+I
J+’
{Se, St,. . . . &,,_I}
defined by a given syndrome
Sk Sk+, . . . &_I
of the (II - 1) x (II - 1) Jacobi matrix k= 1,2 ,...,
11.
for a large number K. It would be interesting to find an effective Goppa polynomial from the view point of zeros of orthogonal polynomials.
His algorithm is based on the qd algorithm (cf. Ref. [ 31) for finding eigenvalues of A over Iw. There does not appear a square root in ck. However, it is known that the qd algorithm over Iw is numerically unstable, namely, Ai becomes zero in some case. Hence Faybusovich’s algorithm cannot avoid such a difficulty. On the other hand, the finite nonperiodic Toda equation (1) is solved by the QR decomposition of (the exponential of) the Jacobi matrix L [2] where the QR algorithm is stable. The algorithm designed here would be related with the QR algorithm. Indeed, as was mentioned in Remark 3.1, since Al #O, . . . , dk j 0, our BCH-Goppa decoding algorithm solves any BCH-Goppa k-error decoding problem. This regularity condition of the tau function can be regarded as the Toeplitz condition for the moment problem over finite fields.
5. Discussions
Acknowledgment
In this paper we consider unexpected applications of a tau function of integrable systems and a moment problem to the BCH-Goppa decoding problem. Recently, Faybusovich [4] gave a Goppa decoding algorithm. He considers the nonsymmetric tridiagonal matrix of the form
The author would like to thank to Professor L. Faybusovich for stimulating discussions. Thanks are also due to Mr. E. Teranishi for programming the decoding algorithm designed in this paper. This work was supported in part by Grant-in-Aid for Scientific Research no. 07210105 from the Japan Ministry of Education;
L0l-l)
=
On the other hand, any Goppa polynomial M(Z) should hold M( aji) $0 for any j, where Cyj are zeros of the generator polynomial G( z ). It is to be noted that a,i appear in our algorithm as zeros of the characteristic polynomial p,,(z) = det( z,l - Lr”)) of the Jacobi matrix L’“) over GF(q) Thus we can choose a candidate for the Goppa polynomial M ( z ) through the condition M(L~)‘)) #O.
j = 1,.
.,n,
rz = 1,. ..,K
(30)
Y. Nu!umuru/Physic.s
Science, and Culture and Aid of Doshisha University’s Research Promotion Fund.
[ 71 N.I. Ahiezer and M. Krein. Transl. Math. Monographs, Vol. 2. Some questions in the theory of moments (Am. Math. Sot.. Providence,
[ 8 1 E.R. Berlekamp,
References III J. Moser. in: Lecture Notes in Physics, Vol. 38. Dynamical systems, theory and applications, ed. J. Moser (Springer, Berlin. 1975) p. 467. 121 W.W. Symes, Physica D 4 (1982) 275. 131 H. Rutishauser, Arch. Math. 5 (1954) 132. 141 L. Faybusovich, in: Linear algebra for control theory, eds. P Van Dooren and 8. Wyman (Springer, Berlin, 1994) p. 87. 151 Y. Nakamura and Y. Kodama, Acta. Appl. Math. 39 ( 1995) 435. [61 R. Hirota, Y. Ohta and J. Satsuma. J. Phys. Sot. Japan 57 (1988) 1901.
81
Letters A 223 (1996) 75-81
1962). Algebraic
coding theory (McGraw-Hill. New York, 1968). 191 F.R. Gantmacher, The theory of matrices, Vol. 2 (Chelsea. New York, 1959). I IO] Y. Nakamura, Phys. Len. A I95 ( 1994) 346. [ II ] G. Szego, Colloq. Publ., Vol. 23. Orthogonal Polynomials, 4th Ed. (Am. Math. Sot., Providence, 1975). 1121R.E. Blahut, Theory and practice of error control codes (Addison-Wesley, Reading, MA, 1983). I 131 E.R. Berlekamp. IEEE Trans. Inform. Theory IT-19 ( 1974) 590.
[ 141 Y. Sugiyama, M. Kasahara, S. Hirasawa and T. Namekawa, IEEE Trans. Inform. Theory IT-22 ( 1976) 518.
a
25 November
1996
PHYSICS
ELX3’IER
LETTERS
A
Physics Letters A 223 (1996) 75-81
The BCH-Goppa decoding as a moment problem and a tau function over finite fields Yoshimasa Nakamura ’ Department
cfl Electronics,
Received 21 November
Doshisha
Universi~,
Tunabe. Kyoto 610-03. Japarl
1995; accepted for publication Communicated by A.P. Fordy
18 September
1996
Abstract A tau function of the finite nonperiodic Toda equation over finite fields is introduced. A new decoding algorithm of the BCH-Goppa codes is presented being based on a moment problem over finite fields. A relationship of the zeros of orthogonal polynomials to Goppa polynomials is also discussed.
1. Introduction
The finite nonperiodic Toda equation [ 1] is one of the fundamental equations in nonlinear sciences. It is well known that the Toda equation can be regarded as a continuous limit of the QR algorithm for computing eigenvalues of a Jacobi matrix [ 21. An expression of the Toda equation was already found by Rutishauser [ 31 in 19.54 as a continuous limit of the qd algorithm. Recently the application of the qd algorithm to coding theory was for the first time discussed in Ref. [ 41. The reduced form of the Toda equation called Moser’s dynamical system [ I] also appears in many areas. In Ref. [ 51, a hierarchy of Moser’s system is introduced as a deformation equation of discrete and continuous Stieltjes measures. An integrable deformation of Stieltjes measures can be described explicitly. The key is that the tau function for the Toda equation (cf. Ref. [ 61) is essentially a positive Hankel determinant ’ Present address: Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka 560, Japan.
of moments. The positivity therefore guarantees the Toeplitz condition for the existence of a solution to the moment problem. Here the moment problem is the inverse problem of constructing a Stieltjes measure from a given set of moments [ 71. In this paper we consider an analogue of the moment problem over finite fields GF( 4). The aim is to investigate the intrinsic structure of the BCH-Goppa decoding in coding theory [ 81 in terms of the tau function of the finite nonperiodic Toda equation. In Section 2, we review the tau function and a moment problem. As an application a new decoding algorithm for the BCH-Goppa codes is designed in Section 3. Here moments and Stieltjes measures correspond to coefficients of a syndrome polynomial and error location polynomials in BCH-Goppa decoding, respectively. It is proved that the algorithm solves any BCH-Goppa decoding problem. Two decoding examples are given. In Section 4, it is shown that any Goppa polynomial should be coprime with an orthogonal polynomial defined by a Jacobi matrix. This is a theoretical contribution of our algorithm.
0375-9601/96/$12.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SO375-960 1(96)007 18-9
Y. Nukumura / Physic> Letters A 223 (1996175-81
76
2. The finite Toda equation
and a moment
problem
ho h,
hl h2
...
..
H,= First of all, we note a relationship among four sets of variables to the finite nonperiodic Todaequation which will be the starting point of this paper. The equation of motion for n particles has the Lax representation dL dt:
L - L+T,
= rL+
h
al
..
0
.
’ a,_~
h, .
I
.
(6)
h2p2
In our case H,, is a positive definite matrix [ 51. Let us set & = detHk,
‘.. ‘.
h,_, h,
k = 1,2,.
,
3
0
al b2
L=
L]
i h,_l
.
(1)
ho hl . hl 112. . & = det .. . hk_, hk . r
d, = h,,
a,-] 6,
hk_2 hk_1
hk hk+,
h2k_3 h2k-, (7)
k=2,3,....
where a; > 0, [A,B] = AB - BA, and L+ indicates the strictly upper triangular part of the Jacobi matrix L. Moser [ I] introduced the rational function of degree N
Now we proceed to a tau function. Define new variablesg(t) >Oandgk(r),k=l,2,...,through
f(r:)=E,T(Zf-L)-lE,,
Let us introduce an infinite sequence of k x k Hankel determinants rk ( t) ,
E,=
(
1,o )...(
; EC.
0 T, > (2)
The numerator and the denominator have no common zeros. Since L is symmetric and aj > 0, the poles of f( z ), denoted by Aj, are real and mutually distinct. Moreover, every residue of f( z ) at z = Aj is positive. Hence f( z ) admits the partial fraction expansion
f(z)=-fpL ,j=I
ZpAj’
-5j2=1.
Let us introduce the Markov parameters tional function f(z) by
f(Z)
=g-$.
(3)
j=I
hk of the ra-
(4)
I;=0
$ iogg
= -2h,,
/
Tk(t)
=det
@ = hk. g
r!?
I
gl
. . .
.!?I g2 .
. ..
.
gk-I
gk
(8)
gk-l
\
gk
(9)
. . &k-z . I
wherek= 1,2,.... Since gk come from the Markov parameters hk of f( z ) , Tk = gkAk. Hence 7k ( t) > 0 for k = 1,2,. . , II. In Ref. [ IO] it is shown that the sequence rk( t) can be regarded as a tau function for the finite Toda equation. We can construct a solution of the Toda equation ( 1) by the direct formula
,
k=1,2
,...,
n-1,
Tk
The residues r12 and the poles Al directly determine the Markov parameters hk by
(5)
ho = 1, j=l
It is known (cf. Ref. [ 91) that the wherek= 1,2,.... condition deg f( i ) = n is equivalent to the regularity of the II x II Hankel matrix
If We substitute dk = gdkTk and dk = gek?k into ( 10) we obtain
Y. Nakumura/Physics
LII,=
xl%zG
,
k= 1,2 )...,
rz-
k= I,2 ,...I
II,
(12)
where we set A0 = 1 and & = 0. This is an alternative proof of the classical identity [ 111 in orthogonal polynomials. We here call Ak and & the tau function for the finite Toda equation. In conclusion we have the diagram {ak.
bk}
(2)
A
lhd’
(7)
Four sets of variables to the Toda equation are related to each other in a purely algebraic manner. Though each arrow in ( 13) can be found in the literature (cf. Refs. [ 1,lO,l 1 ] ) , the cyclic relationship helps us to design a new algorithm for computing {T,~, Aj} from {hk} in this paper. Next we discuss a moment problem. First we consider the following expression of f(z),
f(z)=
Dci@u(v) J
d/*(V) = 2 rj26( v - hi)dv, ,j=l
(14)
where 6( z ) is the delta function. Let (z k, be moments defined by the discrete Stieltjesmeasure p(z), namely (:“) =
d&z)
= $,‘S(z
- A.i)dz
(17)
j=l
The condition dk > 0, k = I, 2,. is known as the necessary and sufficient condition by Toeplitz for the existence of solution to moment problems [ 71. Since Ak(t) > 0, k= l,...,n, themomentproblem (16) associated with the finite nonperiodic Toda equation is always solvable. Here we have established a method for solving the moment problem ( 16) explicitly.
3. Tau function over finite fields and BCH-Goppa decoding algorithm
z-v
-32
Remark 2.1. Let {hk; k = 1, . ,2n - 1} be a given set of moments. If Al > 0, . , A,, > 0, then p(z) defined by
is a Stieltjes measure which gives rise to {hk; k = 1,2,. . .} as a set of moments, namely, hk = k dp( z). Conversely, the Stieltjes measure J-“, ,’ (17) always gives a sequence of positive Hankel determinants Al,. . . , A, through (3) and (14).
{r,*,Aj}
{Adk} -
77
Our idea is to find a way through {dk, &} and {ak, bk} to { rj*, Aj) by using the diagram ( 13). As was shown in this section we have
I,
Ak _
_
Letrers A 223 (1996) 75-81
J--3o 2ER. m zkd&),
Let us briefly review the BCH-Goppadecoding [ 81. Let k be a positive odd number and let a be a root of unity such that ak = I. Let GF(2”‘) be a simple extension field of GF(2) having the primitive element (Y.We write a generator polynomial of a BCH code I as G(X). Let ffl’, a’+’ ,...,a Tfr-’ be roots of I. Then the check matrix H can be expressed as the Vandermonde matrix
(15)
Noting (z’) = cy=, hikrj2, k = 0, 1,. . . and (5), we see that the moments (z”) are nothing but the Markov hk of f(z).Let us consider a moment problem for finding the Stieltjes measure ,u( z) from a givensetofmoments(zk)=hkfork=1,2,...,2n-
paKm%XS
H=[;??;
(f;f;,j.
(18)
1, {rj”*Aj;j=l....,?Z} T
a moment problem
{hl; k=0,...,2n-
1)
(16)
Suppose that one sent the code word c(x) a(x)G(x) andtheotherreceived b(x) = c(x)+e(s), where e(x) is the unknown error. We set s(x)
E b(x)
(modG(x)).
=
(19)
78
Y. Nu!umuru/Physics
Letters A 223 (1996175-81
The decoding problem is the problem of finding e(x) from the syndrome s(x). In vector notation, s = bHT = eHT. Define the syndrome polynomial
S(Z)
Sj=S((Cl’)‘+‘).
=CSjZ",
(20)
.i=o
’
The basic idea of the BCH-Goppa decoding is as follows (cf. Ref. [ 81). Let 1J be Gauss’s symbol. If one can find polynomials a(z) and w(z) such that
w(z)
ff(z)
E S(z)
(modz’),
degg(z)
where P is a permutation of k-words. (ii) Introduce the Jacobi matrix over GF( 2”)
< [i-J, L=
degw(z)
< lki
- 1,
where the elements are defined by
ak
=
JAk- I Ak+I ,
k=1,2
,...,
n-1,
&
W(C) E c U(f)
5
(22)
(modz’),
jCJ
then we find the error e(.r) = cjEJ xj. It should be noted that the decoding problem is very similar to the moment problem (16) of finding {rj*,Aj;j = l,... , n} from given {hk; k = 0,. ,2n - l}. See diagram (13). In this section we present a new algorithm for the BCH-Goppa decoding problem by extending Remark 2.1 over the finite field GF(2”‘). Our decoding algorithm is as follows. (i)For a set of given syndrome {Se, St , . &n-1 }, where Sk E GF( 2”‘)) calculate the determinants
, Sk-t
k=2,3
(25)
(21)
one can know the error location d by factoring a( z ) into p(z) = njEJ( z - cd), where J = {jle, $ 0). Namely, if
.&=s,,
(24)
,...,
rz,
& . . . &k-2
&=det[‘,
,...,
k=1,2
;;;;
;;si,j,
n.
Here we define the determinant
(23) over GF(2”‘) by
I
_
b+
Ak- I -1 &I
k= 1,2 ,...,
?I,
(26)
where A0 = 1 and & = 0. (iii) Compute the rational function
w(z)
= EIT(d - L)-‘E,, c(z) El = (I, 0,. . , O)T.
(27)
(iv) Factor the polynomials a(z) and w( z ) and calculate the partial fraction expansion of w( z) /a( z ) -o(z) u(z)
E CL jEJ
z - ,i
(modz’).
(28)
One obtains the error polynomial e(x) = C,EJ XJ and then an n-error decoding of BCH-Goppa code has been finished. RemarkJ.1. Since L is symmetric, ak always appear in (T( z ) and w (z ) as the quadratic form ~2. Hence the square root in the definition of ak (26) does not cause any problem. It is known [ 121 that Ak $0 if k is equal to II, the number of errors that actually occurred, as the central theorem of Peterson’s decoding algorithm. Moreover, dk = 0 if k is greater than II. To know the
Y. Nukumura / Physics Lerters A 223 (1996) 75-81
79
number of errors II we need computing the tau function &, k = K, K - 1,. . . , where K is a large number. If
S(Z) = &+SIZ modz* .Usingcr’= 1 anda =a+l, we have S( z ) = cr4 + cyz. Consequently, we get
dl io,
Al = a4,
....
d”#O,
(29)
then our algorithm solves any n-error decoding problem. When Ak $ 0 and Ak+l = 0 for some k E {I.. . , II - 1}, we set n = k and consider the k-error decoding problem. Remark 3.2. We can generalize this algorithm to a suitable extension of any finite fields GF(p”‘), where p is a prime number. Moreover, we can present a Goppa decoding algorithm when we replace cuj by mutually distinct elements LY.,E GF(p”‘) and replace Z” by a Goppa polynomial M( z ) of degree r satisfying MC&j)
i0
[
131.
Finally we give two decoding examples. Example 3. I ( 1-error decoding). Let G(x) = x3 + x + 1 be a generator polynomial and let z2 be a Goppa polynomial. We here consider the extension field GF( 2’) = (0, 1, a,cu+l,a2,Cu2+1,(Y*+~,*2+ (US- I}, where cr’ = 1. Since the corresponding BCHGoppa code I : c = (CO,cl,. . . , cf,) is a cyclic Hamming code, c satisfies
j=O
F c.i = 0 2 - aJ
(1, l,O, quence and b = sequence. Problem (O,O, O,O, 1, 0,O) the corresponding Let
c
=
Since 61 = dl/Al = a -3 = cr4, the Jacobi matrix L and the rational function w( z ) /CT(z 1 are simply L =
cx4
w(z) _
(a4),
a(i)
-
(mod:*).
Z -cX4
This indicates e = (O.O,O, 0, l,O, 0). Example 3.2 (3-error decoding). Let G(x) = xi0 + x8 + x5 + x4 + x2 + x + 1 be a generator polynomial and let z6 be a Goppa polynomial. We here consider the extension field GF( 2”)) where CX” = 1. Let S(Z) = l+~+a’0~2+~“+cu’0~J+~s~s
The sum of the numbers of time of products and additions of this algorithm for 2-error decoding is less than that of the Euclidean algorithm [ 141 which is known as a useful BCH-Goppa decoding algorithm. However, with respect to the computational complexity, our algorithm is not superior than the BerlekampMassey algorithm [ 131 and the Euclidean algorithm for multi-error decoding. The merit is a theoretical contribution on Goppa polynomials discussed in the next section.
e
d, = a.
(modz6)
be a given syndrome polynomial. Note that cuJ+cu+ 1 = 0 and cy” + cy5 + 1 = 0. The determinants are then Al=l,
A2=a5,
A3=1,
f&=1,
&=a5,
&=l.
The Jacobi matrix L takes the following
form,
L=(+?Jj).
This gives a solution of the finite Toda equation over the finite field GF( 24). The rational function defined by L is -w(z) r(z) _
z2 +a5
=
23 + 22 + cy5
a3 Z -CXs
I
cl5 + Z -ff5
ff’2
12 Z-Lu
(modt6).
In
conclusion we obtain the error polynomial = x3 + x5 + x1*, or equivalently, e = e(x) (0,0,0,1,0,1,0,0,0,0,0,0,1,0,0).
(modz*).
l,O,O, 0) be a transmitted se(1, l,O,l,l,O,O) be a received in to know the error vector e = and c from b. We first derive syndrome polynomial S(Z). Set
4. Goppa polynomial polynomials
and zeros of orthogonal
One of the most challenging problems in coding theory [ 81 is to find an asymptotically good Goppa polynomial M(z). In Section 3 we set simply M(z) = zr.
80
Y. Ndumura
/ Physics Lertrrs A 223 (1996) 75-81
Namely, we formulate our decoding algorithm only for the BCH-Goppa code. As was mentioned in Remark 3.2, there is no essential difficulty extending the algorithm to other Goppa polynomials, however, the choice of good Goppa polynomials is still an open problem. Here let us consider how to test a candidate for the Goppa polynomial. This is a theoretical contribution of our algorithm. The characteristic polynomial pn ( z ) of the fz x II Jacobi matrix L(“) = L in ( 1) over R is an orthogonal polynomial of degree n (cf. Refs. [ 5.1 I ] ) Here the eigenvalues a, (‘I) of L’“), or equivalently, zeros of the orthogonal polynomial pn (Z ) are mutually distinct < a!“’ < real numbers. We set . . < a;!‘, < al”) Jfl . . Then there is the following property of orthogonal polynomials [ 1 I]: . . . < ai”, < a;“-‘) < cxjn’ < &-I) < dfl) < , . . , where crj(‘-‘) are eigenvalues
A=
(31)
ck =
dk-
d,=$
I bi+ I
An2
,
k=
I,2 ,...,
A.!- I A: dk = p+ AkA;-,
as
I1 - 1, AkA;_,
Ak- I A;_,
k = 2,3, . . . , n,
A; =O,
AL = det
/+I
J+’
{Se, St,. . . . &,,_I}
defined by a given syndrome
Sk Sk+, . . . &_I
of the (II - 1) x (II - 1) Jacobi matrix k= 1,2 ,...,
11.
for a large number K. It would be interesting to find an effective Goppa polynomial from the view point of zeros of orthogonal polynomials.
His algorithm is based on the qd algorithm (cf. Ref. [ 31) for finding eigenvalues of A over Iw. There does not appear a square root in ck. However, it is known that the qd algorithm over Iw is numerically unstable, namely, Ai becomes zero in some case. Hence Faybusovich’s algorithm cannot avoid such a difficulty. On the other hand, the finite nonperiodic Toda equation (1) is solved by the QR decomposition of (the exponential of) the Jacobi matrix L [2] where the QR algorithm is stable. The algorithm designed here would be related with the QR algorithm. Indeed, as was mentioned in Remark 3.1, since Al #O, . . . , dk j 0, our BCH-Goppa decoding algorithm solves any BCH-Goppa k-error decoding problem. This regularity condition of the tau function can be regarded as the Toeplitz condition for the moment problem over finite fields.
5. Discussions
Acknowledgment
In this paper we consider unexpected applications of a tau function of integrable systems and a moment problem to the BCH-Goppa decoding problem. Recently, Faybusovich [4] gave a Goppa decoding algorithm. He considers the nonsymmetric tridiagonal matrix of the form
The author would like to thank to Professor L. Faybusovich for stimulating discussions. Thanks are also due to Mr. E. Teranishi for programming the decoding algorithm designed in this paper. This work was supported in part by Grant-in-Aid for Scientific Research no. 07210105 from the Japan Ministry of Education;
L0l-l)
=
On the other hand, any Goppa polynomial M(Z) should hold M( aji) $0 for any j, where Cyj are zeros of the generator polynomial G( z ). It is to be noted that a,i appear in our algorithm as zeros of the characteristic polynomial p,,(z) = det( z,l - Lr”)) of the Jacobi matrix L’“) over GF(q) Thus we can choose a candidate for the Goppa polynomial M ( z ) through the condition M(L~)‘)) #O.
j = 1,.
.,n,
rz = 1,. ..,K
(30)
Y. Nu!umuru/Physic.s
Science, and Culture and Aid of Doshisha University’s Research Promotion Fund.
[ 71 N.I. Ahiezer and M. Krein. Transl. Math. Monographs, Vol. 2. Some questions in the theory of moments (Am. Math. Sot.. Providence,
[ 8 1 E.R. Berlekamp,
References III J. Moser. in: Lecture Notes in Physics, Vol. 38. Dynamical systems, theory and applications, ed. J. Moser (Springer, Berlin. 1975) p. 467. 121 W.W. Symes, Physica D 4 (1982) 275. 131 H. Rutishauser, Arch. Math. 5 (1954) 132. 141 L. Faybusovich, in: Linear algebra for control theory, eds. P Van Dooren and 8. Wyman (Springer, Berlin, 1994) p. 87. 151 Y. Nakamura and Y. Kodama, Acta. Appl. Math. 39 ( 1995) 435. [61 R. Hirota, Y. Ohta and J. Satsuma. J. Phys. Sot. Japan 57 (1988) 1901.
81
Letters A 223 (1996) 75-81
1962). Algebraic
coding theory (McGraw-Hill. New York, 1968). 191 F.R. Gantmacher, The theory of matrices, Vol. 2 (Chelsea. New York, 1959). I IO] Y. Nakamura, Phys. Len. A I95 ( 1994) 346. [ II ] G. Szego, Colloq. Publ., Vol. 23. Orthogonal Polynomials, 4th Ed. (Am. Math. Sot., Providence, 1975). 1121R.E. Blahut, Theory and practice of error control codes (Addison-Wesley, Reading, MA, 1983). I 131 E.R. Berlekamp. IEEE Trans. Inform. Theory IT-19 ( 1974) 590.
[ 141 Y. Sugiyama, M. Kasahara, S. Hirasawa and T. Namekawa, IEEE Trans. Inform. Theory IT-22 ( 1976) 518.