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Encoding via Grobner bases and discrete Fouriertransforms for several types of algebraic codes

Hajime MatsuiDept. of Electronics and Information Science

Toyota Technological InstituteHisakata 2-12-1, Tenpaku, Nagoya 468-8511, Japan

hmatsui@toyota-ti.ac.jp

Seiichi MitaDept. of Electronics and Information Science

Toyota Technological InstituteHisakata 2-12-1, Tenpaku, Nagoya 468-8511, Japan

smita@toyota-ti.ac.jp

Abstract— We propose a novel encoding scheme for algebraiccodes such as codes on algebraic curves, multidimensional cycliccodes, and hyperbolic cascaded Reed–Solomon codes and presentnumerical examples. We employ the recurrence from the Grobnerbasis of the locator ideal for a set of rational points and thetwo-dimensional inverse discrete Fourier transform. We generalize thefunctioning of the generator polynomial for Reed–Solomon codesand develop systematic encoding for various algebraic codes.

I. I NTRODUCTION

Heretofore, there have been some researches on the encod-ing of codes on algebraic curves, although they are fewerthan researches on the decoding of codes. Heegardet al.[1] proposed an encoding for linear codes with nontrivialautomorphism groups by using Grobner bases for modulesover polynomial rings, which was applied by Chenet al. [3].Matsumotoet al. [4] proposed another encoding for codes oncurves, based on the linear combination of extended Reed–Solomon (RS) codes by the work of Yaghoobianet al. [5].

In this research, we propose a novel encoding scheme forvarious algebraic codes; this scheme is considered to be thenatural generalization of the well-known encoding for RScodes. We first establish a simple but non-systematic encodingthat employs two-dimensional (2-D) inverse discrete Fouriertransforms (IDFT) and that generalizes the encoding for RScodes by using one-dimensional IDFT (that is, the Mattson–Solomon polynomial). Since the syndromes correspond to thediscrete Fourier transform (DFT), we also obtain a concisedecoding via Berlekamp–Massey–Sakata (BMS) algorithm.Next, we establish systematic encoding in the sense of theseparation of given information and generated redundant inaresulting code-word. This second method of encoding employsa Grobner basis and its 2-D linear feedback shift-registerand corresponds to the Euclidean division by the generatorpolynomial in the case of RS codes.

Both the methods often employ the enlargement of thefinite-field arrays to the entire plane by the elements ofGrobner bases, typically, the defining equation of the algebraiccurves. As a more essential idea of our encoding and decodingscheme, we can mention the following duality for substitution

(xiyj)(αr, αs) = (xrys)(αi, αj) = αir+js.

Then, the rational points having any zero are exceptional;however, they can be treated similarly to the case of lengthenedRS codes as shown in section VIII.

II. CODES ON ALGEBRAIC CURVES

LetZ0 denote the set of non-negative integers. LetX denotea non-singular Cba algebraic curves overK := Fq for a, b ∈ Z0

with a < b and gcd(a, b) = 1. Then, the genus ofX is givenby g := (a − 1)(b − 1)/2, andX has only oneK-rationalpoint at infinityP∞. We fix a primitive elementα of K. LetP = {Ph}0≤h<n denote a set ofK-rational points of the formPh = (αr, αs), i.e., non-zero coordinates. We construct codesof symbol-fieldK on Ph’s in P ; K-rational points whosecoordinates include zero are considered in section VIII. Wedefine a subsetΦm of Z2

0 as

Φm := {(i, j) ∈ Z20 | i < q − 1, j < a, ai+ bj ≤ m},

whereai + bj is equal to the pole ordero(xiyj) of xiyj atP∞. In this study, we consider codes on algebraic curves

C(m) :={

(ch) ∈ Kn∣

∣ c(αi, αj) = 0, (i, j) ∈ Φm

}

, (1)

wherec(x, y) :=∑n−1

h=0 chzh with monomialszh := xrys forPh = (αr , αs). For simplicity, we assumem > 2g − 2; then,we obtainn− k = m− g + 1 = ♯Φm.

Elementary encoding: The condition{c(αi, αj) = 0} in (1)is equivalent to the ordinary linear system

(ch)0≤h<n [zh(Ql)]0≤h<n, 0≤l<n−k = 0, (2)

whereQl := (αi, αj) for (i, j) ∈ Φm with orderl ≤ l′ ⇔ ai+bj ≤ ai′+bj′. An encoding method forC(m) is the use of thegenerator matrixG :=

[

Ek −H]

, whereEk is the(k×k)

identity matrix andH is obtained from

[

HEn−k

]

by the row

transform of [zh(Ql)] and, if needed, the order-changing ofP . Then, we can systematically encode information symbols(Iκ)0≤κ<k to a code-word(ch) := (Iκ)G. However, thisrequires the multiplication of (k × (n − k)) matrix H . Thus,our goal should be to eliminate the matrix-multiplication fromthe encoding algorithm.

On the other hand, with regard to the computing of syn-dromes, the situation is similar but better than the aboveencoding because of (1). We suppose that an error-vector

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Fig. 1. Flow chart of a non-systematic encoding by IDFT and a decoding by BMS algorithm with DFT for Hermitian codeC(11) over GF(32); the shadedvalues in (c) and (d) indicate the values on theK-rational points with non-zero coordinates. Array (c) represents a code-word and array (g) indicates thatthree errors have been corrected.

(eh) has occurred during the transmission of(ch) and wehave received a word(rh) := (ch) + (eh). Then, the syn-drome decoding requires the(n − k) values of syndrome(rh) [zh(Ql)], which agree with(r(Ql)) for our expression ofC(m). This generalizes the syndrome-calculation for RS codesby the substitution of the roots of the generator polynomial.Hence, we consider an effective encoding method based onour definition (1) ofC(m).

III. E NCODING BY 2-D DISCRETEFOURIER TRANSFORM

In this section, we provide the example of a Hermitian codeover K := F9 with defining equationy3 + y = x4 of genusg = 3, the minimal pole order (first non-gap)a = 3, and 24K-rational points ofxy 6= 0 and finite. The primitive elementα is fixed to satisfyα3+α+1 = 0, and the non-zero elementαi (0 ≤ i < 8) is simply denoted asi (resp. zero as−1). Notethat −α0 = α4 6= α0. We represent 24F9-rational points asmonomials{xrys | (αr , αs) ∈ P}, which correspond to theshaded boxes of (c) in Fig. 1.

Let Φ ⊂ Z20 be the support of a Grobner basis of the ideal

IP := {f ∈ K[X ] | f(Ph) = 0, Ph ∈ P}, i.e., Φ correspondsto the set of monomial representatives ofK[X ]/IP , whereK[X ] = K[x, y]/(y3 + y − x4). Then, we have♯Φ = ♯P .For Hermitian codes on non-zero coordinates,Φ agrees with{(i, j) ∈ Z

20 | i < q−1, j < a}; in general,Φ is its subset. We

arrange the information symbols(I(i,j)) on Φ\Φm, and thenobtain(I(i,j))(i,j)∈Φ by consideringI(i,j) := 0 if (i, j) ∈ Φm,as (a) in Fig. 1. Furthermore, the Grobner basis ofIP extends(I(i,j))(i,j)∈Φ into (I(i,j)) for (i, j) ∈ Z

20 with 0 ≤ i, j <

q− 1; for Hermitian codes,I(i+a+1,j−a) − I(i,j−a+1) =: I(i,j)from the defining equation, as shown in Fig. 1(b), wherei :=imod (q − 1) if i ≥ q − 1.

To encode(I(i,j)), we perform the 2-D IDFT for(I(i,j)):

c(r,s) :=∑

0≤i,j<q−1

I(i,j)α−ri−sj =

∑

0≤i,j<q−1

I(q−1−i,q−1−j)αri+sj .

Then, by substitutingPh = (αr, αs) into I(x, y), we have

ch := c(r,s) =∑

0≤i,j<q−1

I(i,j)x(Ph)−iy(Ph)

−j (3)

= I(Ph) for I(x, y) :=∑

0≤i,j<q−1

I(q−1−i,q−1−j)xiyj .

Theorem 1: We havec(r,s) = 0 if there is noPh ∈ Pwith Ph = (αr, αs). Moreover, the transform (3) defines aninjective linear map and a code-word(ch)0≤h<n ∈ C(m).

We omit the proof and discuss RS-code case in section VI.The received word(rh)0≤h<n is viewed as(r(i,j)) for 0 ≤

i, j < q − 1 and r(i,j) := rh if there isPh ∈ P with Ph =(αi, αj); otherwiser(i,j) := 0 (cf. Fig. 1(d)).

The syndromes from the received word(rh) can be obtainedby the substitution of(αi, αj) for (i, j) ∈ Φm into r(x, y), asdescribed in Section II. In our framework, it is convenient,asshown in Fig. 1(e), to substitute the entire{(i, j)}0≤i,j<q−1,which can be considered as the DFT

(

r(αi, αj))

0≤i,j<q−1.

Then, we haver(αi, αj) = e(αi, αj) + I(i,j) through theerror polynomiale(x, y) :=

∑n−1h=0 ehzh and the extended

information symbols(I(i,j)) because of our encoding and the2-D Fourier inversion formula

∑

0≤r,s<q−1

∑

0≤i,j<q−1

I(i,j)α−ri−sj+ri′+sj′ = (q − 1)2I(i′,j′).

We notice that the values(e(αi, αj)) are not yet knownfor (i, j) ∈ Φ\Φm since I(i,j) 6= 0 outside Φm. To ob-tain and subtract all syndrome-values(e(αi, αj)) for 0 ≤i, j < q − 1 from (r(αi, αj)), we run the BMS algorithmto calculate the Grobner basis of the idealIE , where Edenotes the set of error locations. Since the Grobner basisprovides the 2-D linear recurrence formula for syndromes,we can extend(e(αi, αj))(i,j)∈Φm

to the entire plane, wherethe array (f) represents the result. Finally, as illustrated in

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Fig. 2. Flow chart of systematic encoding by Grobner basis,decoding by BMS algorithm with DFT for codeC(11); array (k) represents a systematiccode-word and array (o) indicates that the correct information has been obtained.

Fig. 1(g), the information(I(i,j))(i,j)∈Φ\Φm(and its extension

(I(i,j))0≤i,j<q−1) is obtained by (e) minus (f).Thus, DFT is utilized for both the encoding and computing

of syndromes; the decoding consists of two steps, i.e., thisDFTand BMS algorithm to remove the syndrome-values, withoutChien search and error-evaluator formula.

IV. SYSTEMATIC ENCODING

Since the conventional RS codes are usually encodedsystematically, it is natural to consider effective systematicencoding for codes on algebraic curves. However, while theroots of the generator polynomial can be considered a subsetof locations in RS code-words, it does not hold in general forour C(m) since actually{Ql}0≤l<n−k 6⊂ P . In this section,we apply Theorem 1 and its argument to this problem andobtain a satisfactory solution.

As preliminaries, we choose℘ and℘′ so that℘ ∪ ℘′ = P ,℘ ∩ ℘′ = ∅, and ♯℘ = n − k; ℘ is the redundant-point setand is considered to satisfy℘ = {Ph}0≤h<n−k without lossof generality, and℘′ is the information-point set. We calculatethe Grobner basis of the idealI℘ in advance. In the exampleof Fig. 2, the shaded boxes in (h) indicate℘′, and we obtainthe Grobner basis by the BMS algorithm as follows:

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GH I J KL MN

O

PQRS

where, for example, the leftmost array represents the polyno-mial 1 + x4. For simplicity, we assume that the support ofthe Grobner basis forI℘ is generic [6], that is, the supportcorresponds toL(mP∞); this assumption is not very strongsince the generic support has the probability(q − 1)/q.

Then we representk information symbols{I(i,j)}(i,j)∈℘′

as shown in Fig. 2(h). To generate the redundant part of the

code-word, we first compute its DFT{I(i,j)}0≤i,j<q−1 by

I(i,j) := I(αi, αj) for I(x, y) :=∑

(αi,αj)∈℘′

I(i,j)xiyj , (4)

and then, we extend{I(i,j)}(i,j)∈Φm(nine values in Fig. 2(i))

on the support into{I(i,j)}0≤i,j<q−1 on the entire plane bythe Grobner basis (∗), or more precisely, by its recursiveformula (6) in section VII. If we perform IDFT for thenegative{−I(i,j)} of the extended array, the redundant partcan be obtained since the resulting values on℘′ are zeroby Theorem 1 and their DFT (i.e., syndrome) agrees with{−I(i,j)}(i,j)∈Φm

. If we perform IDFT for the subtraction{I(i,j) − I(i,j)}0≤i,j<q−1 (Fig. 2(j)), i.e., we compute

c(r,s) :=∑

0≤i,j<q−1

(

I(i,j) − I(i,j)

)

α−ir−js,

then {c(r,s)} is a code-word inC(m) since c(αi, αj) =

I(i,j) − I(i,j) = 0 for (i, j) ∈ Φm. Moreover, it is systematic,as observed at Fig. 2(k), and in fact we havec(r,s) = I(r,s)for (αr , αs) ∈ ℘′ since the IDFT ofI(i,j) vanishes at℘′ byTheorem 1.

While the error-value estimation was performed by usingthe IDFT of (n) in [8], it is efficiently incorporated into ourprocedure. In the encoding of Section III, each procedureof encoding and decoding contains either the DFT or IDFT.Although in this section, each step of encoding and decodingrequires both the DFT and IDFT, we can use only one DFTcalculator for all transforms in practical circuits for theencoderand decoder.

Recall that the systematic matrix-encoding described inSection II requires multiplications of thek × (n− k) matrix;our method requires the calculators of the 2-D feedbackshift-registers and memory-elements for at mosta × (n − k)coefficients, which correspond to the(n − k) coefficients ofthe generator polynomial for RS codes.

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Fig. 3. Flow chart of systematic encoding by Grobner basis,decoding by BMS algorithm with DFT for a hyperbolic cascadedRS code; array (s) representsa systematic code-word. The shaded values in (t) denote the values with errors added in the channel. Array (w) indicates that four errors have been corrected.

V. A PPLICATION TO HCRSCODES

Our encoding and decoding scheme is widely applied to var-ious algebraic codes such as 2-D cyclic codes and hyperboliccascaded RS (HCRS) codes; for these codes, the encodings insections 3 and 4 are similarly performed except for the totalorder in the BMS algorithm. Here, we deal with the systematicencoding of HCRS codes.

In this section letΦm := {(i, j) ∈ Z20 | (i+1)(j+1) < m}.

A HCRS code [2] overK := Fq is defined as

C(m) :={

(cr,s)0≤r,s<q−1

∣

∣ c(αi, αj) = 0, (i, j) ∈ Φm

}

,

where c(x, y) :=∑

0≤r,s<q−1 cr,sxrys. Then, the minimum

distanced of C(m) is bounded asd ≥ m. In Fig. 3,C(9) overF9 is demonstrated for four-error correction.

The non-systematic encoding is equal to the IDFT of (p).To encode systematically, we first compute a Grobner basisof an ideal{f ∈ R | f(αi, αj) = 0, (i, j) ∈ Φm}, whereR := K[x, y]/(xq−1 − 1, yq−1 − 1, ), with respect to a totalorder(i, j) ≺ (i′, j′) ⇐⇒

(i+ 1)(j + 1) < (i′ + 1)(j′ + 1), or

(i+ 1)(j + 1) = (i′ + 1)(j′ + 1) ∧ j < j′.

The elements of the basis that is needed for the extension inthe systematic encoding are shown below.

´ µ ¶ · ¸ ¹ º »¼ ½ ¾ ¿ À Á Â

Ã Ä Å Æ Ç È É ÊË Ì Í Î Ï Ð Ñ

Ò Ó Ô Õ Ö × Ø ÙÚ Û

Ü Ý

Þ ß

Then, the values after DFT onΦm in (q) are extended bythe recurrence formula similar to (6). Thus, the IDFT of (r),where (r) equals (q) minus the extended array, is a systematiccode-word (s).

To decode a received word (t) from the channel, we perform,for syndrome values onΦm in (u), the BMS algorithm with

respect to the total order (≺) [2]. In the case of our example,the error-locator polynomials are expressed as follows.

à á â ã ä å æ

ç èé ê ë

ì í î

ï

The recurrence similar to (6) by the above basis extends thesyndrome values onΦm to the entire plane, as Fig. 3(v).Finally, the IDFT of (u − v) in Fig. 3 provides the correcttransmitted word Fig. 3(w).

VI. T HE CASE OFRS CODES

Recall the encoding for RS codes by Euclidean division:

c(x) := I(x) −R(x) = Q(x)G(x), deg(R) < n− k, (5)

where I(x) =∑

0≤κ<n Iκxκ is an information polynomial

with Iκ = 0 for 0 ≤ κ < n − k; G(x) = (x − 1) · · · (x −αn−k−1), the generator polynomial;R(x), the remainder ofthe division with quotientQ(x). Then, it is apparent that(ch)from c(x) =

∑

0≤h<n chxh is a code-word of the RS code

C(m) :={

(ch)0≤h<n ∈ Fnq

∣

∣ c(αi) = 0, 0 ≤ i ≤ m}

with n := q−1 andm := n−k−1. This method issystematic,i.e., ch = Ih for n− k ≤ h < n.

If we have received a polynomialc(x) =∑

0≤h<n chxh =

c(x)+e(x) containing an error polynomiale(x) in the channel,the values of syndromes{e(ακ)}0≤κ<n−k can be computedas {c(ακ)} by substituting the roots ofG(x) into c(x). Wenotice thatc(ακ) =

∑

0≤h<n chακh can be also considered

to be the DFT of{ch}. Thus, we obtain another encodingmethod (non-systematic) by the IDFTch :=

∑

0≤i<n Iiα−ih.

Then,(ch)0≤h<n is another code-word ofC(m) since

c(αi′) =∑

0≤h<n

∑

0≤i<n

Iiα−ih+i′h = (q − 1)Ii′ .

It is possible to systematically encode by using an alter-native procedure. From (5), we obtainI(ακ) = R(ακ) for

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Fig. 4. Flow chart of systematic encoding by Grobner basis and DFT for codeC(11) on all finite GF(9)-rational points (including zero components) of theHermitian curvey3 + y = x4; array (aa) represents a systematic code-word inC(11).

0 ≤ κ < n − k. Moreover, we define array(dh)0≤h<n

inductively by

dh :=

{

I(αh) 0 ≤ h < n− k,

−∑n−k−1

i=0 Gisi+h−(n−k) n− k ≤ h < n,

whereG(x) =∑n−k−1

i=0 Gixi + xn−k. Then, it follows that

dh =∑n−k−1

i=0 Riαih with R(x) =

∑n−k−1i=0 Rix

i not onlyfor 0 ≤ h < n−k but also forn−k ≤ h < n. Thus, the IDFT(

−d(α−i))

for (dh)0≤h<n is observed to agree with(Ri) ofR(x) by using Fourier inversion formula;c(x) := I(x)−R(x)again indicates the encoding, and moreover we obtain twoways of calculatingR(x), i.e., a commutative diagram.

���������

����

����

� !

"#

$% &'

()*+,-./0

123

4567

89:;<=>?@

ABCD

EFGH

IJK

LM

NO PQ

RSTUVWXYZ

[\]

^_`a

Thus, we obtain two encoding methods for RS codes, whichwe have generalized.

VII. R ECURSIVE FORMULA FROMGROBNER BASIS

We generate{I(i,j)} for 0 ≤ i, j < q − 1 recursively from{I(i,j)}(i,j)∈Φ as in the encoding at Section III and IV, whichis stated here more precisely. It may be assumed that we havethe supportΦm ⊂ {(i, j) | 0 ≤ i < q− 1, 0 ≤ j < a} and thateach Grobner basis consists ofa+ 1 elements{f (ι)}0≤ι<a ∪

{g}, wheref (ι) =∑

(i,j)∈Φmf(ι)(i,j)x

iyj + xiιyι with iι :=

max{i + 1 | (i, ι) ∈ Φm} and g =∑

(i,j)∈Φ g(i,j)xiyj + ya.

Then the recurrence for(r, s) 6∈ Φm is defined as follows:

I(r,s) :=

{

−∑

(i,j)∈Φmf(ι)(i,j)I(i,j)+(r,s)−(iι ,ι) s = ι,

−∑

(i,j)∈Φmg(i,j)I(i,j)+(r,s)−(0,a) s ≥ a.

(6)

The resulting values do not depend on the order of thegeneration because of the property of Grobner bases.

VIII. T REATMENT OF LOCATIONS INCLUDING ZERO

With regard to the systematic encoding in Section 4, wecan treat locations including zero in a manner similar tothe case of non-zero locations. There are threeF9-rationalpoints (−1,−1), (−1, 2), and (−1, 6) for our example ofHermitian codes, which are denoted by the shaded boxesin the top row of Fig. 4(x). Although we cannot computethe DFT for three information symbols at these locations,we note that if an error1 = α0 has occurred on, e.g.,(−1,−1), then the syndrome values are all−1 except forα0

at (0, 0) ∈ Φ since they are computed by the substitution of(−1,−1) into {xiyj}0≤i,j<q−1. We also note that the IDFTof {xiyj}|(x,y)=(−1,−1) equals an8× 8 all-α0 array. Thus theanalogue of DFT is obtained for information 7 at(−1,−1) asonly 7 at (0, 0) ∈ Φ. Similarly, the analogue is obtained forinformation 5 at(−1, 2) as only[5, 7, 1, 3, 5, 7, 1, 3] in the firstrow of Φ, which also equals the one-dimensional (1-D) DFTfor [−1,−1, 5,−1, · · · ,−1]; for information 2 at(−1, 6), theanalogue is obtained as[2, 0, 6, 4, 2, 0, 6, 4] at the top, which isalso the 1-D DFT for[−1, · · · ,−1, 2,−1]. Hence, we obtainthe analogue of DFT for the array (x) as the array (y) bysumming and obtain the redundant part of the code-word (aa)by the IDFT of the extended array (z). Notice that the IDFTof (y− z) such as in Section 4 provides the sum of the partsof the code-word onΦ and the IDFT of the analogue arraysof DFT, i.e., all-α7 array, all-(−1) array except[1, · · · , 1]T inthe third column, which is the IDFT of{α5xiyj}|(x,y)=(−1,2),and all-(−1) except[6, · · · , 6]T in the seventh column.

Thus, we have completed the treatment of locations includ-ing zero components.

ACKNOWLEDGMENT

This work was partly supported by the Academic FrontierProject for Future Data Storage Materials Research by theJapanese Ministry of Education, Culture, Sports, Science andTechnology (1999–2008), Toyota Physical and Chemical Re-search Institute, and Storage Research Consortium (SRC).

REFERENCES

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[2] K. Saints, C. Heegard, “Algebraic-geometric codes and multidimensionalcyclic codes: a unified theory and algorithms for decoding using Grobnerbases,”IEEE Trans. Inf. Theory, vol.41, no.6, pp.1733–1751, Nov. 1995.

[3] J.-P. Chen, C.-C. Lu, “A serial-in-serial-out hardwarearchitecture forsystematic encoding of Hermitian codes via Grobner bases,” IEEE Trans.Communications, vol.52, no.8, pp.1322–1332, Aug. 2004.

[4] R. Matsumoto, M. Oishi, K. Sakaniwa, “Fast encoding of algebraic ge-ometry codes,”IEICE Trans. Fundamentals, vol.E84-A, no.10, pp.2514–2517, Oct. 2001.

[5] T. Yaghoobian, I. F. Blake, “Hermitian codes as generalized Reed–Solomon codes,”Designs, Codes and Cryptography, vol.2, pp.5–17, 1992.

[6] H. Matsui, S. Mita, “Footprint of polynomial ideal and its applicationto decoder for algebraic-geometric codes,”Proc. Int. Symp. InformationTheory and Its Applications (ISITA), pp.1473–1478, Oct. 2004.

[7] H. Matsui, “Efficient encoding methods for codes on algebraic curves,”Proc. 29th Symp. Information Theory and Its Applications (SITA), pp.97–100, Nov. 28–Dec. 1, 2006.

[8] S. Sakata, H. E. Jensen, T. Høholdt, “Generalized Berlekamp–Masseydecoding of algebraic geometric code up to half the Feng–Raobound,”IEEE Trans. Inf. Theory, vol.41, no.6, Part I, pp.1762–1768, Nov. 1995.