Important algebraic calculations for n-variables polynomial codes

Discrete Mathematics North-Holland

255

56 (1985) 255-263

IMPORTANT ALGEBRAIC CALCULATIONS n-VARIABLES POLYNOMIAL CODES

FOR

A. POLI A.A.E.C.C. Received

lab., L.S.I., University P. Sabatier,

December

Toulouse, France

1984

We deal with codes which are ideals in a finite abelian give constructive proofs to obtain n-variables polynomial construction for the primitive idempotents of A.

group algebra A, over finite field. We codes. In particular, we give an easy

Introduction Finite abelian group codes were studied by several authors: J. MacWilliams, S.D. Berman, P. Delsarte, P. Camion, A. Poli, M. Ventou, P. Charpin. . . . Generalizing Berman [0], we study codes which are ideals in the following algebra :

A = F,[X,, . . . , X,,]/(X:l’l-

1, . . . , Xp-

l),

where [F, is the Galois field or cardinality q (q = pr, p prime), qi is a power Of p (lsisn), qrs* * . sq,,. We will denote A = [F,[Xr, . . .,x”]/(x~~-l)...) For several

years we have been

and where (e,, p) = 1, by ;i the algebra

x>-1). studying

the algebraic

properties

of codes.

In

this paper, our purpose is to present some calculations which appear important to us for a constructive study of n-variable polynomial codes. Our paper is divided into two parts. In the first one we indicate how to simplify the algebraic study of finite abelian group codes by considering ideals in a very easy kind of algebras (denoted by D). In the second one we show how to construct codes in A when we have chosen ‘good’ ideals in algebras D. In particular we give an easy polynomial construction for the primitive idempotents of A. As we give theoretical proofs, we also develop an example.

Part1 The key point, in this part, is the use of decomposition (Proposition 1 and 3), as well as a result we have (Proposition 2). 0012-365X/85/$3.30

@ 1985, Elsevier

Science

Publishers

of ideals by intersection recently published [lo]

B.V. (North-Holland)

A. Poli

256

The chosen example is the group algebra FJG], where G is the Cartesian product 2/20x L/6. We will represent lF,[ G] by A, equal to F,[X, Y]/(X2”1, Y6- l), and A will be equal to F,[X, Y]/(X51, Y3- 1). Note that here q1 is greater

than

Proposition

qz. 1. In F,[X,,

. . . , X,,], the ideal (Xqlel- 1, . . . , Xy-

intersection of all the ideals (~71, . . . , p$),

1) is equal to the

where pi is a prime divisor of X7-

1 in

1.

Fq[xi

A is isomorphic to the Cartesian product: nF,[X,, tended to every (pl, . . . , p,). n is 1, then

This is proved in [7]. When remainder theorem).

. . . , X,,]/(pyl,

it is a very common

. . . , p”,-), ex-

result

(Chinese

Example. xzo - 1 = (X - 1)4(x4 + x3 + x2 + x + 1)4 = ptpll”, Y6-1=(Y-1)*(Y2+Y+1)*=p;p~. We have: Yll(p;l,

A = F,[X,

~$1 xF,[X

WP;,

Proposition 2. The algebra algebra B defined by B = FJX1, This is a result

F,[XI,

P;“> X F,[X

Y]/(P;“,

. . . , X,,]/(pyl,

P:, X ‘Fz[X

. . . , p>)

Yll(pl;‘,

is isomorphic

pi2).

to the

. . . , X,,,-G, . . . , -T,ll(p,, . . . , P,,,Z?, . . . , Z>h we have recently

published

[lo].

Let p+ be a root of pi (lsiin). 3. The ideal (pI, . . . , p,,, Zql, . . . , Z>) is equal to the intersection of all where ideals Z?, . . . >z”), (PI, w,w,, m, . . . 3 wnw,, . . . , XJ,

Proposition the

is a prime divisor of pi in Fq(pl,. . . , pi_,)[Xi]. to the Cartesian product nF,[X,, . . . , X,,, Z1, . . . , ZJ Z>), extended to every (pl, W,, . . . , W,,). . . > W,, Z;ll,. . . ,

wi(cL1>.. . > F,_~,X~) B (PI,.

Proof.

is

isomorphic

Suppose

We will prove Zyl,. . . , Z>). W2b1,

X2),

we have

that Let

. . . , W-1(PI,

Pitxi)

= II

we can decompose be wI,. . . , pi_-1 . . . , WI-21

wi(cL1, . . . 7 Pi-13

Xi-J.

xi)

the ideal respective One

in

(pl, . . . , Wi-l, pi,. . . , p,,, roots of Pl(XI),

has

~q(Pl,

. . . ,

Pi--l)[xil~

Calculationsfor n-uariablespolynomialcodes

257

and, then pi(Xi)sn Consequently

Wi

we have, for each

pi(X

. . . , Wz, pr).

mod(W,_,, Wi

Wz, . . . , Wi, pi+l,

mod(p,,

0)

.

. . , p,,, Z;ll, . . . , Z>).

We can deduce (pi, W2, . . . 2 wi-l,

pi, . . . , pn9 Zf’, . . . 9 Z”) E

n (PI,. . . , W,

But [7] ideals (pi, . . . , Wi, P~+~,. . . , p,,, Zql, . . . , Z$) are pairwise Then, their intersection is equal to their product. Finally, from (I), we have: n

W = 0

mod(pi,

W-1,

pi+19 . . -1.

relatively

prime.

. . . > Wz, PI),

and then: I-I (PI,. . . > wi, Pi+17 . . .T z>:-) s (pl, wI27 . . .7 wi-l~ which achieves Ex=ple.

the proof

of Proposition

(~1, p2, Z?, Z&

for intersection.

h,

pi,. . . > Pm . . .7 zZ”)

Cl

3.

~5, ZT, .%>, (PL ~2, Z?, Zf> are not decomposable

But we have:

Y2+Y+1=(Y+~L:+~:)(Y+l+~~+~~)

in[F2(p1)[Y],

and thus: Y2+Y+1=(Y+X2+X3)(Y+1+X2+X3) Consequently,

mod(p;)

in F,[X, Y].

we have

(p’l, p;, z;,

z’,,=

(p’l, y+x2+x3,

z;,

Z$(p’l,

Y+ 1+x2+x3,

z:,

Z’,),

and F,]X, Y, =

F,[X

Z1, Z,ll(PL

z,,z,MP;,

y,

x F,[X, Y, z,,

Proposition (PI,. . . , K,

4. The -w,.

D = FJZ,, where

Pi, z,

Y + 1 +x2+x3, C,

is isomorphic

. . . , Z,,]/(Zql,

[F,, is the field Fcl(pl,.

Y+x2+x3,z;,z3

Z&p{,

algebra

. . , Z$),

z:,

z:,

z;,.

equal

to F,[Xl, . f . , xv z,, . . . , -cl/ to the algebra D defined by:

. . . , Z>),

. . , p,).

This is a direct consequence of the fact that p1 is irreducible as Wi(pr,. . . . h-19 xi> in FQ(CLIY.. . , Fi-l)lIxil*

in E,[X,],

as well

258

A. Pofi

The study of ideals in C is equivalent to the study of ideals in D. More precisely, an ideal in C which is a k vector space over IF, is in isomorphic correspondence with a (k/a)iF,, subspace in D. (Here a means dimIF,, over ff,). We can sum up the results of Part I in the next theorem. Theorem

1. A is ring isomorphic

to a Cartesian product of algebras D of the

following type: D = ff,,[Z,,

. . . , Z,,]/(Z‘$,

. . . , z:;-)

where IF,, is an extension field of F, which depends on D. Example.

A is ring isomorphic

to the Cartesian product

For studying algebraic properties of codes, it is sufficient to consider ideals in algebras D. In the particular case where qi is equal to p and ei is equal to 1 (1 G i s n), A is isomorphic to [F,[X,, . . . , X,, J/(X: - 1, . . . , XE- l), which is a p-elementary group algebra. In this algebra, the powers of the nil-radical are the GRM codes [O]. In the second part we show how to construct codes in A when we have ‘good’ ideals in algebras D.

Part II There are three points in this part. The first one give a decomposition

of A into

a direct sum of principal ideals. The second one given an easy construction for primitive idempotents in A. The last one gives an isomorphism which enables us to ‘come back’ to A from algebras D. Let Hi be the set of roots of X:- 1 (1~ isn). We will say that (F,, . . . , CL,,) and (pi, . . . , FL) are conjugate if and only if there exists a positive integer r such that we have CL;= CL:’ (1~ i
=

-

P?

1 . .

.nX

se”

-

1

‘I,+qz-1.

PZ

372

.

q,+...+q"-n+l . rn

>

Calculations for n-variables polynomial codes

where

ni(Xl,

PCxi)l

wi(cLl,

. . . , Xi)

is obtained

f . . > Pi-l>

xi).

by substituting

is proved

Xi for pi in the polynomial

(or, . . . , pn) as zero.

Note that no generator of (gk) admits This direct sum is another representation possible to construct codes, particularly This decomposition

259

of the decomposition of A. It makes it by designing an appropriate software.

in [7]. Number

N is also given in [7].

Example. g,=(x16+x1~+x8+x4+1)(Y4+Y~+1), g*= (x’6+x12+x8+x4+ g, = (X4+ 1)(Y4+

1)(Y2+

l),

Y2+ l),

g,=(x4+1)(Y2+1)(Y+x2+x3)5, gs=(X4+1)(Y~+1)(Y+1+X2+X3)s. In particular,

we have

xzo-1

Y6-1 &

g4=(x4+x3+x2+x+1)4(Y2+Y+1)2 where 372bl,

Y)

=

Y+

CL:+

P?

=

(Y’+

Y+

l)/(Y+

1+ ILLI+ pL:)

(in F,(II.,)[YI),

and, where or and p2 are respective roots of pi and Y + 1 + p,:+ CL:. No generator of (g4) admits (pl, CL.,)as a root. Using Lemma

the same notations

as in Proposition

5 we have

1. For every i (2 s i c n) one has, in A XT,-- 1 ___. Pl

. .x>-1 Pm

7r2. . * 7Tiwi = 0,

and in A

y*)q’.

. . (3_$)q”r;,+q2-l.. .

This lemma is proved in [7]. We will now indicate how to construct

Tf,+~~~+q,-i+lw~=O.

primitive

idempotents

in A.

Lemma 2. Let e be a primitive idempotent in A, which does not admit (kl,. . . , CL,,) as a root. Then, eq= is a primitive idempotent in A which does not admit (plr . . . , p,) as a root.

260

A. Poli

Proof.

Let e be an idempotent e2-e=O

in A. In F,[X,,

. . . , X,,], we have

mod(X’,1-l,...,X>-l),

and mod(XT+l-

e*a - eQ, = 0

1, . . . , Xp-

1).

Then e4* is an idempotent in A. Moreover, if the product of two idempotents ei, e, in A is zero, then we have in A eFe>=O. Thus, if e is a primitive idempotent in A then e% is a primitive idempotent in A. Finally, e&r, . . . , p,) is null if and only if eq-(pt, . . . , p,) is null, which achieves our proof. 0 Now, recall that the decomposition of A can be denoted by (El)@. . *Cl3(&). Using the same notations as in Proposition 5, we can consider an element gk. Then we have the following proposition: Proposition (PI,

w2,.

. *

6. Let h, be the multiplicative inverse of gk modulo the ideal W,,). Then the primitive idempotent of (gk), in A, is (hk&>4-. >

Proof. Of course, if we have h& - 1 = 0 mod(p,, . . . , W,,), then (Lemma 1): &&(jE& - 1) = 0 in A. Lemma 2 ends this proof, because hkgk does not admist (pl,. . . , F,,) as zero (Proposition 5). Cl Example.

(1) Construction

of e, (primitive

Sl=((X’-1)(Y3-l))/((X-l)(Y-l)), Then

hl=

1, and e,=

(2) Construction

Et= (X’6+X12+X8+X4+

idempotent

in (gi)).

g,=lmod(X-1, 1)(Y2+

Y4+ 1).

of e2.

g*=((X5-1)(Y3-1))/((X-1)(Y2+Yf1)), &=(Y-l)mod(X-1,

Y*+Y+l).

Then &= Y, and e2= (X16+X12+X8+X4-t (3) For e3. g3=X+1mod(X4+X3+X2+X+1, Then h3=X+X3, (4) For e,.

1)(Y4+

Y”).

Y+l).

and e3=(X16+X12+Xs+X4)(Y4+Y2+1).

g4=(X+l)(Y+1)(Y+X2+X3), g4=1+X+X3mod(X4+X3+X2+X+1,

Y+1+Xz+X3).

Y-l).

Calculationsfor n-variables

Then

&=X+X2+X3,

and

polynomial codes

261

e4= Y4(X12+X8)+Y2(X16+X4)+X16+X12+Xs+

X4. (5) for e5. &=(X+1)(Y+1)(Y+1+X2+X3), g,=X3mod(X4+X3+X2+X+1, Then

Y+X2+X3).

&, = X2, and e5 = Y 4(X16 + X4) + Y *(X1’ + X8) + Xl6 + X1* + X8 + X4.

Now, ideal in back’ to Using

we will develop the last point of this paper. an algebra C. We give now an isomorphism

Suppose we have a ‘good’ which enables us to ‘come

A. the same notations

as in Proposition

5, we consider

now an ideal

2. The application q from F,[X,, . . . , x,, G into (gk), defined by the set of substitutions (PI, . . . , Wm. . . , Z>) (=C)

Theorem

(gk).

. . . , Z,l/

Xi -+ XFe, +

zi

(lSi,jSn)

pie,

is an isomorphism. Proof. First,

it is a direct

q(pl)=.

consequence

of Lemma

. . = cp(W”) = q(Zf1) = * * * = cp(Z>) = 0.

We now prove that cp is injective. C, such that we have cp(P) = 0. Let

Let P(X,,

P = c pG,(X,, . . . , X,,)Z’, * * * Z$, 0’) We can suppose

that no term

in P is zero, jk
PO)& (PI, w2, . . . 7 wJ, From

the hypothesis (

;

P&P’;‘..

. . . , X,,) be a non-null

element

in

with (j) = (jI, . . . , j,,). i.e., (lskcn).

q(P) = 0, we may write in [F,[X1,. . . , X,,]: mod(X’jl’l-

* p’,-) ek= 0

Let the set of (j)‘s be ordered of this set. We have:

1,. . . , XF-

lexicographically.

((P($P> . . * p$+ R)p> + Sp>+‘)ek = 0 We know

1 that

1).

Let (t) be the minimal

mod(Xyl’l-

element

1, . . . ) xp-

1).

1, . . . ) xp-

1).

[7] that we can deduce: (Pc3P2 * . . pi+

Substituting

~~ for X,, we can deduce

(P$&

x29

mod(p~l-‘l,

R + SP&k = 0

. . . , x,h>

XFZ-

in [Fq(p1)[X2,.

’ ’ ’ pk+R(pl,

x2y

. . , X,,]:

. . . , Xd)ek(p17 =

0 mod(Xqzez-

. . . , x,,>

1,. . . , Xgen-

1).

262

A. Poli

Iterations

of this process PP;I&,

give

. . . , cL,)ek(Fr,.

But (Proposition P&%,

finally

. . , CL,) = 0.

_ . , p,) is not zero.

5) ek(pl,.

So we have:

. . . ? /Jr%>= 0,

which means that pet) is an element of (pi, W,, . . . , W,). This is impossible hypothesis. cp is then injective, and so, cp is the isomorphism we want. 0 Example.

Let D5 be FIJZ1, Z,]/(Z$ Zz). Let, in D5, the (Z:, Z:, Zz, Z:Z,). This ideal has dimension 3 over [F,,. It corresponds, (equal to F,[X, Y, Z,, Z&(p;, Y+X2+X3, ZT, Zz)), to the ideal (Z:, XZ:, By using

X2Z:,

the isomorphism

{p’l, x4p;,

. . . , ZTZ,,

X3Z:,

defined

. . . ) x12p;,

. . . , X3Z:Z,)

in theorem

(dimension

by

ideal in C,

12).

2, we have a basis of (g5)

. . . ) xi2p;3p$}.

Conclusion We have proved in the first part of our paper that, for studying algebraic properties of n-variables codes, it is only necessary to study very simple kind of algebras. In a second part we gave tools to ‘come back’ to A when we have ‘good’ ideals in algebeas D. As one can see, the proofs developed are all constructive. They permit easily to design appropriate Our technique

softwares. for constructing

Kung

algorithm

and Tong’s

idempotents

give directly

a generalization

of

(4 bis).

References [0] SD. Berman, Abelian codes, Kibernetika 3 (1967) 31-39. [l] E.R. Berlekamp, Algebraic Coding Theory (McGraw-Hill, New York). [2] P. Camion, Etude de codes abeliens modulaires autoduaux de petites logueurs, Rev. CETHEDEC (1979) 3-24. [3] P. Charpin, Codes ideaux de certaines algbbres modulaires, These de specialit&, Univ. de Paris VIII (1982). [4] C.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras (Wiley, New York, 1962). [4bis] H.T. Kung and D.M. Tong, Fast algorithm for partial fraction decomposition, SIAM .I. Comput. 6 (1977) 582-592. [5] D. Lazard, Resolution des systkmes d’kquations algibriques, Theoret. Comput. Sci. 15 (1981) 77-110.

Calculations [6] A. [7] [8] [9]

[ 101 [ 111 [12] [13] [14]

for n-oariables

polynomial

codes

263

Poli, Codes stables sous le groupe des automorphismes isometriques de A = F,[X,, . . .1 X,,]/(Xl;- 1,. . ,X{- l), Comptes Rendus de I’Acad. des Sciences (1980). A. Poli, Codes dans certaines algebres modulaires, These d’Etat, Univ. P. Sabatier, Toulouse (1978). A. Poli, Quelques resultats sur les codes polynomiaux 9 n variables, Rev. CETHEDEC (1981) 23-33. A. Poli and M. Ventou, Codes autoduaux principaux et groupe d’automorphismes de l’algebre lF,[Xi, , X,1/(X:1,. . ,X?;- 1) (q = p’), European J. Combin. 2 (1981) 179-183. A. Poli, Ideaux principaux nilpotents de dimension maximale dans l’algtbre F,[G] d’un groupe abelien fini G, Comm. Algebra 12(4) (1984) 391-401. A. Poli and C. Rigoni, Codes polynomiaux a n variables, Rev. CETHEDEC (1984) to appear. P. Samuel and 0. Zariski, Commutative Algebra (Van Nostrand, New York, 1958). M. Ventou and C. Rigoni, Self dual circulant codes over IF,, this issue. F.J. MacWilliams and N.J. Sloane, The Theory of Error Correcting Codes (North-Holland, Amsterdam, 1977).