37X IEEE TRANSACTIONS 0]'; INFORMATION THEORY, VOL. 41, NO.2, MARCH 1995

The Automorphism Groups of the Generalized Quadratic Residue Codes

W. Cary Huffman

Abstract-In this paper, we give the full automorphism groups as groups of semiaffine transformations, of the extended gener­alized quadratic residue codes. We also present a proof of the Gleason-Prange Theorem for the extended generalized quadratic residue codes that relies only lin their definition and elementary theory of linear characters.

Index Terms-Automorphism groups, quadratic residue codes, Gleason-Prange Theorem.

1. INTRODUCTION

THE FAMILY of quadratic residue codes, or QR codes, has

a significant place in the literature of algebraic coding

theory, in particular as important examples of cyclic codes.

These codes, of prime block length, and their extensions

include the binary and ternary Golay codes and the quaternary

hexacode. The QR codes wcre first generalized, at least in the

binary case, by Delsarte [6], using inversive planes, to codes of

prime power block length called generalized quadratic residue

codes, or GQR codes. Later Ward [15] and Camion [4] gave

other descriptions of GQR codes which applied to nonbinary

fields as well. In this paper we present these codes using

linear characters as described by van Lint and MacWilliams

[II]. The GQR codes are generally not cyclic, but in using

linear characters to define the codes. we have a tool to study

them which allows one to prove many properties analogous to

properties of thc cyclic QR codes. In Section II we define the

GQR codes and their extensions.

The GQR codes are defined over a coordinate set equal

to a finite field. The codes are invariant under translation

of this coordinate set; in thc case that the field has prime

order, translation is the same as cyclic shift. These codes are also invariant under multiplication of the coordinate set by

a quadratic residue. Showing that these maps indeed fix the

codes is rather trivial. When the codes are extendcd, there are

other maps which move the cxtcnded coordinate and leave

thc extended code fixed. To show that these maps indeed fix

the extended code is nontrivial. In the case of the extended

QR codes, this fact is the Gleason-Prange Theorem. The first

published proof of the Gleason-Prange Theorem appeared in

[I]. In [4], Camion indicated that with appropriate notation,

the proof of the Gleason-Prange Theorem in [1] is valid

for the extended GQR codes also. These proofs used the

trace function and Fourier transforms (which are connected

Manuscript received November iii. 1993; revised July I. 1994. The author is with the Department of Mathematical Sciences. Loyola

University, Chicago, IL 60626 USA. IEEE Log Numhcr 9407923.

to linear characteristics) and relied on the duality of the

extended QR or GQR codes. Van Lint and MacWilliams,

in generalizing the codes, generalized the Gleason-Prange

Theorem as well. Their proof relied on knowledge of the

idempotent generator and generator matrix of the GQR code.

In Ward's construction [15], the extended GQR codes arise

from induced representations of the two-dimensional general

linear group, following work of Gleason, giving the Gleason­

Prangc Theorem for "free" (but not without work). Blahut [2]

gave another proof of this theorem for extended QR codes

using Fourier transforms. His proof did not make explicit use

of either the generator matrix or the duality. In Section IV of

this paper we prove the Gleason-Prangc Theorem for extended GQR codes directly from the linear character definition of the

codes. The proof has characteristics similar to the proof given

by Blahut.

The translation maps, multiplication by quadratic residues,

and the map defined in thc Gleason-Prange Theorem gen­

crate the two-dimensional projective linear group over the

coordinate set. It has been suspected that these indeed are

"essentially" all of the automorphisms of all GQR codes except the extended binary and ternary Golay codes, the extendcd

binary Hamming code of length 8, and the hexacode. (A remark to this effect, without proof, was made in [9].) In

Section V we prove that this is indeed the case provided

we throw in a few more automorphisms that arise from the

field automorphisms of both the coordinate set and the ground

field over which thc GQR codes are defined. (This was proved

in [10, Theorem 6] for the binary extended QR codes.) The

group-theoretic setting in which the automorphisms are defined

will be discussed in Section III.

II. THE GENERALIZED QUADRATIC RESIDUE CODES

In this section we define the GQR codes following [11],

describing these codes as subspaces of a certain group algebra.

We will also show how to extend these codes.

Let q = pr where p is an odd prime and r :.> 1. Let I be the additive group of Fq, the finile field of order (j, and let K be a finite field of characteristic 8 not equal to p. Let K[I] denote the algebra of formal polynomials

K[I] = {x = LxgXq I Xg E K for all 9 E I} gET

where X is an indeterminate. The algebra K[I] has operations

0018-9448/95$04.00 © 1995 IEEE

HUFF\1At\", THE AUTOMORPHISM GROUPS OF THE GENERALIZED QUADRATIC RESIDUE CODES 379

given by

and

a LxgX9 + b LygXg = L(aXg + bYg)Xq gET .qET yET

for a, bE K. The zero and unit of K[ll are 2:gET OX9 and Xo, respectively. The algebra K[ll is isomorphic to the group algebra of I over the field K. In effect the set I will serve as the set of coordinate labels for the GQR codes.

A GQR code is the intersection of the kernels of certain linear characters of I extended to K[I]. Let � be a primitive pth root of unity in an extension field of K. A linear character 1,& of Iis a map 1,&: I ---> K(O such that 1,&(g+h) = 1,&(g)1,&(h), All linear characters can be easily described as follows. Let ( E I be a primitive element of Fq = I. Every element 9 E I can be written uniquely as

1'-1

where gi E Fp for ° ::; z < r - 1. Define the function WI: I -> K(O by

r-l 1h(g) = �90, for 9 = L.'JiC E I,

i=O

Clearly, 1,&1 is a linear character of I. For all h E I define 1,&h: I ...... K(O by

1,&h(g) = wl(hg), for 9 E I.

1,&h is also a linear character of I, and {1f>h Ih E I} is the set of all linear characters of I. Obviously, 1/Jh can be extended to a ring homomorphi,m from K[I] into K(�) by defining

Let Q be the set of nonzero squares in .F� = I, and let N be the set of nonsquares. Then I is the disjoint union of Q, fl, and {OJ. Define the generalized quadratic residue (or GQR) codes, CdK) and C)o/(K) of length q = pr over K to be

CdK) = {c E K[I] I'I/'g(c) = 0, for all 9 E Q}

and

CN(K) = {c E K[I] I'l/'q(C) = 0, for all 9 EN}.

As q is odd, I QI = INI = � (q - 1). With the restriction on K given later in (*), CQ(K) and CN(K) are both of dimension Hq + 1). If q = p, these codes are the QR codes.

The ground field K of a GQR code is of characteristic s not equal to p. Further restrictions are usually placed on K so that

CQ(K) and CN(K) have generating idempotents over K and

have dimension ! (q + 1). To do this we define the Legendre symbol ;((g) for 9 E I as

{O,

;((g) = 1, -1,

if 9 = 0 if 9 E Q ifg E N

Notice that X is multiplicative; that is, X(gh) = X(g) x(h) for g, h E I. With I* = I \{O}, we have

L x(g) = L X(g) = O. gET gET'

Define the Gaussian suml (j for I by

8 = L X(g)1,&I(g). gET'

Finally, let Og, h be the Kronecker delta function on I, where Og, h is I if 9 = h and is 0 if 9 i- h. Some basic facts about linear characters and about 8 are necessary to determine K, and to construct the extended GQR codes and their automorphism groups.

Lemma 2, J : The following hold:

i)

ii)

iii) If

for g, hE I.

L?JJg(h) = L1/Jh(9) = qOg,O, hET hET

c = LcgXg E K[I], gET

for 9 E I.

then

for 9 E I.

iv)

X(h)8 = L X(g)1/Jl (gh), for h E I, gET'

v)

82=X(-1)q.

I In [t1]. there is a different definition given for e; the definitions agree if q = )1. Our definition gives a more compact description of the extended UQR codes and arises naturally in the proof of our extended version of the Gleason-Prange Theorem,

380 IEEE TRANSACfIONS UN INFORMATION THEORY. VOL. 41. NO.2. MARCH 1995

Proof" 1/Jg(h) = 1/Jl(gh) = 1/Jh(g) for g, h E I, yielding part i).

In part ii), if 9 = 0, then

L1/Jo(h) = L1 = q. hEI hEI

Suppose go/D. Then

L1/Jg(h) = L1/Jl(gh) = L1/Jl(h) "EI hET hEI

as gI = I. Let

Io = {g E Ilg = gl( + ... + gr_lC-1}.

As p-l

I = Uu + Io), j=O

p-l L1/JI(h) = pr-lLe = O. hEI j=o

This proves ii) using i).

as

For part iii)

L1/Jh(C)1/Jh( -g) = L1/Jh (LCkXk) �)h( -g) hEI hEI kEI

= LCkL1/Jh(k - g) = qCg kET hEI

L1/Jh(k - g) = qDk-g, O hEI

by ii). Part iv) holds if h = 0 because 1/11 (0) = 1 and

L X(g) = 0 = x(O). gEI'

Suppose h 0/ O. Then

X(h)B = X(h) L X(g)1/Jl(g) = L X(*)1/Jl(9) gEI' gET'

because X(h)X(g) = X(h2)x(g/h) = X(g/h) since X(h2) = 1. So

as I'(l/h) = I* verifying iv). Using the definition of (I and part iv)

= L X(g) L ',jJl«g + l)h). gEI' hET'

If 9 = -1, then

L 1/Jl«g + l)h) = L 1,/11(0) = q - l.

If 9 0/ -1, then

L �)I«g + l)h) = L l/Jl(k) = -1/Jl(O) = -1 hEI' kEI'

using part ii) and the fact that (g + 1 )I' = I'. Thus

(P=x(-1)(q -1)- Lx(g)=x(-l) q

because

Lx(g) = O. gET'

This verifies v).

CdK) has generating idempotent

q-l (q; 1 XO _ au LX9 - al LX 9)

9EQ yEN by [11, Lemma 2], where ao, al are as follows: Using Lemma 2.1 v) if s > 2, then

{ao, ad = gC-1- vxC-1)q), �(-1 + JX(-l)q)}

= {�(-l -B), �C-l+B)}.

If s = 2 and 2 E Q, then {ao, ad = {O, I} and if 2 E N, then ao, al are the roots of 1 + x + x2 implying F4 � K. An easy number theoretic argument (similar to [11, Lemma 3]) shows that K has no restrictions except in the following cases:

If K has odd characteristic, then K must be an extension field of F82 if l' is odd and s is a quadratic nonresidue of p. If K has characteristic 2 and 2 E N, then K must be an extension field of F4• ( * )

The extensions of CQ(K) and CN(K), denoted CQ(K) and CN(K)"also involve (I, These extensions each have index set I = 'I u {DO} and are defined by

CQ(K) =

and

{LCgX9 I LCgX9 E CQ(K), C= = - �LC9 } gET gET q gET

CN(K) = {LCgX91 LCgxg E CNCK), gET gEL

= C=X(-1)�LC9 } ' q gEL

If -1 EN, the extended coordinate is defined using the same equation for both CQ(K) and CN(K), whereas the extensions

HUFFMA:-l: THE AUTOMORPHISM GROUPS OF THE GENERALIZED QUADRATIC RESIDUE CODES 381

differ if -1 E Q. If -1 EN, then CQ(K) and C}/(K) are self-dual, and if -1 E Q, then CQ(K) and C}/(K) are duals of each other by [1 1, Lemma 4).

Ill. THE SETTING FOR AUTOMORPHISMS

A number of different notions of automorphism groups of codes have appeared in the literature. One very natural group from which to choose our automorphisms is the group of all Hamming weight preserving semilinear bijections of linear codes. We first discuss this group.

Let C be an [n, k] linear code over a field K. Let Sn be the permutations of an n-element set, and let K* =

K\{D}. By [12) the group of all Hamming weight preserving semi linear bijections of a linear code is the set of monomial transformations of Kn extended by Gal (K), the Galois group of K.2 This group is isomorphic to Wn(K) = (K*)" ><I (Sn x Gal (K)), the semidirect product of (K*t extended by Sn x Gal (K), and it is this group from which we will choose our automorphisms3 Multiplication in Wn(K) is defined by

(a; (T, �()(b; T, ti) = (c; (fT, �(ti) Ci = (a;fj)bi<T' for 1 :s; i :s; n (I )

where a, bE (K*)", fT, T E Sn, and" {j E Gal (K). W,,(K) acts on x = Xl ... Xn E Kn by

for 1 :s; i :s; n.

(2)

We interpret this action on the GQR codes or extended GQR codes in the natural manner given by

where!1 = I for GQR codes or !1 = I = I U too} for extended GQR codes.

Define the automorphism group Aut (C) of C to be the set of elements in Wn(K) which map each codeword of C to a codeword of C under the action given in (2). The image of the homomorphism 7[': Wn(K) --> Sn given by

(a: 0', ,)7[' = 0'

will be called the underlying permutation group of C and denoted by Per (C). Clearly if K = F2, Aut(C) and Per (C) are essentially the same.

We conclude this section with the verification that certain elements of Wq(K) are indeed automorphisms of CQ(K) and C}/(K). Let In denote the vector of all ones of length n. Define the following elements of Sq acting on I. For !} E I, let Tg E Sq be given by hTg = h + 9 for h E I. For 9 E I*, let A1g E Sq be given by hMg = hg for hE I. Notice that an

2 In this paper maps in S" and Gal (I\) will act on the right of elements in their domain and hence multiplication of two e1ements 0' and /J in one of these groups is given by .r(n;3) = (.rn),'.

'In the coding theory litemture. there are two other groups from which automorphisms are sometimes chosen: the group Sn and the group of n X 1l monomial matrices. In the case of binary codes. these are essentially the sarne as )IV" (/'). If I, has prime order, then }V" (I, ) is essentially the group of monumial matrices.

element T E Gal (Fq) is a permutation of 7 = Fq and hence T ESq.

Theorem 3.1,' Let C be either CQ(K) or C}/(K) where K has characteristic s and satisfies (*). Suppose that q = pT. The following conditions hold:

i) If 9 E I, then (lq; Tg, 1) E Aut(C). ii) If 9 E Q, then (lq; Mg, 1) E Aut (C) . If 9 EN, then

(lq; Mg, 1) interchanges CQ(K) and C}/(K). iii) If T E Gal (I), then (lq; T, 1) E Aut (C). iv) Let, E Gal (K) be given by X, = X' for X E K. If

r is even or s is a quadratic residue modulo p, then (lq; 1, 1) E Aut (C). If 'I' is odd and s is a nonresidue modulo p, then (lq; 1,,) interchanges CQ(K) and C}/(K); furthermore (lq; 1, ,2) E Aut (C).

Proof' Let

Then

C = LChXh E K[I]. hET

c(l; Tg, 1) = LChXh+!1 hET

implying 1/}k(r:(l; Tg, 1» = 1/1k(g)1/1k(C) for k E I. Also

c(l; Mg, 1) = LChXhg hET

implying 1/1k(c(l; Mg, 1» = 1/1kg(C). These yield i) and ii). Let T E Gal (I) and 9 E I. Define 1/1 : K[I] --> K(�) by

1/) (11,) = 1/Jg(hT-l) for all h E I. 1/1 is a linear character of I, and therefore .//) = 'ljJg, for some gT E I. As

1/1gy(h) = 1/1g(hT-l) = 1/Jl(g(hT-l)) = 1/}l(((!}T)h)T-1) = 1/1d(gT)h) = 1/1l, (gT) (h)

for all h E 7, we have

gT = 17'(gT). (4) Clearly

for {j E Gal (I). (5)

We next show that 1T E Q. By induction using (4) and (5),

we obtain ;-1

IT' = II (1r)Tj, for 1 :s; i. (6) j=O

Elements of Gal (I) map Q to Q and N to N; hence 1 T' E Q if IT E Q. Thus to show that IT E Q, it suffices to show this for the generator T of Gal (I) given by XT = xP for all X E I = Fpr. By (6) and the fact that Tr is the identity map

1 = 1T" = (IT)1+P+p2+.+pr-l = (IT)(p"-lJ!(P-1J.

Thus IT has multiplicative order a divisor of (pr - 1) I (p - 1) implying that 17' E Q as p is odd. SO 1T E Q for an arbitrary T E Gal (I). By (4), 9 E Q if and only if gT E Q. As

382 IEEE TRANSACTIONS ON INFORMATION THEORY. VOL. 41. NO.2. MARCH 1995

For part iv), extend, E Gal (K) to be an element of Gal (K(�» by defining x, = x' for all x E K(O. If 9 E I

where

1/Jy(C) = L:;Ch1/J1(gh) = L:;Ch�(9h)o hEY hEI

r-1 gh = L:;(9h)i(i

i=O

r-l s-lgh = L:;(s-lgh)i(i

;=0

Wg(c(lq; 1, I»� = LCh�(gh)o hET

= (LCh�(8-'9h)O)8 = (1/J,-lq(C»"'

hET

Hence (lq; 1, ,) E Aut (C) if s is a square inI, and (lq; 1, ,) . interchanges CQ(K) and CN(K) if s is a nonsquare in I. Part iv) follows immediately.

IV. THE GENERALIZED GLEASON-PRANGE THEOREM

The automorphisms of CQ(K) and CN(K) given in Theorem 3.1 can be extended to automorphisms of CQ(K) and CN(K). The aulomorphisms (lq; Tg, 1), (lq; Mg, 1), (lq; T, 1), and (lq; 1, ,,) extend to the automorphisms (lq+1; Tg, 1), (lq+1; My, 1), (lq+1; T, 1), and (lq+l; 1, ,) where, naturally, ooll.1g = XlTg = ooT = Xl. (The map (a: Mg, 1) where aoo = -Xl - I) and lJ.i = 1 for I E I interchanges CQ(K) and CN(K) if 9 E N as does the map (b; 1, ,) where boo = Xl-I) and bi = 1 for i E I if r is odd and s is a nonresidue modulo p.)4 There are elements of Aut (CQ(K») and Aut (CN(K» which move ex!. Let 1" = (t; v, 1) E Wq+1(K) where tg =X(g) for 9 E I*, to = 1, toe = X( -1), and l/ is the involution interchanging o with 00 and 9 with -1/ 9 for 9 E I*. The Gleason-Prange theorem shows that 1" is in Aut(CQ(K» for the ordinary extended QR codes. Van Lint and MacWilliams [11, sect. 41 proved this for the extended GQR codes using shifts of the generating idempotent. We prove this result directly from the definition.

Theorem 4.1: The map 1" E Wq+1(K) is in Aut (CQ(K». If n E N, then (a; MT" 1)-11"(a; Mn, 1) E Aut(CN(K» where aoc = -Xl-I) and (Ii = 1 for·j. E I.

4 An easy argument using Lemma 2.1 v) shows that e E Fs unless r is odd, ,I; is mId, and s is a nonresidue modulo p in which case e E Fs2' Hence (), = -f! if r is odd and s is a nonresidue modulo p again by Lemma 2.1 vJ. This allows us to show that (b: 1.;) interchangesCQ(J() and CX(K).

Proof: By Theorem 3.1 ii) and the previous paragraph. the second statement follows from the first.

Let

and

Then

and

C = L:;cgXg E CQ(K) gET

c = L:;cgXY E CQ(K) gET

d=LdgXg=CT. gET

I1g = X( �1 )Cl/9, if 9 E r

do = X( -l)coo

It suffices to show that a) 't/Jk(tl) = 0 for all h E Q where

d= LdyXg gET

and b) doc = -() /q LgET dg• We verify a) first. By Lemma 2.1 iii), if 9 E I*

dg = x(-I) C_1/g = �x(-I )L't/Jh(C)1/Jh (�) ' 9 q 9 hEY 9

So for k E Q

1/Jk(d) = du1/Jk(O) + L (�X(-l) gEY' q 9

But

. L:;1/JhCC)1/Jh (�) )1/Jk(9) hET 9

= xC -1)coo + � L:; 1/Jk(g)X (-1) 1/Jo(c)1/Jo (2.)

q gET' 9 9

+ � L 't/Jk(9)x (-I) L 't/Jh(C)1/Jh (�). q gET' 9 hET' 9

1/Ju(C) = L:; Cg = -(q/O)c= gET

and 't/!o{I/.q) = 1. So 't/Jk(d) = CocAk + (l/q)Rk where

Ak = X( -1) - � L:; 't/;k(9)X( �l) gET" 9

and

Bk = LI/Jk(g)x(-I ) L:; 1/Jh(C}l/Jh (�). gET' 9 hET' 9

Part a) is verified provided Ak = () and Bk = O.

HUFFMA:-l: THE AUTOMORPHISM GROUPS OF THE GENERALIZED QUADRATIC RESIDUE CODES 383

As X(-l/g) = X(-I)x(g) = X(-I)x(kg) for k E Q

Ak = X( -1) (1 - � L x(k!JNJl (kg») . gET'

But (J is by definition

as kI* = I*. So Ak = O. Substituting X( -1) x( h / g) X( h) for x( -1 / g) and noting that

X(h)l/Jh(C) is 0 if hE Q and is -l/Jh(C) if hEN, we have

Bk = -xl -1) L I/Jh(C) L I/Jl(k9)x('2)1/J1 ('2). hOi gEP 9 9

Thus Bk = 0 provided

for hEN.

But

When t' = (kh)/t, t' runs through N as t runs through Q and (kh)/t + t = (kh)/t' + t'. Hence Ck,h = 0 completing the verification of a).

To prove b), we have

Ldg = x( -l)coo + L x(-I) C_lj9 gET gEP g

=x(-l)coo+ LX(g)cg• gEP

Using Lemma 2.1 iii) and reversing the order of summation, we obtain

But

and using Lemma 2.1 iv), we ha�e

Ldg = x( -l)coo + � L 4Jh(C)X( -h)e. gET q hET'

If h E Q,4Jh(e) = 0 and if hEN, Xl-h) = -X(-I). Therefore

Expanding 'l/Jh(C) and reversing the order of summation yields

L, dg = X( -1) (coc - � [eo L I/Jh(O)

gET q hEP

+ LCg Ll/Jh(9)] ) ' gET' hET' By Lemma 2.1 ii)

Ll/Jh(9) hET'

is q - 1 if 9 = 0 and is -I/Jo (g) = -1 if 9 E I*. So

as

Ldg = X(-I) (coo -Iko + �Lcg) gET q9ET

= -X(-l)eco = -X(-1)(Jdoo

() Coo = -- LCg

q9ET and doo = co. Using Lemma 2.1 v), b) is verified, completing the proof.

V. THE AUTOMORPHISM GROUPS

In this section we give the complete automorphism groups of the extended GQR codes.

Before proceeding, we give the notation for the groups that will arise. GLn (q) is the general linear group of all nonsingular 11 x 11 matrices over Fq, and SLn ( q) is the special linear group consisting of those matrices in GLn (q) with determinant 1. The projective general linear group PGLn(q) is the quotient group GLn(q)/Z where Z = {alia E Fq, a of- O} is the center of GLn (q) , I being the identity matrix. Analogously, PSLn(q) is the projective special linear group SLn(q) modulo its scalar matrices. The group A ><l B is the semidirect product of the group A extended by the group B. In particular, �Ln ( q) � SLn(q) ><l Gal (Fq), rL,,(q) � GLn(q) ><lGal (Fq), P�Ln(q) � PSLn(q) ><lGal (Fq), and prL,,(q) � PGLn(q) )<lGal (Fq). The group GA,,( q) ss the set of all invertible affine maps from F:: onto F::; that is, GA,,(q) � F:: ><l GLn(q). The groups M12 and A124 are the 5-fold transitive Mathieu groups on 12 and 24 letters, respectively. Finally, the group An is the alternating group on 11 letters.

Let 9 be the subgroup of Sq+l, acting on T, generated by Tg for 9 E I, Mg ror 9 E Q, and the involution II where II is the permutation part of T. By Theorems 3.1 and 4.1, 9 <:::: Per (Cd K». It is well known that 9 � PSL2 (q) where the following identifications are made:

Tg� [� n Ma2-[� a�l ] and 1I�[�1 �l In particular, Per (CQ(K» is doubly transitive on 't. We can use the classification of the doubly transitive groups [3J to determine Per (Cd K». Thc classification of the rank 3 permutation groups5 will also play a role.

5 A transitive permutatiun group is rank 3 if the subgroup fixing a point has three orbits (including the fixed point).

384 IEEE TRANSACTIONS ON INFORMATION TIlEORY. VOL. 41. NO.2, MARCH 1995

Lemma 5.1 : Let q � 5. Per (CQ(K)) has a minimal normal subgroup N where one of the following holds:

i) IV = Y � PSL2(q) and Per (CdK)) � P where P is eithcr P:EL2(q) or prL2(q), or

ii) q = 5 and N � A6, or iii) q = 7, N � Fl, and Pcr(CdK)) � GA3(2), or iv) q = 11 and N � )'.1112, or v) q = 23 and N � )\.124.

Pro(j( Let H = Per (CQ (K)). As H is doubly transitive, by [3, Proposition 5.2], II contains a unique minimal normal subgroup N which is either elementary abelian6 and regular7 or is nonabelian and simple.

Assume first that N is elementary abelian and regular. Thus IV must have order q + I, and H = N H= where Hoo is the subgroup of H fixing 00: also N n Hoo = {I}. Hence there is a surjective homomorphism p: N Hoo -t Hoo, with kernel N, givcn by gp = g* where g* is the unique element of the coset gN lixing DC. As Y � PSL2(q) is nonabelian and simple for q � 5, pig must be injective; in particular, Hoc contains a subgroup yp which is isomorphic to PSL2(q). As Tqp = Tg for 9 E I and Ivhp = Mh for h E Q and because the subgroups of yp generated by {Tqlg E I} and {Mh Ih E Q} have orbits {{oo}, I} and {{oo}, {OJ, Q, N}, respectively, then yp must be either doubly transitive or rank 3 on the q-element set Y. Therefore, yp contains a doubly transitive or rank 3 permutation group isomorphic to PSL2(q) acting on the q points of I. By [8, Theorem 1.2], PSL2(q) does not have a rank 3 permutation representation on q points. By l3, table following Theorem 5,3], PSL2(q) has a doubly transitive permutation representation on q points only if q = 7 (in which case PSL2(7) � GL3(2) . This gives the possibility listed in iii).

Assume now that N is nonabelian and simple. Possibilities for N are listed in [3, table following Theorem 5.31. As the centralizer of N in H is normal in H, it must be trivial. Thereforc, there is a group HI with H � H' where N � Inn (N) S H' S Aut (N), Inn (N) being the group of inner automorphisms of N. As Q is simple, Q n N is either {l} or Q. If Q n N = {I}, then there is a subgroup of Aut (N)/Inn (N) isomorphic to PSL2(q), which is impossible as Aut(N)/Inn (N) is solvable for every possible N listed in [3]. Therefore, y c:;; N. The only possibilities for N of degree (I + 1 which contain a copy of PSL2 (q) are those listed in iJ. iv), or v) together with one further possibility, namely N � Aq+l• We show the latter is impossible if IJ � 7 and so only ii) is possible if N � Aq+1'

Suppose Aq-rl c:;; Per(CQ(K)) for some q � 7. Then A =

(d; (CX), 0, 1), 0) E Aut (CQ(K)) for some d = doododl , . , E (K*p+1 andb E Gal(K). By Theorem 3.1 iv),after replacing A by A4, we may assume that b = 1. Suppose CQ(K) has minimum distance I or 2. Because a doubly transitive group acts on CQ(K), the minimum weight vectors span a subspace of dimension q or q -t:: 1, a contradiction as CQ(K) has dimensions �(q + 1). So C dK) has minimum distance

6 A group is elementary abelian if it is abelian of order ab where a is a prime and every nonidentity element is of order (L

7 A permutation group is regular if it is transitive and the only element fixing a point is the identity.

3 or more, Let i oj:. j be in I\{ 0, I}. By the Singleton bound and (q - I)-fold transitivity of Aq+1, there is a minimum weight codeword e in CQ(K) such that c= = Co = Cl = ° with ei and Cj nonzero. But then wt(dic - cAl < wt(e) and so die = eA yielding di = dj for all i, j E I\{ 0, I}. We now choose another minimum weight codeword e' in Cd K) with e� oj:. 0 and e� = e� = O. But if i E I\{ 0, I}, then wt (die' - e' A) = 2, a contradiction.

Consider the case N = 9 as in i), Because the centralizer of N in H is trivial, the remainder of i) follows by Theorem 3.1 iii) and the fact that Aut (PSL2(q)) � PrL2(q).8

The next reyult examines the action of the diagonal maps (d; 1, 1) on CQ(K).

Lemma 5.2: i) Suppose (d: 1, 1) E Aut (CQ(K»). Then d = d1q+l for

some d E K. ii) There does not exist (d; 1, 1) E WqH(K) such that

CQ(K )(d; 1, 1) = C.N(K). Proof' Let 9 and h be distinct elements of I. As

Per(CQ(K) is doubly transitive on I, there is a minimum weight vector C E CQ(K) with c9 and Cit both nonzero. Then wt(dgc - c(d; 1, 1)) < wt(c) implying dgc = c(d; 1, I). Thus dg = dh and i) follows.

For 9 E I let

(e; 1, 1) = (lq+l; Tg, 1)-I(d; 1, l)(lq+l;Tg, l)(d; 1, 1) -1.

If CQ(K)(d; 1, 1) = C.'f(K), then (e; 1, 1) E Aut(CQ(K)) by Theorem 3.1 i). So eoo = dood;;} = 1 and fh = dh_gd/:1

for h E Y, By part i), elL = eoo = 1, and so dh-g = dh for all g, h E Y. Thus dk = do for all kEY. But then (do1q; 1, 1) maps CQ(K) to C.N(K), which is clearly impossible.

We can now give explicitly the full automorphism groups of all extended GQR codes,

Theorem 5.3: Let q = pr where p is an odd prime. Let K be a field of characteristic s relatively prime to p satisfying (*); suppose K, is a primitive element of K. Let 'Y E Gal (K) be given by X7 = x"', and let T be the Glea­son-Prange automorphism (t; lJ, 1). Let T(q, m) be the group generated by (dq+l; 1. 1), (lq+1; TI, 1), (lq+1; Mm, 1), T, and (lq+1; 1,7). The following hold (where the generator polynomial in i)-iv) depends on an appropriate choice of �, the primitive pth root of unity in an extension field of K).

i) Let q = 5, s = 2, and let K be an extension field of F4 = {O, 1, {3, {32} where {32 = 1 + (j. The cyclic code CQ(K) has generator polynomial 9 = 1 + {1X + X2 and &.enerating idempotent 1 + {jX + {j2 X2 + 82 X3 + (3X4. CQ(K) has basis {{3X= + gXilO s i :S 2}. Aut(CQ(K) is generated by (1)16; 1, 1), (16; (0, 1, 2, 3, 4), 1), and (1,1, {3, {32, (32, (3); (00,0),7-1). This group has order IK* I' 360 ·IGal (Kll· In addition Per (CdK )) � S6.

8The group PSL2(Q) is isomorphic to the Chevalley group A I (q) (see [5, p. xl). The outer aUlumorphism group of .4, (q) has order dt q where d = g�d(2, q - 1)= 2 (as q is odd), '1 = pI = pro and g = 1 by [5, pp. XV-XVII. The centraltz�r ofPSL2(Q) in prL2( q) is trivial. and hence rrL2 ('1) acts fmthfully by conjugation as a group of autumurphisms of PSL2(q). As PSL2(Q) has index 2r = dgg in prL2(Q). prL2(Q) must act as the full automorphism group of PSL2 ('t).

HUHMAN, THE AU'IDMORPHIS,>\ GROUPS OF THE GENERALIZED QUADRATIC RESIDUE CODES 385

ii) Let q = 7 and s = 2. The cyclic code CQ(K) has gener­ator polynomial 9 = 1 + X + X3 and generating idempotent gX. CdK) has basis {X"" + gXilO �i � 3}. Aut (CdK)) is generated by T(7, 2) (where v = (00, 0)(1, 6)(2, 3)(4, 5)) and (Is; P, 1) where p = (2,4)(5,6). This group has order IK *I ' 1344 ·IGal(K)I. In addition Per(CQ(K)) -:::: GA3(2).

iii) Let q = 11 and s = 3. Thc cyclic code CQ(K) has generator polynomial 9 = -1 + X 2 - X3 + X4 + X5 and generating idempotent _ X2 _XG _ X7 _XB_XlO. CQ(K) has basis {-X"'" + gX' 10 � i � 5}. Aut (CQ(K)) is generated by T(IL 3) (where // = (oc, 0)(1,10)(2,5)(3,7)(4,8)(6,9)) and (In; f, 1) where E = (1. 9)(4, 5)(6, 8)(7, 10). This group has order I K* I 95040· IGal (K) I. In addition Per(C:Q(K)) "-' M 12 .

iv) Let q = 23 and s = 2. The cyclic code Cd K) has gen­erator polynomial 9 = 1 + X + XG + XG + X7 + X9 + Xli and generating idempotent X + X2 + X3 + X4 + X6 + X8 + X9 + X 12 + X13 + X16 + X18. CQ(K) has basis {XOC + gXilO � i � II}. Aut(CQ(K)) is generated by T(23,2) (where v = (00,0)(1, 22)(2,11)(:3.15)(4.17)(5,9)(6,19)(7,13) (8,20)(10,16)(12,21)(11,18) and (hi: 7/,1) where ." = (1,3,13,8,6)(4,9.18,12,16)(7,17,10,11,22)(14,19, 21, 20, 15). This group has order IK* I· 244823040·IGal (K)I. In addition Per (CQ(K)) -:::: M24.

v) Let 1/, s, and K be any case not occurring in parts i)-iv). Let T be the group generated by (K.1q+1: 1, 1), (lq+1: Tg, 1) for 9 E I, (lq+1; 1I-1h, 1) for h E Q, (lq+1; T, 1) for T E Gal (Il, and lr. There are two possibilities.

a) If r is cven or s is a quadratic residue modulo p, then Aut (CQ(K)) is the group generated by T and (lq+l: 1,1'). Also Per(CQ(K)) "-' P2;L2(q).

b) If T is odd and s is a quadratic nnnresidue module p, then Aut (CQ (K)) is the group generated by T and (c; M", ,) for any n in }If where Cou = -1 and Ci = 1 for i E I. Also Per (CQ(K)) -:::: prL�(q).

In both cases Aut (CQ(K)) has order IK* I . �(q3 - q)r . IGal(K )I·

In all cases Aut (C;.r(K)) = (a; Mn. 1)-1 Aut (CQ(K))(a; 11-1",1) for any n EN where (1= = - X(- I ) and (Ii = 1 for i E I.

Proof' Using Theorem 3.1, Theorem 4.1, the discussion beginning Section IV, and simple computation, Aut (Cd K») contains the elements claimed in each parL By Theorem 3.1

iil, Aut (C;.r(K) is as asserted. The case IJ = 3 satisfies v) also by direct computation noting that PSL2(3) -:::: A1 and PGL2(3) -:::: 54· Thus we assume that q 2': 5.

Consider the case q = 5 arising in Lemma 5. 1 ii). We first show that 8 = 2. The generator polynomial for CdK) is 9 = 1 - fiX + X2 where /J = � -I- �4 and (J2 = 1 - !-J. Per(CQ(K) contains a subgroup isomorphic to A5• Hence there is an element A E Aut (C Q (K)) of the form A = «(do, dl,' " d4); (0, 1,2), b) where b E Gal (K). Replacing A by A4 we may assume that h = 1 by Theorem 3.1

iv). By rescaling, we may assume that d2 = 1. So gA = d2 + doX - di/:JX 2 must be a multiple of g, giving do = -(3 and (h = _(i- I . (gX)A = -/1 - (j-l X2 + d:1X3 which must be -('3.'1 - (32 (gX) = -/:1 + (-(j + ({3)J{2 - (�2 X3. Thus B-1 = (J - ;P, and using /)2 = 1- P, we obtain 2(1- (1) = o.

As (3 cannot equal 1, 2 = 0 and K must have characteristic 8 = 2. When s = 2, by (*) , K must contain F4, its primitive element being /3. The code CQ(K) has basis {(3xoo+gXiIO ..s; i � 2}. Assume that (d; (J', 8) E Aut(CQ(K)) . As (00,0) and (0, 1, 2. 3, 4) generate 56, we may assume that (J' = 1. As «1, 1, P, 132, (32, (3); (oc, 0), 1'- 1)2 = (16; 1, "1-2), we may assume that either tJ = 1 or 8 = "1-1, a simple calculation excluding the latter. Part i) follows by Lemma 5.2 i).

Recall that Q is the group generated by Tg for 9 E I, Mh for h E Q, and v.

In [14], Shaughnessy showed that if an extended GQR code9 of length 8, 12, or 24 has an automorphism of the form (d; (J', 1) where (J' rf. Q, then K has characteristic 2, 3, or 2 for the respective lengths 8, 12, or 24. Suppose that we have an automorphism (f!; T, 8), where T rf. Q. By Lemma 5.1 iii)-v), Per (CQ(K») has a subgroup isomorphic to GA3(2), A1l2, or M24, all of which are their own commutator group. Hence as Gal (K) is abelian, we can find an element (d; (J', 1) in the commutator group of Aut (CQ (K» where (J' rf. Q. Thus hy Shaughnessy's result, K has characteristic 2, 3, and 2,

respectively. As Aut(Ff) = GL3(2) and Aut(M24) = M24, parts ii) and iv) follow. Since IAut (M12)1 = 2 . IM121 but Aut (M12) cannot be realized as a subgroup of 512, part iii) follows. to

Finally, we consider the case arising in Lemma 5.1 i). Suppose first that T is even or s is a quadratic residue modulo p. Suppose that (d; (J'Mh, til E Aut (CdK) for some (J' E Q, It E N, and b E Ga\(K). By Theorem 3.1, we may assume that (J' = 1 and 8 = 1. So by Theorem 3.\ ii), (d'; Mf, 1) = (d; Nh, 1)(a; Mh, 1) maps CQ(K) to C;.r(K) where aoo = - x( - 1) and (li = 1 for i E I. As Mf = A1h2 we obtain a contradiction from Theorem 3.1 ii) and Lemma 5.2 ii). Now assume that (d; (J', 8) E Aut (CQ(K)) for some (J' E Q and 8 E Gal (K). By Theorem 3.1, we may assume that (J' = 1 and 8 = 1. By Lemma 5.2 i), d = dlq+1 and v) a) is verified.

Suppose now that r is odd and 8 is a quadratic nonresidue modulo p. Suppose that (d; (J', 8) E Aut(CQ(K) for some a in the group generated by Q and 1'vh for h E I*, and b E Gal (K). By Theorem 3.1, we may assume that (J' = 1 and 8 = 1 or h = "I. If 8 = 1, d = dlq+1 by Lemma 5.2

i). By Theorem 3.1 iv), (b; 1, "1-1) , where boo = x( - 1) and bi = 1 for i E I, maps CdK) to C;.r(K) . SO if 8 = l' we obtain a contradiction to Lemma 4.2 ii). This completes the proof of v) b).

Remarks:

i) In Theorem 5.3 i) there is a subgroup of index 2 in Aut (CQ(K») which is isomorphic to P )<I Gal (K: F4) where P is a central product of K* with the nonsplitting central extension of a group of order 3 by A6 (i.e, the triple cover of A6).

9The extended GQR codc, of lengths 8, 12, and 24 are actually extended QR codes.

lOll' Aut(,'vt12) is a subgroup uf 812. then the permutation character of Aut (,'vt12) is the sum of the trivial character and an irreducible character of degree II as Aut (.\,1,,) is doubly transitive (see, for example, [7, Theorem 11.3]). By 15, p. 331, AUt(,·'vt,2) has nu irreducible character of degree 11; the twu characters of degree 11 for ,'vt,2 fuse to form an irreducible character of degree 22 in AUt(.'vt,2).

386 IEEE TRANSACTIONS or-; INfORMATION TIIEORY, VOL. 41, NO.2, MARCH 1995

ii) In Theorem 5.3 ii) Aut (CdK» is isomorphic to (K* x GA3(2» >l Gal (K).

iii) In Theorem 5.3 iii) Aur(CdK» is isomorphic to P ><I Gal (K) where P is a central product of K* with the nonsplitting central extension of a group of order 2

by M12 (i.e., the double cover of M12). iv) In Theorem 5.3 iv) Aut (CQ(K) is isomorphic to

(K* x M24) >l Gal (KJ. v) In Theorem 5.3 v) a), when B = 2 or -1 E Q,

Aut (Cd K» is isomorphic to P ><I Gal (K) where P = K* X PEL2(q). If S > 2 and -1 E N, Aut (CQ(K» is isomorphic to P >l Gal (K) where P is a central product of K* with �L2(q).

vi) In Theorem 5.3 v) b), when s = 2 or -1 E Q, Aut (Cd K» has a subgroup of index 2 isomorphic to P ><I Gal (K: Fs') where P = K* X PEL2(q). If S > 2 and -1 EN, Aut(CdK» has a subgroup of index 2 isomorphic to P >l Gal (K: F,2) where P is a central product of K* with �L2(q).

vii) The codes in Theorem 5.3 i) when K = F4, The­orem 5.3 ii) when K = F2, Theorem 5.3 iii) when K = F3, and Theorem 5.3 iv) when K = F2 are the [6, 3, 4] quaternary hexacode, the [844] extended Hamming code, the [12, 6, 6] ternary Golay code, and the [24, 12, 8] binary Golay code, respectively.

viii) Using the computer algebra program GAP [13], we verified that the permutations listed in Theorem 5.3 iii) and iv) in fact generate simple groups of orders 95040 and 244823040, respectively. We also verified that the permutations of Theorem 5.3 ii) form a group of order 1344 with stabilizer a simple group of order 168.

ACKNOWLEDGMENT

The author wishes to thank E. Assmus, P. Carnion, S. Mattson, and T. Ward for helpful comments, particularly relating to the history of the development of the GQR codes and the Gleason-Prange Theorem.

REl'ERENCE5

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