Type II Codes over Z4Alexis Bonnecaze, Patrick Sol�e �, Christine Bachoc y, Bernard Mourrain zABSTRACTType II Z4-codes are introduced as self-dual codes over the integers modulo 4 containingthe all-one vector and with euclidean weights multiple of 8: Their weight enumerators are char-acterized by means of invariant theory. A notion of extremality for the euclidean weight isintroduced. Their binary images under the Gray map are formally self-dual with even weights.Extended quadratic residue Z4-codes are the main example of this family of codes. They areobtained by Hensel lifting of the classical binary quadratic residue codes. Their binary imageshave good parameters. With every type II Z4-code is associated via construction A modulo 4an even unimodular lattice (type II lattice). In dimension 32, we construct two unimodularlattices of norm 4 with an automorphism of order 31. One of them is the Barnes-Wall latticeBW32.Index Terms | Self-dual codes, codes over rings, weight enumerators,Z4-codes, lattices.�CNRS I3S, Route des Colles, BP 145, 06903 Sophia Antipolis, Cedex, France (bonnecaz@essi.fr,sole@alto.unice.fr)yLaboratoire A2X, 351 Cours de la Lib�eration, F-33405 Talence, France (bachoc@math.u-bordeaux.fr)zINRIA, Route des Lucioles, BP 93, 06902 Sophia Antipolis, Cedex, France (mourrain@sophia.inria.fr)1

I. IntroductionThe conditions satis�ed by the weight enumerator of self-dual codes, de�ned over the ring ofintegers modulo four, have been studied by Klemm [1], then by Conway and Sloane [2]. TheMacWilliams transform determines a group of substitutions, each of which �xes the weightenumerator of a self-dual code. This weight enumerator belongs to the ring of polynomials�xed by the group of substitutions, called the ring R of invariants. Finding an explicit basisfor this ring is possible since R is Cohen-Macaulay [3]. Each invariant is written uniquelyinto this basis. (For more information about this theory, see [4] and [5]).Among all of the self-dual Z4-codes, some have the property that all euclidean weights aremultiples of 8. These codes are called type II codes by analogy with the binary case. Anupper bound on their minimumeuclidean weight is given, thereby leading to a natural notionof extremality akin to similar concepts for type II binary codes and type II lattices. The mostinteresting examples of type II codes are perhaps the extended quadratic residue Z4-codes[6]. This class of codes includes the octacode [8; 4; 6] and the lifted Golay [24; 12; 12]. Otherclasses of interest comprise a multilevel construction from binary Reed-Muller and lifteddouble circulant codes. The paper is organized as follows. Section II contains backgroundinformation onZ4-codes, the restrictions on weight enumerators provided by invariant theory,and recalls the results of Conway and Sloane in [2]. Section III de�nes type II codes andgives the conditions satis�ed by their complete weight enumerator. Section IV gives threeexamples of families of type II codes with their general construction. Section V shows therelation between type II codes and even unimodular lattices.2

II. Z4-Codes and Invariant TheoryBy a Z4-code C of length N we shall mean a linear block code over Z4, that is an additivesubgroup of ZN4 . We de�ne an inner product on ZN4 by (a; b) = a1b1 + � � � + aNbN (mod 4),and then the notions of dual code (C?), self-orthogonal code (C � C?) and self-dual code(C = C?) are de�ned in the standard way. We shall say that two Z4-codes are equivalentif one can be obtained from the other by permuting the coordinates and (if necessary)changing the signs of certain coordinates. The automorphism group Aut(C) consists of allmonomial transformations (coordinate permutations and sign changes) that preserve the setof codewords. We shall say that a code C is isodual if it is equivalent to its dual C?.Several weight enumerators are associated with a Z4-code C. The complete weight enu-merator (or c.w.e.) of C iscweC(W;X; Y; Z) = Xa2CW n0(a)Xn1(a)Y n2(a)Zn3(a) ;where ni(a) is the number of components of a that are congruent to i modulo 4. Sincea monomial transformation may change the sign of a component, the appropriate weightenumerator for an equivalence class of codes is the symmetrized weight enumerator (or s.w.e.)given by sweC(W;X; Y ) = cweC(W;X; Y;X) :The MacWilliams identity over Z4 expresses the weight enumerator of the dual code C? interms of the weight enumerator of C:cweC?(W;X; Y; Z) = 1jCjcweC(W+X+Y +Z;W+iX�Y�iZ;W�X+Y�Z;W�iX�Y+iZ);3

which impliessweC?(W;X; Y ) = 1jCjsweC(W + 2X + Y;W � Y;W � 2X + Y ) :The Gray map � provides a one to one correspondence between a Z4-code and a binarycode. Hammons et al. [8] explored the Gray map which is a distance preserving map orisometry from (ZN4 , Lee distance) to (Z2N2 , Hamming distance). Recall that the Lee weights ofthe elements 0; 1; 2; 3 of Z4 are respectively 0; 1; 2; 1, and that Lee weight of a vector a 2 ZN4is just the rational sum of the Lee weights of its components. This weight function de�nesthe Lee metric on ZN4 . We de�ne maps �; from Z4 to Z2 byc �(c) (c)0 0 01 0 12 1 13 1 0and extend them in the obvious way to maps fromZN4 to Z2N2 . The Gray map � : ZN4 !Z2N2is given by �(c) = (�(c); (c)).The binary image �(C) of a Z4-linear code C under the Gray map need not be Z2-linear,so that the dual code may not even be de�ned. We de�ne the Z4-dual of �(C) to be the code4

C? = �(C?). Thus C ��! �(C)dual ????yC? ��! C? = �(C?) ;note that the formal duality of the binary codes implies that the diagram is non commuting.The binary image under the Gray map of a self-dual code over Z4 is formally self-dual (see[8]) and distance invariant. Furthermore, its weights are even, since the residue code mod 2of its preimage is self-orthogonal. Using this simple observation it follows immediately, usinga well-known result on binary type I codes (see [9] Chapter 7).Theorem 1 The minimum Lee distance dL of a self dual Z4 code of length N is at mostdL � 2(bN4 c+ 1):This bound is not tight in general (except for n = 8 in the case of the octacode). It is adi�cult open problem to sharpen this bound using the following Gleason-type theorems.In [1], Klemm has studied the conditions satis�ed by the complete weight enumerators ofself-dual codes overZ4. Conway and Sloane deduced analogous theorems for the symmetrizedand Hamming weight enumerators (theorems 6-9 of [2]). These results come from a "new ap-plications of a nineteenth-century technique" [10]: invariant theory. The weight enumeratorsare invariants of a group of substitutions. The number of linearly independent homogeneousinvariants of degree i is given by the coe�cient of �i in the Molien series. The followingtheorem characterizes the symmetrized weight enumerator of a self-dual code of length N5

over Z4 containing a vector �1.Theorem 2 The symmetrized weight enumerator of a self-dual code of length N over Z4containing a vector �1 belongs to the ringS � 8S � 28S ;where S is the ring of polynomials in�4 = W 4 + 6W 2Y 2 + Y 4 + 8X4 ;�8 = (W 2Y 2 �X4)((W 2 + Y 2)2 � 4X4) ;�12 = X4(W 2 � Y 2)4 ;and 8 is the polynomial 8 = X4(W � Y )4 :This ring has Molien seriesS(�) = 1 + �8 + �16(1 � �4)(1� �8)(1� �12)= 1 + �4 + 3�8 + 4�12 + 7�16 + 9�20 + 13�24+ 16�28 + 21�32 + 25�36 + 31�40 + 36�44 + 43�48 + � � � :Example 3 The so-called lifted Golay QR24 is a remarkable self-dual code over Z4. It is anextended quadratic residue Z4-code. The expression of the symmetric weight enumerator ofthe \lifted Golay" in terms of the basic polynomials given in theorem 2 issweQR24 = �64 � 36(�44�8) + 390�24�28 � 1056�38 � 84�34�12 + 588�4�8�12 � 76�212+644 8�4�12 + 28(�322�24 + 5152�8):6

III. Type II Z4-CodesAZ4-code of type II is a self-dualZ4-code containing a vector �1 and which has the propertythat all euclidean weights are multiples of 8.Theorem 4 The complete weight enumerator of a type II code of length N over Z4 belongsto the ring S � 16S � 32S � 16 32S;where S is the ring of polynomials in �8; �8; �16; �24 where�8 is the cwe of the octacode O8 ;�8 is the cwe of Q8 ;�16 is the cwe of RM(1; 4) + 2RM(2; 4);�24 is the cwe of the lifted Golay ; 16 is the cwe of K16 = RM(0; 4) + 2RM(3; 4), the self-dual code introduced by Klemm [1],and 32 is a homogenious polynomial of degre 32 which doesn't belong to the algebra S� 16S.This ring has Molien seriesS(�) = 1 + �16 + �32 + �48(1� �8)2(1� �16)(1� �24)= 1 + 2�8 + 5�16 + 9�24 + 16�32+ 25�40 + 39�48 + � � � :7

The complete weight enumerator of the code is left invariant by the matrix group G, of size768, generated by M1; M2; M3 withM1 = 12 266666666666664 1 1 1 11 i �1 �i1 �1 1 �11 �i �1 i 377777777777775 ;M2 = 266666666666664 1 0 0 00 w 0 00 0 �1 00 0 0 w 377777777777775 ;M3 = 266666666666664 0 0 0 11 0 0 00 1 0 00 0 1 0 377777777777775 ;where w is a 8th root of unity.Remark 1 All the cwe of the codes we have considered in this paper belong to the algebraS� 16S: In degre 32, the dimension of this algebra is 15 and the dimension of the invariantalgebra is 16. The algebra generated by the cwe of the type II codes is included into theinvariant algebra, but we don't know if we have equality.Corollary 5 A Z4-code of type II can only exist in lengths multiple of 8:Theorem 6 The symmetrized weight enumerator of a type II code of length N over Z4belongs to the ring S � 16S ;where S is the ring of polynomials in �8; �8; �24 where�8 is the swe of the octacode O8 ;�8 is the swe of Q8 ;�24 is the swe of the lifted Golay ;and 16 is the swe of RM(1; 4) + 2RM(2; 4).8

This ring has Molien seriesS(�) = 1 + �16(1� �8)2(1� �24)= 1 + 2�8 + 4�16 + 7�24 + 10�32 + � � � :The symmetric weight enumerator of the code is left invariant by the matrix group G, ofsize 768, generated by M1; M2; M3 withM1 = 12 2666666664 1 2 11 0 �11 �2 1 3777777775 ;M2 = 2666666664 1 0 00 w 00 0 �1 3777777775 ;M3 = 2666666664 0 0 10 1 01 0 0 3777777775 ;where w is a 8th root of unity.Remark 2 Results similar to theorems 3.1 and 3.2 have been obtained in [11, theorem 4 and corol-lary 5], with di�erent bases and of course the same Molien series.IV. Examples of Type II CodesWe give three examples of families of type II codes with their general construction.A. Extended Quadratic Residue Z4-CodesRecently, Bonnecaze, Sol�e and Calderbank investigated extended quadratic residue Z4-codes[6]. They obtained the parameters of the lifted Golay. These codes represent a remarkableclass of type II. The image by the Gray map of the octacode is the Nordstrom-Robinsonand the image of the lifted Golay is a (48; 24; 12) non linear binary code. In this section,9

we describe QR32 and QR48 the extended quadratic residue Z4-codes of length respectively32 and 48. Recall (see [6]) that a quadratic residue Z4-code is de�ned to be the cyclic codegenerated by h(x) 2 Z4[x], the Hensel lift of the generator polynomial h2(x) 2 Z2[x] of thebinary quadratic residue code.The code QR32 is a [32; 16; 14] code with minimum euclidean distance 16. It is generated bythe polynomialh(x) = x15 + 3x12 + 2x11 + 2x9 + 2x8 + 3x7 + x6 + 2x3 + 3x2 + 3x+ 3and extended by a parity check symbol. The complete weight enumerator of QR32 in termsof the basic polynomials is1405349 �84 + 996916 �84 � 236165168 �83 �8 + 287715112 �82 �82� 374515880 �82 �16 + �8 ��82934 �83 + 5363420 �8 �16 + 3�24�+ 4=49�162 � 5513840 �16 �82 � 53 �8 �24+ 16��827240 �82 + 1127120 �8 �8 � 1427240 �82� :And the symmetrized weight enumerator of QR32 in terms of the basic polynomials is63136 �48 � 121160 �48 � 12367360 �38�8 � 21160 �28�28 + 19945 �8�24 + 4823120 �8�38 � 13945 �8v�24+ 16(� 71120�28 � 6920�8�8 + 9724�28):10

Its image by the gray map �(QR32) is a (64; 232; 14) non linear formally self-dual code witha weight enumerator symmetric with respect to 32Number of words Weights1 0; 6439680 14; 50127596 16; 481944320 18; 469443840 20; 4432347136 22; 42136671808 24; 40232207360 26; 38603596288 28; 36643625472 30; 34974960294 32; 32In length 48, the binary extended quadratic residue code has very good parameters. It is a[48; 24; 12] F2-code with the following weight enumeratorW 48 + 17296W 36X12 + 535095W 32X16 + 3995376W 28X20+7681680W 24X24 + 3995376W 20X28 + 535095W 16X32+17296W 12X36 +X48: 11

The Hensel lift of this code is a [48; 24; 18] Z4-code generated byh(x) = x23 + 2x21 + x19 + x18 + 2x16 + x14+3x13 + 3x12 + 2x11 + 3x10 + 3x9 + x7+3x6 + 3x5 + 2x4 + x3 + x2 + 3x+ 3:Its minimum euclidean distance is 24 and its symmetric weight enumerator in terms of thebasic polynomials is�3190583120960 �68 � 1363994996048000 �58�8 + 16377833336048000 �48�28+�38(�130629443432000 �38 � 7747619252000 �24) + �28(�11991809126000 �48 + 41678737756000 �24�8)+�8(18290296400 �58 � 41747321600 �28�24)� 633311957600 �68 � 9607121600�38�24 + 6532700�224+ 16(�1617613224000 �48 + 231511163000 �38�8 � 218263167200 �28�28 + �8(�1239477200 �38 + 2095312600�24) + 155335180640 �48 � 19092520�24�8):12

Its image by the gray map �(QR48) is a (96; 248; 18) non linear formally self-dual code witha weight enumerator symmetric with respect to 48Number of words Weights1 0; 96138368 18; 781660416 20; 7617434368 22; 74169418832 24; 721698605568 26; 708017318656 28; 6852482567296 30; 66207498156471 32; 64742015419264 34; 622370501481216 36; 605453161473024 38; 5812741529744944 40; 5620713663863552 42; 5434296818992128 44; 5240358425379712 46; 5047582973403024 48; 48Remark 3 In [12], V. Pless and Z. Qian give the explicit complete weight enumerator (with-out using invariant theory) of QR32 and QR48.13

B. Cm;r codesLet Cm;r denote RM(r;m) + 2RM(m � r � 1;m):Proposition 7 For 0 � r � m�13 and m � 3 the code Cm;r is a type II Z4-code.Proof The linearity of the code is trivial. Self-duality follows by orthogonality of RM(r;m)and RM(m � r � 1;m). To prove that all codewords have euclidean weights divisible by 8,write a generic codeword of Cm;r as c+ 2d, with c 2 RM(r;m) and d 2 RM(m� r � 1;m).Identifying binary words with their supports, we get n1 = jcj; n2 = jdj � jc \ dj:By MacEliece's theorem ([4], p 447), for r > 0, n1 is a multiple of 2bm�1r c � 8 (the case r = 0is trivial).By orthogonality of c and d, jc \ dj is even. Since RM(m � r � 1;m) � RM(m;m); jdj isalso even. Hence, the euclidean weight of c+ 2d, that is n1 + 4n2 is a multiple of 8. 2Remark 4 The cwe of Cm;r is computed easily from the joint weight enumerator of RM(r;m)and RM(m� r � 1;m) (See [4], p. 149).Example 8 C5;1 is a type II code with parameters [32,16,8] and minimum euclidean distance16. The complete weight enumerator of C5;1 in terms of the basic polynomials is268490147 �48� 598607 �8�38+ 20979314 �28�28� 232892 �38�8+3384�48 � 2873147 �16�28+ 170542 �16�8�8�29714 �16�28 + 449�216 � 583 16�28 + 1393 16�8�8 � 27 16�28 + 403 �24�8 � 12�24�8Note that Cm;0 is the Klemm code with parameters [32,16,4]. The complete weight enumer-ator of C5;0 in terms of the basic polynomials is14

�5080627882 �48+4654007180 �8�38�4559608105 �28�28+1294094 �38�8�36098140 �48�3509098820 �16�28+2483 �16�8�8�13456315 �16�28� 475880�216+143719630 16�28�25795 16�8�8+1313845 16�28� 31210 16�16+128245 �24�8�134645 �24�8C. Double Ciculant Z4-CodesIn [11] Calderbank and Sloane study the double circulant codes DN . This code is similar tothe code B of [4] p.507, but it is read modulo four.De�nition 9 Let n = 2p + 2 where p is a prime congruent to 3 (mod 8). The code DN hasgenerator matrix M = 266666666666664 1 1 : : : 1 1 1 : : : 11 0... I ... I + 3R + 2N1 0 377777777777775if p � 11 (mod 16), or M = 266666666666664 1 1 : : : 1 1 1 : : : 11 0... I ... b(I +R)1 0 377777777777775if p � 3 (mod 16), where R = (Rij); Rij = 1 if j � i is a nonzero square mod p, or 0otherwise; N = (Nij); Nij = 1 if j � i is not a square mod p, or 0 otherwise; and b can beeither +1 or �1. 15

Theorem 10 (Calderbank and Sloane) DN is a self-dual code over Z4 in which all normsare divisible by 8.Example 11 For p = 19, the code D40 is a type II code with minimal norm 16.V. Even Unimodular LatticesAn N-dimensional lattice � in RN is the set of integer linear combinations of N linearlyindependent vectors v1; : : : ; vN . The dual of an N -dimensional lattice � is denoted ��, andis given by �� = fx 2 RN j (x; a) 2 Z, for all a 2 �g. A lattice � is integral if the innerproduct of any two lattice points is integral, or equivalently, if � � ��. If � is an integrallattice, then � � �� � �=det�. An integral lattice � with det� = 1 (or equivalently� = ��) is called unimodular. If (x; x) is an even integer for all x 2 � then � is called even.The class of even unimodular lattices includes the Gosset lattice E8, the Leech lattice �24and the Barnes-Wall BW32. The theta series ��(q) of the integral lattice � gives the numberof points which are at equal distance from the origin. It is the formal power series��(q) = Xx2� q(x;x) = 1Xm=0Nmqm ;where Nm is the number of vectors x 2 � with norm m.The kissing number is the number of spheres touching one sphere in the packing or equiva-lently the number of points on the �rst layer of the lattice.If C is a Z4-linear code with block length N then the quaternary lattice �(C) is given by�(C) = fx 2 ZN j x � c (mod 4) for some c 2 Cg16

or equivalently �(C) = C + 4ZN:This is Construction A ([9, Chapter 5]) applied to Z4-codes. We call it Construction A4.Remark 5 This construction limits the minimum norm of the lattice to four. To constructlattices with minimum norm greater than four we should consider codes over Z2a (a > 2).The metric generally used in lattice theory is the euclidean metric. The theta series of �(C)=2can be found by replacingW;X; Y by f0 = Px24Zqx2=4; f1 = Px24Z+1 qx2=4; f2 =Px24Z+2 qx2=4in the symmetrized Lee weight enumerator of the Z4-code C.Theorem 12 Let C be a self-dual code over Z4 such that all Euclidean weights in C aredivisible by 8. Then �(C)=2 is an even unimodular lattice.Proof. See [6]. 2Corollary 13 The minimum euclidean distance dE of a type II Z4-code of length N is atmost dE � 8(bN24c+ 1):Proof. Let C be a type IIZ4-code of lengthN . The lattice �(C)=2 being even unimodular by[6, theorem 4.1], its theta series �C , of weightN=2, belongs to the polynomial algebra C [E4;�]([9], chapter 7 ), where E4 is the theta series of the root latticeE8, and � = q2Qr�1(1�q2r)24.Furthermore, �(C)=2 contains the norm 4 lattice �0 = (4ZN )=2 with theta series �0. Ifj = N=8, we have : 17

�C = [j=3]Xs=0 asEj�3s4 �s =Xr�0Nrqr = �0 +Xr�1�rqr:With these notations, the smallest r such that �r is non zero is r = dE=4.We then follow the same line as in [13]. The linear system with unknowns a0; ::; a[j=3]de�ned by the equations �0 = 0; �2 = 0; ::; �2[j=3] = 0 has got a unique solution �� =�0+Pr�2([j=3]+1) ��rqr, which is the theta series of the lattice associated to an extremal code,if it exists.The inequality dE � 8([N=24]+ 1) is equivalent to the assumption ��2([j=3]+1) 6= 0. All theseries are in q2 = t, and B�urman's formula shows that��2([j=3]+1) = �b2([j=3]+1)where b2s = 1s! ds�1dts�1 ( ddt(E�j4 �0)(�=E34)s)ft=0gwhich gives, using the fact that �0 = �j1, where �1 is the theta series of the lattice (4Z)8=2,b2s = �js! ds�1dts�1 (E3s�j�14 �j�11 (�1E 04 � �01E4)(t=�)s)ft=0gIt is know su�cient to show that the coe�cients of �j�11 (�1E 04 � �01E4) are positive up tothe index [j=3] (here f 0 is the derivation of f with respect to t = q2).Let C8 be any type II Z4-code of length 8, such that �(C8)=2 is the lattice E8. Then18

E4 = WC8(f0; f1; f2) and �1 = f80 . Deriving E4=�1 = WC8(1; f1=f0; f2=f0), we �nd :�j�11 (�1E04 � �01E4) = @WC8@Y (f0; f1; f2)f8j�10 (f0f 01 � f 00f1)+@WC8@Z (f0; f1; f2)f8j�10 (f0f 02 � f 00f2):We are reduced to show that f8j�10 (f0f 01 � f 00f1) and f8j�10 (f0f 02 � f 00f2) have positivecoe�cients up to [j=3].t(f0f 01 � f 00f1) = Xx;y2Z(1 + 4y)2 � (4x)28 t (1+4y)2+(4x)28and tf s0(f0f 01 � f 00f1) = Xx;y;x1;::;xs2Z(1 + 4y)2 � (4x)28 t (1+4y)2+(4x)2+(4x1)2+:::+(4xs )28 :Let k be a �xed integer. If (1 + 4y)2 + (4x)2 + (4x1)2 + ::: + (4xs)2 = k, one can, bysymmetry, put together the terms corresponding to the di�erent choices of x. One gets, asa coe�cient of tk=8,(s+ 1)(1 + 4y)2 � (4x)2 � (4x1)2 � :::� (4xs)28 = (s+ 2)(1 + 4y)2 � k8 :which is positive as soon as s+ 2 is greater than k, because 1 + 4y is never zero. 2A code meeting that bound with equality is called extremal. The codes O8; Q8;K16 andD24 are extremal and generate extremal type II lattices via construction A4: This is notnecessarily so. For instance QR48 is extremal even though the lattice it generates containsvectors of norm 4 by de�nition of construction A4:19

Theorem 14 Let E8; �24 and BW32, be respectively the Gosset the Leech and the Barnes-Wall lattice. Then I: 2E8 = O8 + 4Z8II: 2E8 = Q8 + 4Z8III: 2�24 = QR24 + 4Z24IV: 2BW32 = RM(1; 5) + 2 RM(3; 5) + 4Z32Proof. see [6]. 2Since QR32 is a code of type II, the lattice obtained by the same construction with the codeQR32 is an even unimodular lattice. Furthermore, the minimum euclidean weight of QR32is 16. Then, the produced lattice is of minimum norm 4 and its theta series is1 + 146880q16 + 158757681973184q64 + 64757760q24 + 137695887360q40+4844836800q32 + 2121555283200q48 + 21421110804480q56 + 928985325895680q72+18845727679406080q88 : : :Let us call this lattice 2BSBM32: We see that BSBM32 is an even unimodular lattice ofnorm 4; therefore extremal. There are exactly two such lattices possessing furthermore anautomorphism of order 31: This is proved in [14]. One of them is BW32. It can be prooved[15] that BW32 and BSBM32 are non equivalent. The lattice BW32 can be constructed fromconstruction A4 applied to the Reed-Muller Z4-code of order 2; QRM(2; 5) in the notationof [8]. This code has the same cwe as QR32 (electronic computation due to Quian) and canbe considered as an extended lifted duadic code [16]. Both constructions of BSBM32 andBW32 appear in [7, p. 226]. 20

A. A Table of Codes and LatticesThe �rst two columns of Table 3 give the block length of the code de�ned over Z4, and theblock length of the binary image under the Gray map. The minimum distance (Lee distancein the Z4 world, Hamming distance in the Z2 world) appears in the third column. The Z4description of the code appears in column 4, and the parameters of its binary image underthe Gray map appear in column 5 (here square brackets are used to indicate a binary linearcode). The last column describes the lattice obtained from the Z4-code by Construction A4:Block Block Minimum GrayLength Length Distance Code C Image Lattice(Z4) (Z2) D �(C) �(C)=28 16 6 O8 (16; 28; 6) E88 16 4 Q8 [16; 8; 4] E88 16 4 C3;0 (16; 28; 4) E824 48 12 QR24 (48; 224; 12) �2432 64 14 QR32 (64; 232; 14) BSBM3232 64 8 C5;1 (64; 232; 8) BW3232 64 14 QRM(2; 5) (64; 232; 14) BW3248 96 18 QR48 (96; 248; 18) ?Table 1: Z4-Codes, their Binary Images under the Gray Map and Associated Lattices.21

AcknowledgementWe wish to thank Iwan Duursma for helpful discussions and A.R. Calderbank and N.J.ASloane for sending us a copy of [11], as well as Vera Pless and Z. Qian for helpful discussionsand sending us a copy of [12]. More results on type II Z4-codes will appear in [17]. Theauthors would also like to thank the two anonymous referees for their detailed comments onthe manuscript.A Complete weight enumerators of the basic polynomialsThe cwe of O8�8 = W 8 +X8 + Y 8 + Z8 + 14X4Z4 + 56WXY 3Z3 + 56WX3Y 3Z+56W 3XY Z3 + 56W 3X3Y Z + 14W 4Y 4The cwe of Q8�8 = Z8 + Y 8 + 4X2Z6 + 22X4Z4 + 4X6Z2 +X8 + 48WXY 3Z3 + 48WX3Y 3Z+4W 2Y 6 + 48W 3XY Z3 + 48W 3X3Y Z + 22W 4Y 4 + 4W 6Y 2 +W 8The cwe of K16 = RM(0; 4) + 2RM(3; 4) 16 = W 16 + 120Y 2W 14 + 1820Y 4W 12 + 8008Y 6W 10 + 12870Y 8W 8 + 8008Y 10W 6+1820Y 12W 4 + 120Y 14W 2 +X16 + 120Z2X14 + 1820Z4X12 + 8008Z6X10+12870Z8X8 + 8008Z10X6 + 1820Z12X4 + 120Z14X2 + Y 16 + Z1622

The cwe of RM(1; 4) + 2RM(2; 4)�16 = Z16 + 30Y 8Z8 + Y 16 + 140X4Z12 + 420X4Y 8Z4 + 448X6Z10 + 870X8Z8 + 30X8Y 8+448X10Z6 + 140X12Z4 +X16 + 3360W 2X2Y 6Z6 + 6720W 2X4Y 6Z4 + 3360W 2X6Y 6Z2+420W 4Y 4Z8 + 140W 4Y 12 + 6720W 4X2Y 4Z6 + 19320W 4X4Y 4Z4 + 6720W 4X6Y 4Z2+420W 4X8Y 4 + 448W 6Y 10 + 3360W 6X2Y 2Z6 + 6720W 6X4Y 2Z4 + 3360W 6X6Y 2Z2+30W 8Z8 + 870W 8Y 8 + 420W 8X4Z4 + 30W 8X8 + 448W 10Y 6 + 140W 12Y 4 +W 16The cwe of the lifted Golay code QR24�24 = Z24 + Y 24 + 1518X4Y 8Z12 + 6072X6Y 8Z10 + 759X8Z16 + 9108X8Y 8Z8+6072X10Y 8Z6+2576X12Z12+1518X12Y 8Z4+759X16Z8+X24+552WXY 11Z11+ 6072WX3Y 11Z9 + 24288WX5Y 11Z7 + 24288WX7Y 11Z5 + 6072WX9Y 11Z3+ 552WX11Y 11Z + 3036W 2X2Y 6Z14 + 3036W 2X2Y 14Z6 + 36432W 2X4Y 6Z12+6072W 2X4Y 14Z4+166980W 2X6Y 6Z10+3036W 2X6Y 14Z2+267168W 2X8Y 6Z8+166980W 2X10Y 6Z6+36432W 2X12Y 6Z4+3036W 2X14Y 6Z2+6072W 3XY 9Z11+123464W 3X3Y 9Z9+437184W 3X5Y 9Z7+437184W 3X7Y 9Z5+123464W 3X9Y 9Z3+ 6072W 3X11Y 9Z + 1518W 4Y 12Z8 + 6072W 4X2Y 4Z14 + 36432W 4X2Y 12Z6+94116W 4X4Y 4Z12+94116W 4X4Y 12Z4+418968W 4X6Y 4Z10+36432W 4X6Y 12Z2+661848W 4X8Y 4Z8+1518W 4X8Y 12+418968W 4X10Y 4Z6+94116W 4X12Y 4Z4+6072W 4X14Y 4Z2+24288W 5XY 7Z11+437184W 5X3Y 7Z9+1578720W 5X5Y 7Z7+1578720W 5X7Y 7Z5+437184W 5X9Y 7Z3+24288W 5X11Y 7Z +6072W 6Y 10Z8+3036W 6X2Y 2Z14+166980W 6X2Y 10Z6+36432W 6X4Y 2Z12+418968W 6X4Y 10Z4+166980W 6X6Y 2Z10+166980W 6X6Y 10Z2+267168W 6X8Y 2Z8+6072W 6X8Y 10+166980W 6X10Y 2Z6+36432W 6X12Y 2Z4+3036W 6X14Y 2Z2+24288W 7XY 5Z1123

+437184W 7X3Y 5Z9+1578720W 7X5Y 5Z7+1578720W 7X7Y 5Z5+437184W 7X9Y 5Z3+ 24288W 7X11Y 5Z + 9108W 8Y 8Z8 + 759W 8Y 16 + 267168W 8X2Y 8Z6+ 1518W 8X4Z12 + 661848W 8X4Y 8Z4 + 6072W 8X6Z10 + 267168W 8X6Y 8Z2+ 9108W 8X8Z8 + 9108W 8X8Y 8 + 6072W 8X10Z6 + 1518W 8X12Z4+6072W 9X1Y 3Z11+123464W 9X3Y 3Z9+437184W 9X5Y 3Z7+437184W 9X7Y 3Z5+ 123464W 9X9Y 3Z3 + 6072W 9X11Y 3Z + 6072W 10Y 6Z8+166980W 10X2Y 6Z6+418968W 10X4Y 6Z4+166980W 10X6Y 6Z2+6072W 10X8Y 6+ 552W 11XY 1Z11 + 6072W 11X3Y Z9 + 24288W 11X5Y Z7 + 24288W 11X7Y Z5+ 6072W 11X9Y Z3 + 552W 11X11Y Z + 1518W 12Y 4Z8 + 2576W 12Y 12+ 36432W 12X2Y 4Z6 +94116W 12X4Y 4Z4 + 36432W 12X6Y 4Z2 +1518W 12X8Y 4+ 3036W 14X2Y 2Z6 + 6072W 14X4Y 2Z4 + 3036W 14X6Y 2Z2 + 759W 16Y 8 +W 24References[1] M. Klemm, \Selbstduale Codes uber dem Ring der ganzen Zahlen modulo 4," Arch.Math., vol. 53, pp. 201{207, 1989.[2] J. Conway and N. Sloane, \Self-dual Codes over the integers modulo 4," JCT A 62,30-45, 1993.[3] O. Zariski and P. Samuel, Commutative Algebra, vol. 2. van Nostrand, Princeton, 1960.[4] F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes. North-Holland,1977. 24

[5] R. P. Stanley, \Invariants of Finite Groups and their Applications to Combinatorics,"Bull. Amer. Math. Soc, vol. 1, pp. 475{511, 1979.[6] A. Bonnecaze, P. Sol�e, and A. Calderbank, \Quaternary Quadratic Residue Codes andUnimodular Lattices," IEEE Transactions on information theory, vol. 41, pp. 366{377,1995.[7] A.R. Calderbank, G. McGuire, P.V. Kumar, T. Helleseth, \Cyclic Codes Over Z4, Lo-cator Polynomials, and Newton's Identities," IEEE Transactions on information theory,vol 42, pp. 217{226, 1996[8] R. Hammons, P. Kumar, A. Calderbank, N. Sloane, and P. Sol�e, \Kerdock, Preparata,Goethals and Other Codes are linear over Z4," IEEE Transactions on informationtheory, vol. 40, pp. 301{319, 1994.[9] J. Conway and N. Sloane, Sphere Packings, Lattices and Groups. Springer-Verlag, 1988.[10] N. J. A. Sloane, \Error Correcting Codes and Invariant Theory: New Application of aNineteenth-Century Technique," Amer. Math. Monthly, vol. 84, pp. 82{107, 1977.[11] A. Calderbank and N. Sloane, \Double Circulant Codes over Z4 and Even UnimodularLattices," Submitted to J. Alg. Combinatorics, 1995.[12] V. Pless and Z. Qian, \Cyclic codes and quadratic residue codes over Z4," Preprint,1995.[13] C. Mallows, A. Odlyzko, and N. Sloane, \Upper bounds for modular forms, lattices andcodes," J. Algebra, vol. 36, pp. 68{76, 1975.25

[14] H. G. Quebemann, \Zur Klassi�kation Unimodularer Gitter mit Isometrie vonPrimzahlordnung," J. Reine Angew. Math., vol. 326, pp. 158{170, 1981.[15] R. Chapman and P. Sol�e, \Universal Codes and Unimodular Lattices," J. de th. desnombres de Bordeaux, submitted, 1996.[16] V. Pless Private communication, 1994.[17] V. Pless, P. Sol�e, and Z. Qian, \Cyclic self-dual Z4-codes," Finite Fields and Applica-tions, submitted, 1996.

26