Type II Codes over Z4 - CiteSeerX

Type II Codes over Z4 Alexis Bonnecaze, Patrick Sole , Christine Bachoc y, Bernard Mourrain z ABSTRACT Type II Z4-codes are introduced as self-dual codes over the integers modulo 4 containing the all-one vector and with euclidean weights multiple of 8: Their weight enumerators are characterized by means of invariant theory. A notion of extremality for the euclidean weight is introduced. 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 are obtained by Hensel lifting of the classical binary quadratic residue codes. Their binary images have good parameters. With every type II Z4-code is associated via construction A modulo 4 an even unimodular lattice (type II lattice). In dimension 32, we construct two unimodular lattices of norm 4 with an automorphism of order 31. One of them is the Barnes-Wall lattice BW32. 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 ([email protected],

[email protected]) y

Laboratoire A2X, 351 Cours de la Liberation, F-33405 Talence, France ([email protected])

z

INRIA, Route des Lucioles, BP 93, 06902 Sophia Antipolis, Cedex, France ([email protected])

1

I. Introduction

The conditions satis ed by the weight enumerator of self-dual codes, de ned over the ring of integers modulo four, have been studied by Klemm [1], then by Conway and Sloane [2]. The MacWilliams transform determines a group of substitutions, each of which xes the weight enumerator 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 basis for this ring is possible since R is Cohen-Macaulay [3]. Each invariant is written uniquely into 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 are multiples of 8. These codes are called type II codes by analogy with the binary case. An upper bound on their minimum euclidean weight is given, thereby leading to a natural notion of extremality akin to similar concepts for type II binary codes and type II lattices. The most interesting 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]. Other classes of interest comprise a multilevel construction from binary Reed-Muller and lifted double circulant codes. The paper is organized as follows. Section II contains background information on Z4-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 and gives the conditions satis ed by their complete weight enumerator. Section IV gives three examples of families of type II codes with their general construction. Section V shows the relation between type II codes and even unimodular lattices. 2

II.

Z4-Codes

and Invariant Theory

By a Z4-code C of length N we shall mean a linear block code over Z4, that is an additive subgroup of ZN4 . We de ne an inner product on ZN4 by (a; b) = a1b1 +


+ aN bN (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 equivalent if 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 all monomial transformations (coordinate permutations and sign changes) that preserve the set of 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 enumerator (or c.w.e.) of C is

cweC (W; X; Y; Z ) =

X a2C

W n (a)X n (a)Y n (a)Z n (a) ; 0

1

2

3

where ni(a) is the number of components of a that are congruent to i modulo 4. Since a monomial transformation may change the sign of a component, the appropriate weight enumerator 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 ? in terms of the weight enumerator of C :

cweC? (W; X; Y; Z ) = jC1 j cweC (W +X +Y +Z; W +iX ?Y ?iZ; W ?X +Y ?Z; W ?iX ?Y +iZ ); 3

which implies

sweC? (W; X; Y ) = jC1 j sweC (W + 2X + Y; W ? Y; W ? 2X + Y ) : The Gray map  provides a one to one correspondence between a Z4-code and a binary code. Hammons et al. [8] explored the Gray map which is a distance preserving map or isometry from (ZN4 , Lee distance) to (Z22N, Hamming distance). Recall that the Lee weights of the elements 0; 1; 2; 3 of Z4 are respectively 0; 1; 2; 1, and that Lee weight of a vector a 2 ZN4 is just the rational sum of the Lee weights of its components. This weight function de nes the Lee metric on ZN4 . We de ne maps ; from Z4 to Z2 by

c (c) (c) 0

0

0

1

0

1

2

1

1

3

1

0

and extend them in the obvious way to maps from ZN4 to Z22N. The Gray map  :

ZN 4

! Z22N

is 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 code

4

C? = (C ?). Thus

 C ?! (C )

?? dual ??y

;

 C ? ?! 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 2 of its preimage is self-orthogonal. Using this simple observation it follows immediately, using a 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 most dL  2(b N4 c + 1): This bound is not tight in general (except for n = 8 in the case of the octacode). It is a dicult 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 of self-dual codes over Z4. Conway and Sloane deduced analogous theorems for the symmetrized and Hamming weight enumerators (theorems 6-9 of [2]). These results come from a "new applications of a nineteenth-century technique" [10]: invariant theory. The weight enumerators are invariants of a group of substitutions. The number of linearly independent homogeneous invariants of degree i is given by the coecient of i in the Molien series. The following theorem characterizes the symmetrized weight enumerator of a self-dual code of length N 5

over Z4 containing a vector 1.

Theorem 2 The symmetrized weight enumerator of a self-dual code of length N over Z4 containing a vector 1 belongs to the ring

S  8S  82S ; where S is the ring of polynomials in

4 = W 4 + 6W 2Y 2 + Y 4 + 8X 4 ; 8 = (W 2Y 2 ? X 4 )((W 2 + Y 2)2 ? 4X 4 ) ; 12 = X 4(W 2 ? Y 2)4 ; and

8

is the polynomial 8

= X 4(W ? Y )4 :

This ring has Molien series

+ 8 + 16 S () = (1 ? 41)(1 ? 8)(1 ? 12) = 1 + 4 + 38 + 412 + 716 + 920 + 1324 + 1628 + 2132 + 2536 + 3140 + 3644 + 4348 +


:

Example 3 The so-called lifted Golay QR24 is a remarkable self-dual code over Z4. It is an extended quadratic residue Z4-code. The expression of the symmetric weight enumerator of the \lifted Golay" in terms of the basic polynomials given in theorem 2 is

sweQR = 64 ? 36(448) + 39024 28 ? 105638 ? 8434 12 + 5884 812 ? 76212 24

+644 8412 + 82(?32224 + 51528 ): 6

III. Type II

Z4-Codes

A Z4-code of type II is a self-dual Z4-code containing a vector 1 and which has the property that all euclidean weights are multiples of 8.

Theorem 4 The complete weight enumerator of a type II code of length N over Z4 belongs to the ring

S

16S



32S



16 32 S;

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 ; is the cwe of K16 = RM (0; 4) + 2RM (3; 4), the self-dual code introduced by Klemm [1],

16

and

32

is a homogenious polynomial of degre 32 which doesn't belong to the algebra S 

This ring has Molien series 16 32 + 48 S () = (1 ?1+8)2(1 + ? 16)(1 ? 24)

= 1 + 28 + 516 + 924 + 1632 + 2540 + 3948 +


:

7

16S .

The complete weight enumerator of the code is left invariant by the matrix group G, of size 768, generated by M1; M2; M3 with

3 3 3 2 2 2 66 0 0 0 1 77 66 1 0 0 0 77 66 1 1 1 1 77 7 77 77 66 66 66 66 1 0 0 0 777 66 0 w 0 0 77 66 1 i ?1 ?i 77 77 ; 77 ; M = 66 77 ; M = 66 M1 = 12 666 3 6 2 6 7 7 66 0 1 0 0 777 66 0 0 ?1 0 77 66 1 ?1 1 ?1 77 77 77 77 66 66 66 5 5 5 4 4 4

1 ?i ?1 i

0 0 0 w

0 0 1 0

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 algebra S  16S: In degre 32, the dimension of this algebra is 15 and the dimension of the invariant algebra is 16. The algebra generated by the cwe of the type II codes is included into the invariant 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 belongs 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

Z4

This ring has Molien series

16 S () = (1 ? 18+ )2(1 ? 24) = 1 + 28 + 416 + 724 + 1032 +


: The symmetric weight enumerator of the code is left invariant by the matrix group G, of size 768, generated by M1; M2; M3 with 2 66 1 2 66 M1 = 21 66 1 0 66 4

3 3 2 3 2 66 0 0 1 77 66 1 0 0 77 1 77 77 77 66 77 66 7 6 7 6 ; M = ; M = ?1 77 2 66 0 w 0 77 3 66 0 1 0 777 ; 75 75 64 75 64

0 0 ?1

1 ?2 1

1 0 0

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 corollary 5], with dierent bases and of course the same Molien series. IV. Examples of Type II Codes

We give three examples of families of type II codes with their general construction. A. Extended Quadratic Residue Z4-Codes Recently, Bonnecaze, Sole and Calderbank investigated extended quadratic residue Z4-codes [6]. They obtained the parameters of the lifted Golay. These codes represent a remarkable class of type II. The image by the Gray map of the octacode is the Nordstrom-Robinson and 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 respectively 32 and 48. Recall (see [6]) that a quadratic residue Z4-code is de ned to be the cyclic code generated by h(x) 2

[x], the Hensel lift of the generator polynomial h2(x) 2

Z4

[x] of the

Z2

binary quadratic residue code. The code QR32 is a [32; 16; 14] code with minimum euclidean distance 16. It is generated by the polynomial

h(x) = x15 + 3x12 + 2x11 + 2x9 + 2x8 + 3x7 + x6 + 2x3 + 3x2 + 3x + 3 and extended by a parity check symbol. The complete weight enumerator of QR32 in terms of the basic polynomials is 14053  4 + 9969  4 ? 236165  3  + 287715  2  2 49 8 16 8 168 8 8 112 8 8  8293  5363 37451 2 3 ? 5880 8 16 + 8 ? 4 8 + 420 8 16 + 3 24 2 5 + 4=49 16 2 ? 5513  16 8 ? 8 24 840 3  827  1127 1427 2 2 + 16 ? 8 + 240 120 8 8 ? 240 8 : And the symmetrized weight enumerator of QR32 in terms of the basic polynomials is 631 4 36 8

211 2 2 199 4823 4 12367 3 3 139 ? 1211 60 8 ? 360 88 ? 60 8 8 + 45 824 + 120 8 8 ? 45 8 v24

71 2 ? 69   + 97 2 ): + 16(? 120 8 20 8 8 24 8

10

Its image by the gray map (QR32) is a (64; 232; 14) non linear formally self-dual code with a weight enumerator symmetric with respect to 32 Number of words Weights 1

0; 64

39680

14; 50

127596

16; 48

1944320

18; 46

9443840

20; 44

32347136

22; 42

136671808

24; 40

232207360

26; 38

603596288

28; 36

643625472

30; 34

974960294

32; 32

In length 48, the binary extended quadratic residue code has very good parameters. It is a [48; 24; 12] F2 -code with the following weight enumerator

W 48 + 17296W 36 X 12 + 535095W 32 X 16 + 3995376W 28 X 20 +7681680W 24 X 24 + 3995376W 20 X 28 + 535095W 16 X 32 +17296W 12 X 36 + X 48:

11

The Hensel lift of this code is a [48; 24; 18] Z4-code generated by

h(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 the basic polynomials is ?3190583 6 ? 136399499 5 8 + 16377833334 2 + 120960

8

6048000

8

6048000

8 8

3 7747619 2 11991809 4 41678737 38(? 130629443 432000 8 ? 252000 24) + 8 (? 126000 8 + 756000 248)+

6333119 6 96071 3 653 2 5 417473 2 8( 1829029 6400 8 ? 21600 8 24) ? 57600 8 ? 21600 8 24 + 2700 24+

2182631 2 2 123947 3 20953 1553351 4 1909 4 2315111 3 ? 1617613 224000 8 + 63000 8 8 ? 67200 88 + 8 (? 7200 8 + 12600 24) + 80640 8 ? 2520 248 ):

16(

12

Its image by the gray map (QR48) is a (96; 248; 18) non linear formally self-dual code with a weight enumerator symmetric with respect to 48 Number of words Weights 1

0; 96

138368

18; 78

1660416

20; 76

17434368

22; 74

169418832

24; 72

1698605568

26; 70

8017318656

28; 68

52482567296

30; 66

207498156471

32; 64

742015419264

34; 62

2370501481216

36; 60

5453161473024

38; 58

12741529744944

40; 56

20713663863552

42; 54

34296818992128

44; 52

40358425379712

46; 50

47582973403024

48; 48

Remark 3 In [12], V. Pless and Z. Qian give the explicit complete weight enumerator (without using invariant theory) of QR32 and QR48.

13

B. Cm;r codes Let Cm;r denote RM (r; m) + 2RM (m ? r ? 1; m):

Proposition 7 For 0  r  m3?1 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 2b mr? c  8 (the case r = 0 1

is trivial). By orthogonality of c and d, jc \ dj is even. Since RM (m ? r ? 1; m)  RM (m; m); jdj is also even. Hence, the euclidean weight of c + 2d, that is n1 + 4n2 is a multiple of 8.

2

Remark 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 distance 16. The complete weight enumerator of C5;1 in terms of the basic polynomials is 268490 4 147 8

3 209793 2 2 23289 3 4 2873 2 1705 ? 59860 7 8 8 + 14 8 8 ? 2 8 8 + 33848 ? 147 168 + 42 1688 ?

297  2 + 4 2 14 16 8 49 16

? 583 1628 + 1393 1688 ? 27 1682 + 403 248 ? 12248

Note that Cm;0 is the Klemm code with parameters [32,16,4]. The complete weight enumerator of C5;0 in terms of the basic polynomials is 14

360981 4 350909 4 4654007 3 4559608 2 2 129409 3 2 248 ? 5080627 882 8 + 180 8 8? 105 8 8 + 4 8 8 ? 40 8 ? 8820 168 + 3 1688 ?

?

?

?

13138 1282 13456  2 47 2 + 143719 2 2579 2 31 315 16 8 5880 16 630 16 8 5 1688 + 45 168 210 1616+ 45 248

?

1346   45 24 8

C. Double Ciculant Z4-Codes In [11] Calderbank and Sloane study the double circulant codes DN . This code is similar to the 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 has generator matrix

if p  11 (mod 16), or

2 66 66 66 M = 666 66 66 4

3 1 : : : 1 77 77 77 77 7 I + 3R + 2N 777 75

1 1 ::: 1 1 1 .. .

0 .. .

I

1

2 66 66 66 M = 666 66 66 4

0 1 1 ::: 1 1 1 .. .

0 .. .

I

1

0

3 1 : : : 1 77 77 77 77 7 b(I + R) 777 75

if p  3 (mod 16), where R = (Rij ); Rij = 1 if j ? i is a nonzero square mod p, or 0 otherwise; N = (Nij ); Nij = 1 if j ? i is not a square mod p, or 0 otherwise; and b can be either +1 or ?1.

15

Theorem 10 (Calderbank and Sloane) DN is a self-dual code over Z4 in which all norms are divisible by 8.

Example 11 For p = 19, the code D40 is a type II code with minimal norm 16. V. Even Unimodular Lattices

An N -dimensional lattice  in RN is the set of integer linear combinations of N linearly independent vectors v1; : : :; vN . The dual of an N -dimensional lattice  is denoted , and is given by  = fx 2 RN j (x; a) 2 Z, for all a 2 g. A lattice  is integral if the inner product of any two lattice points is integral, or equivalently, if   . If  is an integral lattice, 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 24 and the Barnes-Wall BW32. The theta series  (q) of the integral lattice  gives the number of points which are at equal distance from the origin. It is the formal power series 1 X (x;x) X (q) = q = Nmqm ; x2

m=0

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 equivalently 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 C g 16

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 construct lattices 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 )=2

Z

Zq

can be found by replacing W; X; Y by f0 = Px24 qx =4; f1 = Px24 2

in the symmetrized Lee weight enumerator of the Z4-code C .

+1

x2 =4; f2

Zq

= Px24

+2

x2 =4

Theorem 12 Let C be a self-dual code over Z4 such that all Euclidean weights in C are divisible by 8. Then (C )=2 is an even unimodular lattice.

Proof. See [6].

2

Corollary 13 The minimum euclidean distance dE of a type II Z4-code of length N is at most

N c + 1): dE  8(b 24

Proof. Let C be a type II Z4-code of length N . The lattice (C )=2 being even unimodular by [6, theorem 4.1], its theta series C , of weight N=2, belongs to the polynomial algebra C [E4 ;
] ([9], chapter 7 ), where E4 is the theta series of the root lattice E8, and
= q2 Qr1(1?q2r)24. Furthermore, (C )=2 contains the norm 4 lattice 0 = (4Z N )=2 with theta series 0. If

j = N=8, we have : 17

C =

[X j=3]

s=0

asE4j?3s
s =

X r0

Nr q r =  0 +

X r1

r qr:

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 + Pr2([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 the

series are in q2 = t, and Burman's formula shows that  2([ j=3]+1) = ?b2([j=3]+1)

where s?1 b2s = s1! dtd s?1 ( dtd (E4?j 0)(
=E43)s )ft=0g

which gives, using the fact that 0 = 1j , where 1 is the theta series of the lattice (4Z)8=2, s?1 b2s = ?s!j dtd s?1 (E43s?j?1 1j?1 (1E40 ? 10 E4)(t=
)s)ft=0g

It is know sucient to show that the coecients of 1j?1(1E40 ? 10 E4) are positive up to the 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. Then

18

E4 = WC (f0; f1; f2) and 1 = f08. Deriving E4=1 = WC (1; f1=f0; f2=f0 ), we nd : 8

8

@WC8 8j ?1 0 0 @Y (f0 ; f1; f2 )f0 (f0 f1 ? f0f1 ) + @W@ZC8 (f0; f1; f2)f08j?1(f0f20 ? f00 f2):

1j?1(1E40 ? 10 E4) =

We are reduced to show that f08j?1 (f0f10 ? f00 f1) and f08j?1(f0f20 ? f00 f2) have positive coecients up to [j=3].

t(f0f10 ? f00 f1) =

X (1 + 4y )2 ? (4x)2

Z

and

tf0s(f0f10 ? f00 f1) =

X

8

x;y2

(1 + 4y)2 ? (4x)2 t 8

Z

x;y;x1 ;::;xs2

t

y

x

(1+4 )2 +(4 )2 8

y

x

x

:::+(4xs )2

(1+4 )2 +(4 )2 +(4 1 )2 + 8

:

Let k be a xed integer. If (1 + 4y)2 + (4x)2 + (4x1)2 + ::: + (4xs )2 = k, one can, by symmetry, put together the terms corresponding to the dierent choices of x. One gets, as a coecient of tk=8, (s + 1)(1 + 4y)2 ? (4x)2 ? (4x1)2 ? ::: ? (4xs)2 = (s + 2)(1 + 4y)2 ? k : 8 8 which is positive as soon as s + 2 is greater than k, because 1 + 4y is never zero.

2

A code meeting that bound with equality is called extremal. The codes O8; Q8; K16 and

D24 are extremal and generate extremal type II lattices via construction A4: This is not necessarily so. For instance QR48 is extremal even though the lattice it generates contains vectors 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 BarnesWall lattice. Then

I:

2E8 = O8 + 4Z8

II: 2E8 = Q8 + 4Z8 III: 224 = QR24 + 4Z24 IV: 2BW32 = RM (1; 5) + 2 RM (3; 5) + 4Z32

Proof. see [6].

2

Since QR32 is a code of type II, the lattice obtained by the same construction with the code

QR32 is an even unimodular lattice. Furthermore, the minimum euclidean weight of QR32 is 16. Then, the produced lattice is of minimum norm 4 and its theta series is 1 + 146880q16 + 158757681973184q64 + 64757760q24 + 137695887360q40 +4844836800q32 + 2121555283200q48 + 21421110804480q56 + 928985325895680q72 +18845727679406080q88 : : : Let us call this lattice 2BSBM32: We see that BSBM32 is an even unimodular lattice of norm 4; therefore extremal. There are exactly two such lattices possessing furthermore an automorphism 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 from construction A4 applied to the Reed-Muller Z4-code of order 2; QRM (2; 5) in the notation of [8]. This code has the same cwe as QR32 (electronic computation due to Quian) and can be considered as an extended lifted duadic code [16]. Both constructions of BSBM32 and

BW32 appear in [7, p. 226]. 20

A. A Table of Codes and Lattices The rst two columns of Table 3 give the block length of the code de ned over Z4, and the block length of the binary image under the Gray map. The minimum distance (Lee distance in the Z4 world, Hamming distance in the Z2 world) appears in the third column. The Z4 description of the code appears in column 4, and the parameters of its binary image under the Gray map appear in column 5 (here square brackets are used to indicate a binary linear code). The last column describes the lattice obtained from the Z4-code by Construction A4:

Block Block Minimum Length Length Distance

Gray Code C

Image

Lattice

(C )

(C )=2

(Z4)

(Z2)

D

8

16

6

O8

(16; 28 ; 6)

E8

8

16

4

Q8

[16; 8; 4]

E8

8

16

4

C3;0

(16; 28 ; 4)

E8

24

48

12

QR24

(48; 224 ; 12)

24

32

64

14

QR32

(64; 232 ; 14) BSBM32

32

64

8

C5;1

(64; 232; 8)

BW32

32

64

14

QRM (2; 5) (64; 232 ; 14)

BW32

48

96

18

QR48

(96; 248 ; 18)

?

Table 1: Z4-Codes, their Binary Images under the Gray Map and Associated Lattices.

21

Acknowledgement We wish to thank Iwan Duursma for helpful discussions and A.R. Calderbank and N.J.A Sloane for sending us a copy of [11], as well as Vera Pless and Z. Qian for helpful discussions and sending us a copy of [12]. More results on type II Z4-codes will appear in [17]. The authors would also like to thank the two anonymous referees for their detailed comments on the manuscript. A Complete weight enumerators of the basic polynomials

The cwe of O8

8 = W 8 + X 8 + Y 8 + Z 8 + 14X 4Z 4 + 56WXY 3Z 3 + 56WX 3Y 3Z +56W 3XY Z 3 + 56W 3 X 3Y Z + 14W 4Y 4 The cwe of Q8

8 = Z 8 + Y 8 + 4X 2Z 6 + 22X 4 Z 4 + 4X 6Z 2 + X 8 + 48WXY 3Z 3 + 48WX 3Y 3Z +4W 2Y 6 + 48W 3XY Z 3 + 48W 3X 3 Y Z + 22W 4Y 4 + 4W 6Y 2 + W 8 The cwe of K 16 = 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 + X 16 + 120Z 2 X 14 + 1820Z 4 X 12 + 8008Z 6 X 10 +12870Z 8 X 8 + 8008Z 10X 6 + 1820Z 12 X 4 + 120Z 14X 2 + Y 16 + Z 16

22

The cwe of RM (1; 4) + 2RM (2; 4)

16 = Z 16 + 30Y 8Z 8 + Y 16 + 140X 4 Z 12 + 420X 4 Y 8Z 4 + 448X 6 Z 10 + 870X 8 Z 8 + 30X 8 Y 8 +448X 10 Z 6 + 140X 12 Z 4 + X 16 + 3360W 2 X 2Y 6Z 6 + 6720W 2 X 4Y 6Z 4 + 3360W 2 X 6Y 6Z 2 +420W 4 Y 4Z 8 + 140W 4 Y 12 + 6720W 4 X 2Y 4Z 6 + 19320W 4 X 4Y 4Z 4 + 6720W 4 X 6Y 4Z 2 +420W 4 X 8Y 4 + 448W 6 Y 10 + 3360W 6 X 2 Y 2Z 6 + 6720W 6 X 4Y 2Z 4 + 3360W 6 X 6Y 2Z 2 +30W 8Z 8 + 870W 8 Y 8 + 420W 8 X 4Z 4 + 30W 8X 8 + 448W 10Y 6 + 140W 12Y 4 + W 16 The cwe of the lifted Golay code QR24

24 = Z 24 + Y 24 + 1518X 4 Y 8Z 12 + 6072X 6 Y 8Z 10 + 759X 8 Z 16 + 9108X 8 Y 8Z 8 +6072X 10 Y 8Z 6 +2576X 12 Z 12 +1518X 12 Y 8Z 4 +759X 16 Z 8 +X 24 +552WXY 11Z 11 + 6072WX 3 Y 11Z 9 + 24288WX 5 Y 11Z 7 + 24288WX 7 Y 11Z 5 + 6072WX 9 Y 11Z 3 + 552WX 11 Y 11Z + 3036W 2 X 2Y 6Z 14 + 3036W 2 X 2Y 14Z 6 + 36432W 2 X 4Y 6Z 12 +6072W 2 X 4Y 14Z 4+166980W 2 X 6Y 6Z 10+3036W 2 X 6Y 14Z 2 +267168W 2 X 8Y 6Z 8 +166980W 2 X 10Y 6Z 6 +36432W 2 X 12Y 6Z 4 +3036W 2 X 14Y 6Z 2 +6072W 3 XY 9Z 11 +123464W 3 X 3Y 9Z 9+437184W 3 X 5Y 9Z 7+437184W 3 X 7Y 9Z 5+123464W 3 X 9Y 9Z 3 + 6072W 3 X 11Y 9Z + 1518W 4 Y 12Z 8 + 6072W 4 X 2Y 4Z 14 + 36432W 4 X 2Y 12Z 6 +94116W 4 X 4 Y 4Z 12+94116W 4 X 4Y 12Z 4+418968W 4 X 6Y 4Z 10+36432W 4 X 6Y 12Z 2 +661848W 4 X 8Y 4Z 8 +1518W 4 X 8Y 12 +418968W 4 X 10Y 4Z 6 +94116W 4 X 12Y 4Z 4 +6072W 4 X 14Y 4Z 2+24288W 5 XY 7Z 11+437184W 5 X 3Y 7Z 9+1578720W 5 X 5Y 7Z 7 + 1578720W 5 X 7Y 7Z 5 + 437184W 5 X 9Y 7Z 3 + 24288W 5 X 11Y 7Z + 6072W 6 Y 10Z 8 +3036W 6 X 2Y 2Z 14+166980W 6 X 2Y 10Z 6+36432W 6 X 4Y 2Z 12+418968W 6 X 4Y 10Z 4 +166980W 6 X 6Y 2Z 10 +166980W 6 X 6Y 10Z 2 +267168W 6 X 8Y 2Z 8 +6072W 6 X 8Y 10 +166980W 6 X 10Y 2Z 6+36432W 6 X 12Y 2Z 4+3036W 6 X 14Y 2Z 2 +24288W 7 XY 5Z 11 23

+437184W 7 X 3Y 5Z 9+1578720W 7 X 5Y 5Z 7+1578720W 7 X 7 Y 5Z 5+437184W 7 X 9Y 5Z 3 + 24288W 7 X 11Y 5Z + 9108W 8 Y 8Z 8 + 759W 8 Y 16 + 267168W 8 X 2Y 8Z 6 + 1518W 8 X 4Z 12 + 661848W 8 X 4 Y 8Z 4 + 6072W 8 X 6Z 10 + 267168W 8 X 6Y 8Z 2 + 9108W 8 X 8Z 8 + 9108W 8 X 8Y 8 + 6072W 8 X 10Z 6 + 1518W 8 X 12Z 4 +6072W 9 X 1Y 3Z 11+123464W 9 X 3Y 3Z 9+437184W 9 X 5Y 3Z 7+437184W 9 X 7Y 3Z 5 + 123464W 9 X 9Y 3Z 3 + 6072W 9 X 11Y 3Z + 6072W 10 Y 6Z 8 +166980W 10 X 2 Y 6Z 6+418968W 10 X 4Y 6Z 4+166980W 10 X 6Y 6Z 2+6072W 10 X 8Y 6 + 552W 11XY 1Z 11 + 6072W 11 X 3Y Z 9 + 24288W 11 X 5Y Z 7 + 24288W 11 X 7Y Z 5 + 6072W 11 X 9Y Z 3 + 552W 11X 11Y Z + 1518W 12 Y 4Z 8 + 2576W 12 Y 12 + 36432W 12 X 2Y 4Z 6 + 94116W 12 X 4Y 4Z 4 + 36432W 12 X 6Y 4Z 2 + 1518W 12 X 8Y 4 + 3036W 14 X 2Y 2Z 6 + 6072W 14 X 4Y 2Z 4 + 3036W 14 X 6Y 2Z 2 + 759W 16 Y 8 + W 24 References

[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 62,3045, 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. Sole, and A. Calderbank, \Quaternary Quadratic Residue Codes and Unimodular 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, Locator 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. Sole, \Kerdock, Preparata, Goethals and Other Codes are linear over Z4," IEEE Transactions on information theory, 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 a Nineteenth-Century Technique," Amer. Math. Monthly, vol. 84, pp. 82{107, 1977. [11] A. Calderbank and N. Sloane, \Double Circulant Codes over Z4 and Even Unimodular Lattices," 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 and codes," J. Algebra, vol. 36, pp. 68{76, 1975. 25

[14] H. G. Quebemann, \Zur Klassi kation Unimodularer Gitter mit Isometrie von Primzahlordnung," J. Reine Angew. Math., vol. 326, pp. 158{170, 1981. [15] R. Chapman and P. Sole, \Universal Codes and Unimodular Lattices," J. de th. des nombres de Bordeaux, submitted, 1996.

[16] V. Pless Private communication, 1994. [17] V. Pless, P. Sole, and Z. Qian, \Cyclic self-dual Z4-codes," Finite Fields and Applications, submitted, 1996.

26