Shadow Codes over Z4 Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510 USA Email: [email protected] Masaaki Harada Department of Mathematical Sciences Yamagata University Yamagata 990–8560, Japan Email: [email protected] and Patrick Sol´e CNRS, I3S ESSI, BP 145 Route des Colles 06 903 Sophia Antipolis France Email: [email protected] June 22, 2011

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Running head: Shadow codes over Z4 Name: Steven T. Dougherty, Masaaki Harada and Patrick Sol´e Contact author: Masaaki Harada Address: Department of Mathematical Sciences Yamagata University Yamagata 990–8560, Japan Telephone: +81-236-28-4533 Fax: +81-236-28-4538 E-mail: [email protected]

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Abstract The notion of a shadow of a self-dual binary code is generalized to self-dual codes over Z4 . A Gleason formula for the symmetrized weight enumerator of the shadow of a Type I code is derived. Congruence properties of the weights follow; this yields constructions of self-dual codes of larger lengths. Weight enumerators and the highest minimum Lee, Hamming and Euclidean weights of Type I codes of length up to 24 are studied.

3

1

Introduction

Recently the notion of Type II codes over Z4 and more generally over Z2k has been introduced in [3, 1], respectively, where Zn is the ring of integers modulo n. Consequently, the notion of Type I codes over Z4 makes sense and the question of deriving an upper bound on their minimum Lee, Euclidean and Hamming weights arise. Due to the bivariate nature of the symmetrized weight enumerator the natural notion of optimal weight enumerator is still missing in the Z4 world for Lee and Euclidean weight. The Hamming weight enumerator however is univariate and we shall introduce in Section 5 a notion of a self-dual optimal Hamming weight enumerator. A Gray-map driven upper bound for the minimum Lee weight of self-dual codes over Z4 has been given in [3, Theorem 1]. Some of the techniques used to study binary self-dual codes can be carried over to selfdual codes over Z4 . In this article we generalize the notion of shadows, partly to obtain better lower bounds on the minimum weights of Type I codes over Z4 , and partly to build longer self-dual codes. Shadows for Type I binary codes were introduced by Conway and Sloane in [6] as a tool to derive upper bounds on the minimum distance. They were generalized by Brualdi and Pless in [4] to Type II codes and used to construct longer self-dual codes. The situation is more complicated than in the binary case since the glue group of the even weight subcode is not always isomorphic to the Klein group as in the binary case, but also sometimes to the cyclic group of order 4. The paper is organized in the following way. Section 2 gives the basic notions of self-dual codes over Z4 as well as Type II codes, simplifying the definition of [3]. Section 3 introduces the notion of a shadow of a Type I code by defining even weight subcodes. It contains the main weight congruence and weight enumerator properties of the shadow, as well as the constructions of longer self-dual codes. It also defines a generalized shadow for Type II codes. Section 4 associates to every shadow a relative invariant for the group of order 744 which fixes the symmetrized weight enumerators of a Type II code over Z4 . Section 5 collects the information known to us on the weight enumerators and the highest minimum weights of codes of length up to 24.

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Definitions and Notations

A code C of length n over Z4 is an additive subgroup of Zn4 . An element of C is called a codeword of C. A generator matrix of C is a matrix whose rows generate C. In this paper, we use three different weights for codewords over Z4 , namely the Euclidean weight, the Lee weight and the Hamming weight. The Euclidean weights of the elements 0, 1, 2 and 3 of Z4 are 0, 1, 4 and 1, respectively, and the Lee weights of the elements 0, 1, 2 and 3 of Z4 are 0, 1, 2 and 1, respectively. The Euclidean and Lee weight of a codeword is just the rational sum of the Euclidean and Lee weights of its components, respectively. The Hamming weight of a 4

codeword is the number of non-zero coordinates in the codeword. The minimum Euclidean, Lee and Hamming weights, dE , dL , and dH of C are the smallest weights among all non-zero codewords of C, respectively. Let x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) be two elements of Zn4 . We define the inner product of x and y in Zn4 by x · y = x1 y1 + · · · + xn yn (mod 4). The dual code C ⊥ of C is defined as C ⊥ = {x ∈ Zn4 | x · y = 0 for all y ∈ C}. The code C is self-dual if C = C ⊥ . Type II codes over Z4 were first defined in [3] as self-dual codes containing a ±1 vector and with the property that all Euclidean weights are divisible by eight. However, even if Type II codes do not contain a ±1 vector, the lattices constructed from these codes are even unimodular (see [2, Theorem 4.1]). In this paper, we say that self-dual codes with the property that all Euclidean weights are divisible by eight are Type II. Self-dual codes which are not Type II are called Type I. It will be shown below that the condition of containing an all-2 vector is not justified from the point of view of invariant theory of the symmetrized weight enumerator. It has been shown in [13] that, more generally, the condition of containing a ±1 vector is redundant.

3 3.1

Shadows Even Weight Subcode

The even weight subcode C0 of a Type I code C is the set of codewords of C of Euclidean weight divisible by 8. Lemma 3.1 The subcode C0 is Z4 −linear of index 2 in C. Proof. The first assertion follows by the self-duality of C using the relation (1)

wE (x + y) ≡ wE (x) + wE (y) + 2(x · y)

(mod 8),

where wE (x) is the Euclidean weight of a vector x. The second assertion follows by observing that every codeword y of C has Euclidean weight divisible by 4. By the preceding relation (1) we see that C2 := C − C0 is of the form x + C0 where x is any codeword of C of Euclidean weight congruent to 4 modulo 8 and that translation by x is a one to one map from C0 onto C2 . 2 By the preceding lemma we see that C is of index 2 in C0⊥ and we let C1 , C2 , C3 be S nontrivial cosets of C0 in C0⊥ labeled in such a way that C = C0 C2 , that is if C = hC0 , ti and C0⊥ = hC, si then C1 = (C0 + s) and C3 = (C0 + s + t). With these notations, define S the shadow of C as S := C1 C3 .

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Remark. Recently the notion of Type I and Type II codes over Z2k has been introduced in [1]. Define the even weight subcode C0 of a Type I code C over Z2k as the set of codewords of C of Euclidean weights divisible by 4k. Then we have that the subcode C0 is Z2k −linear of index 2 in C. Moreover, similarly to Type I codes over Z4 , the shadow codes can be defined for Type I codes over Z2k . Unlike the binary case, C0⊥ /C0 is not necessarily isomorphic to the Klein 4-group; it may be isomorphic to either the Klein 4-group or the cyclic group of order 4. We shall refer to the first situation as the Klein case and to the second as the cyclic case. For example, C with generator matrix 2I2 yields the Klein 4-group and the self-dual code of length 1 yields the cyclic case. We shall prove that the cyclic case occurs iff n is odd, and that the Klein case occurs iff n is even (see Propositions 3.5 and 3.9).

3.2

Weight Enumerators of Shadows

We define the symmetrized weight enumerator (swe) of a code C as: sweC (a, b, c) =

X

an0 (v) bn±1 (v) cn2 (v) ,

v∈C

where ni (c) is the number of components of c ∈ C that are congruent to i modulo 4. √ Denote by ζ8 := exp(π −1/4). We begin with the easy lemma: Lemma 3.2 The swe of C0 is obtained from the swe of C as 1 sweC0 (a, b, c) = (sweC (a, b, c) + sweC (a, ζ8 b, −c)). 2 Proof. Direct application from the definition of C0 .

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We are now in a position to state a simple but useful result. Theorem 3.3 The swe of S is related to the swe of C by the relation sweS (a, b, c) = sweC (b + ζ8 (a − c)/2, (a + c)/2, b − ζ8 (a − c)/2). Proof. We proceed as in [6, p. 1323] by computing first by the MacWilliams relation [14] sweC ⊥ (a, b, c) =

1 sweC (a + 2b + c, a − c, a − 2b + c) |C|

the swe of C ⊥ , then the swe of its even weight subcode, the swe of the dual of the latter, and finally the swe of the shadow by the difference of the swe of C0⊥ and the swe of C. 2

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Theorem 3.4 by:

1) The symmetrized weight enumerator of a self-dual code can be given X

αij (a + c)n−4i−8j (2b4 − ac(a2 + c2 ))i (b4 (a − c)4 )j

i,j

and the symmetrized weight enumerator of its shadow is given by: X

αij (−1)j 2n−4i−12j bn−4i−8j (a3 c + ac3 − 2b4 )i (a2 − c2 )4j .

i,j

2) The Euclidean weights of the shadow are congruent to n modulo 8. 3) The Lee weights of the shadow are congruent to n modulo 2. 4) For all i, j as in 1) the quantity αij 2n−4i−12j should be an integer. Proof. The assertion 1) follows from Theorem 3.3 and Theorem 6 in [7]. The second follows by letting a = 1, b = q, c = q 4 to get the Euclidean weights of the vectors in the shadow X αij (−1)j 2n−4i−12j q n−8j (q 8 − 1)i (1 − q 8 )4j . i,j

To prove 3) let a = 1, b = q, c = q 2 yielding the Lee weights of the vectors in the shadow X

αij (−1)j 2n−2i−12j q n−2i−8j (1 + q 4 − 2q 2 )i (1 − q 4 )4j .

i,j

2

The assertion 4) is immediate from 1).

3.3

The Klein Case

Here we consider the Klein case. Proposition 3.5 In the Klein case, the length of a Type I code is even. Proof. Suppose that the length is odd and that C1 + C1 = C0 . Let x be a nonzero element in C1 . By Theorem 3.4, the Euclidean weight of x is odd. By homogeneity of the Euclidean weight wE (2x) ≡ 4wE (x) ≡ 4 (mod 8). Since, by hypothesis, 2x ∈ C0 , the Euclidean weight of 2x must be divisible by 8. 2 We now investigate orthogonality relations among the Ci ’s. Lemma 3.6 Suppose that C is a self-dual code of length n ≡ 2 (mod 4). Then Table 1 holds where the symbol ⊥ in position (i, j) means that x · y ≡ 0 (mod 4) for any vector x ∈ Ci and any vector y ∈ Cj , and the symbol 6⊥ means that x · y ≡ 2 (mod 4) for any vector x ∈ Ci and any vector y ∈ Cj . 7

Proof. Since C0⊥ = C0 C1 C2 C3 , we get the first row and the first column in the table. The remaining cases are similar, and we give details only for i = 1 and j = 1. Let x and y be elements of C1 then x + y ∈ C0 . By Theorem 3.4, wE (x) ≡ wE (y) ≡ n (mod 8) and wE (x + y) ≡ 0 (mod 8). Using (1), we have x · y ≡ 2 (mod 4). 2 S

S

S

Table 1: Orthogonality Relations for n ≡ 2

C0 C1 C2 C3

C0 ⊥ ⊥ ⊥ ⊥

C1 ⊥ 6⊥ 6⊥ ⊥

C2 ⊥ 6⊥ ⊥ 6⊥

(mod 4) (The Klein Case) C3 ⊥ ⊥ 6⊥ 6⊥

Lemma 3.7 Suppose that C is a self-dual code of length n ≡ 0 (mod 4). Then Table 2 holds where the symbol ⊥ in position (i, j) means that x · y ≡ 0 (mod 4) for any vector x ∈ Ci and any vector y ∈ Cj , and the symbol 6⊥ means that x · y ≡ 2 (mod 4) for any vector x ∈ Ci and any vector y ∈ Cj . 2

Proof. Similar to that of Lemma 3.6.

Table 2: Orthogonality Relations for n ≡ 0

C0 C1 C2 C3

C0 ⊥ ⊥ ⊥ ⊥

C1 ⊥ ⊥ 6⊥ 6⊥

C2 ⊥ 6⊥ ⊥ 6⊥

(mod 4) (The Klein Case) C3 ⊥ 6⊥ 6⊥ ⊥

We now give a method for constructing self-dual codes using shadows. For i = 1, 3 define S Si := C0 Ci . Proposition 3.8 If n is a multiple of 4 then Si is a self-dual Z4 -linear code. Moreover if n ≡ 0 (mod 8) then Si is Type II. Proof. Lemma 3.7 gives the self-orthogonality. To prove Z4 -linearity use Si ⊆ Si⊥ and |Si | = 2n . The second assertion follows from Theorem 3.4. 2 Remark. The shadows of S1 and S3 are C2 (mod 8). 8

S

C3 and C2

S

C1 , respectively if n ≡ 4

3.4

The Cyclic Case

Now we deal with the cyclic case. Proposition 3.9 If C is in the cyclic case then its length n is odd. Proof. Suppose n even and C1 + C1 = C2 . Pick a nonzero x ∈ C1 . By hypothesis 2x ∈ C2 . By homogeneity of the norm, wE (2x) ≡ 4wE (x) (mod 8). But wE (x) is even by Theorem 3.4 and wE (2x) ≡ 4 (mod 8). 2

Lemma 3.10 Suppose that C is a self-dual code of length n ≡ a (mod 4) where a is 1 or 3. Then Table 3 holds where the value in position (i, j) is x · y for any vector x ∈ Ci and any vector y ∈ Cj . 2

Proof. Similar to that of Lemma 3.6. Remark. The entries in Table 3 are read modulo 4. Table 3: Orthogonality Relations (The Cyclic Case)

C0 C1 C2 C3

3.5

C0 0 0 0 0

C1 0 a 2 2+a

C2 0 2 0 2

C3 0 2+a 2 a

A Generalized Shadow

In Section 3, the shadow of a Type I code was defined. A similar definition gives no information for Type II codes since the shadow is the code itself. In this subsection, we give a certain generalization in order to define the shadow of a Type II code. This can be easily applied to Type II codes over Z2k . Let C be a self-dual code over Z4 of length n and v = (v1 , . . . , vn ) be any vector in Zn4 not in C. Define a map Ψv : C → Z4 by Ψv (u) =

n X

vi ui ,

i=1

where u ∈ C and u = (u1 , . . . , un ). This map is a group homomorphism from the additive group C to the additive group of Z4 . The map is not identically 0 since v ∈ / C. Set 9

C0 = ker(Ψv ), then C0 is a subcode of C. Since Im(Ψv ) is a subgroup of the additive group of Z4 , we have that |Im(Ψv )| = r where r is a divisor of 4. Hence [C : C0 ] = r. We shall say that C0 is a subcode of index r. We take a vector whose components are 0 or 2 as the vector v. Then r = 2 and C0 is a S S S S subcode of index 2 with C0⊥ = C0 C2 C1 C3 where C0 C2 = C. We define the shadow S of C with respect to v to be the coset S := C1 C3 for not only a Type II code C but also a Type I code C. We often say that this shadow is the generalized shadow with respect to a (2, 0)-vector v in order to distinguish this shadow and the shadow with respect to the even weight subcode. For Type II codes, we can take v = (2, 0, . . . , 0) to get a subcode of index 2. Thus we can define at least one generalized shadow for Type II codes. Now we give some properties of the generalized shadows. Lemma 3.11 For the generalized shadow with respect to a (2, 0)-vector v, C0⊥ /C0 must be isomorphic to the Klein 4-group. Proof. Suppose that C0⊥ /C0 is isomorphic to the cyclic group of order 4. Without loss of generality, we may assume (1) C2 = 2x + C0 , C1 = x + C0 and C3 = 3x + C0 , (2) C2 = x + C0 , C1 = 2x + C0 and C3 = 3x + C0 . In case (1), for any vector c0 ∈ C0 we have v ·c0 ≡ 0 (mod 4) and v ·(c0 +2x) ≡ 2 (mod 4) which gives that v ·x ≡ 1 (mod 2). However v ·x ≡ 0 (mod 2) since v is a (2, 0)-vector. In case (2), since x ∈ C, x · x ≡ 0 (mod 4). Therefore, any vector z ∈ Ci is orthogonal to any S S S vector y ∈ Cj for any i, j. Thus C ⊥ ⊃ (C0 C1 C2 C3 ) which gives the contradiction. 2

Lemma 3.12 For the generalized shadow S with respect to a (2, 0)-vector v, let C0⊥ = S S S S S C0 C2 C1 C3 where C = C0 C2 and S = C1 C3 . Then C1 = v + C0 , C2 = w + C0 , C3 = (v + w) + C0 , where w is a codeword of C2 . 2

Proof. Trivial. The above lemma determines the following orthogonality relation.

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S

Lemma 3.13 Suppose that C is a self-dual code of length n. Let S = C1 C3 be the generalized shadow of C with respect to a (2, 0)-vector v. Then Table 4 holds where the symbol ⊥ in position (i, j) means that x · y ≡ 0 (mod 4) for any vector x ∈ Ci and any vector y ∈ Cj , and the symbol 6⊥ means that x · y ≡ 2 (mod 4) for any vector x ∈ Ci and any vector y ∈ Cj . Proof. Since C0⊥ = C0 C1 C2 C3 , we get the first row and the first column in the table. We have that C1 ⊥ C1 and C2 ⊥ C2 since the Euclidean weights of v and w are congruent to 0 (mod 4). S

S

S

C1 · C2 ≡ (v + C0 ) · (w + C0 ) ≡ v · w ≡ 2

(mod 4),

C1 · C3 ≡ (v + C0 ) · ((v + w) + C0 ) ≡ v · w ≡ 2

(mod 4),

C2 · C3 ≡ (w + C0 ) · ((v + w) + C0 ) ≡ w · (v + w) ≡ 2

(mod 4),

C3 · C3 ≡ ((v + w) + C0 ) · ((v + w) + C0 ) ≡ (v + w) · (v + w) ≡ 0

(mod 4). 2

Table 4: Orthogonality Relations for generalized shadows

C0 C1 C2 C3

3.6

C0 ⊥ ⊥ ⊥ ⊥

C1 ⊥ ⊥ 6⊥ 6⊥

C2 ⊥ 6⊥ ⊥ 6⊥

C3 ⊥ 6⊥ 6⊥ ⊥

Extensions of Self-Dual Codes

If C0 is a proper subcode of index r in C, where r is 2 or 4, with vectors t and s such that hC0 , ti = C and hC, si = C0⊥ , then define the coset Cα,β = (C0 + αt + βs). That is, C0 is the kernel of the map Ψs (v) =

n X

si vi .

i=1

Note that α, β may run over either {0, 1} or {0, 1, 2, 3} depending on r.

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Let C ∗ = {(vα,β , Cα,β )}, giving that |C ∗ | = r|C|. To make C ∗ self-orthogonal we need vα,β · vα0 ,β 0 ≡ −cα,β · cα0 ,β 0 (mod 4) where ci,j ∈ Ci,j . To insure C ∗ is linear we require vα,β = αv1,0 + βv0,1 . Hence we need to chose v1,0 and v0,1 such that v1,0 · v1,0 ≡ −t · t

(mod 4)

v1,0 · v0,1 ≡ −t · s

(mod 4)

v0,1 · v0,1 ≡ −s · s

(mod 4).

We have that C ∗ is a self-orthogonal linear code. If n is the length of C and m is the length of vα,β and |C ∗ | = r|C| = 2n+m then the code C ∗ is self-dual. Otherwise take a self-orthogonal vector w in (C ∗ )⊥ and let C 0 = hC ∗ , wi. The length of vα,β depends on the existence of vectors v1,0 and v0,1 satisfying the above conditions. If M is the generator matrix for C0 , then the generator matrix for C 0 is: 

0 ...   .. . .  . .         

0 .. .

0 ... 0 v1,0 v0,1 w



M t s

      .     

Theorem 3.14 Let C be a self-dual code of length n and C0 the even weight subcode. The above construction for the Klein case gives: 1) If n ≡ 2 (mod 4), let v1,0 = (2, 0) and v0,1 = (1, 1) then C ∗ is a self-dual code of length n + 2. If n ≡ 6 (mod 8) then C ∗ is a Type II code and if n ≡ 2 (mod 8) then C ∗ is a Type I code. In addition, the swe of C ∗ is (a2 + c2 )sweC0 (a, b, c) + 2ac sweC2 (a, b, c) + 2b2 (sweC1 (a, b, c) + sweC3 (a, b, c)). 2) If n ≡ 0 (mod 4), let v1,0 = (2, 0, 0, 0), v0,1 = (1, 1, 1, 1) and w = (0, 2, 2, 0, 0, . . . , 0) 0 then C 0 is a self-dual code of length n + 4. If n ≡ 4 (mod 8) then C is a Type II code 0 0 and if n ≡ 0 (mod 8) then C is a Type I code. In addition, the swe of C is (a4 +6a2 c2 +c4 )sweC0 (a, b, c)+(4a3 c+4ac3 )sweC2 (a, b, c)+8b4 (sweC1 (a, b, c)+sweC3 (a, b, c)). Proof. The orthogonality relations are given in Table 1 and Table 2 and the weight enumerator follows from a direct calculation. 2

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Theorem 3.15 Let C be a self-dual code of length n and C0 the even weight subcode. The above construction for the cyclic case gives: 1) If n ≡ 3 (mod 4), let v1,0 = (2) and v0,1 = (1) and then C ∗ is a self-dual code of length n + 1. If n ≡ 7 (mod 8) then C ∗ is a Type II code and if n ≡ 3 (mod 8) then C ∗ is a Type I code. In addition, the swe of C ∗ is a sweC0 (a, b, c) + c sweC2 (a, b, c) + b(sweC1 (a, b, c) + sweC3 (a, b, c)). 2) If n ≡ 1 (mod 4), then let v1,0 = (0, 0, 2), v0,1 = (1, 1, 1) and w = (0, 2, 2, 0, . . . , 0) and 0 C 0 is a self-dual code of length n + 3. If n ≡ 5 (mod 8) then C is a Type II code and if 0 0 n ≡ 1 (mod 8) then C is a Type I code. In addition, the swe of C is (a3 + 3ac2 )sweC0 (a, b, c) + (3a2 c + c3 )sweC2 (a, b, c) + 4b3 (sweC1 (a, b, c) + sweC3 (a, b, c)). Proof. The orthogonality relations are given in Table 3 and the weight enumerator follows from a direct calculation. 2

S

Theorem 3.16 Suppose that C is a self-dual code of length n. Let S = C1 C3 be the generalized shadow of C with respect to a (2, 0)-vector v. Then the code C 0 constructed above with v1,0 = (2000), v0,1 = (1111) and w = (00220 . . . 0), is a self-dual code of length 0 n + 4. In addition, the swe of C is (a4 +6a2 c2 +c4 )sweC0 (a, b, c)+(4a3 c+4ac3 )sweC2 (a, b, c)+8b4 (sweC1 (a, b, c)+sweC3 (a, b, c)). Proof. The orthogonality relations are given in Table 4 and the weight enumerator follows from a direct calculation. 2 Remark. For binary self-dual codes, constructing methods using the concept of shadows were given in [4] and [19]. Theorem 3.14 corresponds to the method given in [4] and Theorem 3.16 corresponds to the method given in [19]. This technique was explained for finite fields in [8].

4 4.1

Relative Invariants Three Variables

Let us denote the swe of Ci by Wi . Let M3 transform, namely  1 1  M3 =  1 2 1 13

be the 3 by 3 matrix of the MacWilliams 

2 1  0 −1   −2 1

and let





1 0 0   J3 =  0  .  0 ζ8 0 0 −1 Let G3 denote the matrix group with generators are invariants of G3 . Let  0 0 1   U3 =  0 1 0 1 0 0

M3 and J3 . The swe’s of Type II codes   . 

Then the swe’s of Type II codes in the sense of [3] are invariant under the seemingly larger group G03 generated by M3 , J3 and U3 . A computation in Magma shows that |G3 | = |G03 | = 768. Therefore the supplementary condition of containing the all-2 vector does not bring any new constraint on the swe. This is explained in the following easy but unnoticed result. Proposition 4.1 Every self-dual Z4 code C contains an all-2 vector. Proof. Let h denote the all-2 vector. Since x · x, the Euclidean weight of x, is divisible by 4 so is the number of ±1’s that x contains. Therefore x · h = 0 and h ∈ C ⊥ = C by self-duality. 2 Remark. Recently it has been shown in [13] that all Type II codes contain a ±1 vector. On the other hand the condition of containing a ±1 vector is a supplementary constraint on the complete weight enumerator (cwe) (see [7] for the definition of the cwe) as we now explain. Let   1 1 1 1   1 i −1 −i    1 M4 =    2  1 −1 1 −1   1 −i −1 i denote the matrix of the MacWilliams relation acting on the cwe. Let 

J4 =

     

1 0 0 0 0 ζ8 0 0 0 0 −1 0 0 0 0 ζ8

      

denote the operator whose fixed points correspond to cwe’s of codes with Euclidean weight a multiple of 8. Let U4 correspond to addition of the all-one vector. 

U4 =

     

0 1 0 0

0 0 1 0 14

0 0 0 1

1 0 0 0

    .  

Then a Magma computation shows that the order of the group generated by M4 and J4 is 1536 while the order of the group generated by M4 , J4 , and U4 is 6144. Theorem 4.2 If n is odd then W1 = W3 = sweS /2. If n is even the polynomial W1 − W3 is a relative invariant for the group G3 with respect to the character χ defined on M3 and J3 as χ(M3 ) = (i)n and χ(J3 ) = ζ8n . Proof. If we are in the cyclic case (odd n) then write C1 = x + C0 , to get C3 = 3x + C0 ≡ S −(x + C0 ) (mod 4), implying W1 = W3 . Since S = C1 C3 the first assertion follows. If we are in the Klein case (even n) either n is a multiple of 4 and both S1 , S3 are self-dual or n is congruent to 2 modulo 4 and S1 and S3 are dual of each other. Writing the MacWilliams relation for both codes and subtracting the second equality from the first yields M3 (W1 − W3 ) = (i)n (W1 − W3 ). In both cases by Theorem 3.4 2) we get J3 (W1 − W3 ) = ζ8n (W1 − W3 ). 2 When n is a multiple of 8 the polynomial W1 − W3 is an invariant for the group G4 and Theorem 6 of [3] applies. W1 − W3 =

X

βij (a + c)n−4i−8j (2b4 − ac(a2 + c2 ))i (b4 (a − c)4 )j .

i,j

For instance, in the case of the code C16 defined in §5.2, we obtain W1 = 224b8 c8 + 256b16 + 3584ab12 c3 + 8a2 c14 +3584a2 b8 c6 + 896a3 b4 c9 + 3584a3 b12 c + 10304a4 b8 c4 +2688a5 b4 c7 + 56a6 c10 + 3584a6 b8 c2 + 2688a7 b4 c5 +128a8 c8 + 224a8 b8 + 896a9 b4 c3 + 56a10 c6 + a14 c2 . and W3 = 256b8 c8 + 7168(a2 b8 c6 + a6 b8 c2 ) + 17920a4 b8 c4 + 256a8 b8 .

4.2

Two Variables

As in [11] let φ denote the Gray map. Let us denote the Hamming weight enumerator of φ(Ci ) by Hi . Let M2 denote the 2 by 2 matrix of the MacWilliams relation namely 



1 1 1  M2 = √  2 1 −1 15

and let





1 0  J2 =  . 0 −1 Let G2 denote the matrix group of order 16 with generators M2 and J2 . The Hamming weight enumerators of self-dual Type I binary codes are invariants of this group. Theorem 4.3 If n is even, then the polynomial H1 −H3 is a relative invariant for the group G2 with respect to the character χ defined on M2 and J2 as χ(M2 ) = in and χ(J2 ) = 1. Proof. The result follows immediately by the properties of φ with respect to the MacWilliams duality [14] and Theorem 3.4 3). 2 When n is a multiple of 4 the polynomial H1 − H3 is an absolute invariant for G2 and the first Gleason theorem applies. bn/4c

(H1 − H3 )(x, y) =

X

aj (x2 + y 2 )n−4j {x2 y 2 (x2 − y 2 )2 }j .

j=0

5 5.1

The Highest Minimum Weights of Type I Codes Optimal Codes

An upper bound for the minimum Lee weight of self-dual codes over Z4 has been given in [3]. In this section, we give some improved upper bounds by studying shadows. It is useful to determine the highest minimum Lee weights when investigating the highest minimum weights of binary nonlinear codes obtained from self-dual codes using the Gray map. Likewise it is useful to determine the highest minimum Euclidean weight when investigating the corresponding unimodular lattices. We say that Type I codes are Euclidean-optimal, Lee-optimal and Hamming-optimal if the codes have the highest minimum Euclidean, Lee and Hamming weight for that length, respectively. The following upper bounds are useful when determining the highest minimum weights. Proposition 5.1 (Harada [12]) Let dE be the minimum Euclidean weight of a Type I code of length n over Z4 . dE ≤

 

4(b n8 c + 1)  4b n c 8

n = 1, . . . , 7, 12, 14, 15 and 23, otherwise,

for length n ≤ 24.

16

Theorem 5.2 If C is a self-dual Z4 code of length n then dH ≤ (1 + bn/2c). Proof. Recall that the Hamming weight enumerator satisfies the MacWilliams relations WC (x, y) =

1 WC (x + 3y, x − y). 2n

The matrix group of order 2 generated by 



1 1 3  2 1 −1 has Molien series 1/(1 − t)(1 − t2 ), and primary invariants f1 := x + y and f2 := y(x − y). The Hamming weight enumerator is an isobaric polynomial in f1 and f2 . bn/2c

WC =

X

aj f1n−2j f2j .

j=0

The optimal weight enumerator is defined as the one corresponding to a code having the highest possible dH as permitted by the above expression. The result follows in the vein of Corollary 3 of [15]. 2 We observe following [18, p.70] that the analogous result over a quaternary alphabet which is a field follows trivially from the Singleton bound. The situation is subtler over a ring.

5.2

Short Lengths

By computing the possible symmetrized weight enumerators using Theorem 3.4 of self-dual codes and their shadows and insuring that the coefficients of these are non-negative integers, we can establish an upper bound on the minimum Lee, Euclidean and Hamming weights of Type I codes. The method is similar to one described in [6] for binary self-dual codes. We shall adopt the terminology for specific codes given in [7]. Since self-dual codes over Z4 are classified up to length 15 [9] we shall begin with length n = 16. • n = 16: The highest possible minimum Euclidean and Lee weights are 8; there are 3 parameters in the family of weight enumerators with minimum Lee weight equal to 8. Let C16 be

17

the code with generator matrix                  

10000000 01000000 00100000 00010000 00001000 00000100 00000010 00000001

21111111 32113133 33211313 33321131 31332113 33133211 31313321 31131332

         .        

C16 is Type I and the minimum Euclidean and Lee weights of C16 are 8. Thus the highest minimum Euclidean and Lee weights are just 8. The highest attainable minimum Hamming weight is 4 and there is a family of possible weight enumerators with five parameters. The minimum Hamming weight of C16 is 4; thus the highest minimum weight is just 4. • n = 17: The highest attainable minimum Lee weight is 8 and there is a family of possible weight enumerators with three parameters. The highest attainable minimum Hamming weight is 4 and there is a family of possible weight enumerators with five parameters. The highest attainable minimum Euclidean weight is 8. Here we construct a self-dual code with dE = 8 using the concept of the generalized shadows. All self-dual codes of length up to 15 were classified in [9]. By Theorem 3.16, a self-dual code of length 17 is constructed using v = (2, 0, 0, . . . , 0, 0) and C equal to the code [13,5]-d10b in [9] with generator matrix of the form                  

1110000001000 1101000002102 1100100002210 1100010002021 0100001113333 0000002020000 0000000220000 2222222222222

         .        

The minimum Euclidean weights of C0 , C2 and S are 8, 4 and 4, respectively. Thus the extended code C ∗ (we denote this code by C17 ) has the minimum weights dL = 4, dE = 8 and dH = 2. C17 determines a unique 17-dimensional unimodular lattice with minimum norm 2 by Construction A4 in [2]. 18

• n = 18: The highest attainable minimum Lee weight is 8 and there is a family of possible weight enumerators with three parameters. The highest attainable minimum Hamming weight is 4 and there is a family of possible weight enumerators with five parameters. The highest attainable minimum Euclidean weight is 8. Here we construct a self-dual code with dE = 8 using the concept of the generalized shadows. All self-dual codes of length up to 15 were classified in [9]. Self-dual codes of length 18 are constructed by Theorem 3.16. When v = (2, 0, 0, . . . , 0, 0) and C is the code denoted by [14,7]-2e7c in [9], the minimum Euclidean weights of C0 , C2 and S are 8, 8 and 4, respectively. Thus the extended self-dual code C18 of length 18 has the parameters dL = 4, dE = 8 and dH = 3. Therefore the highest Euclidean weight is just 8. The code also produces a unimodular lattice with no vectors of norm 1 by Construction A4 . There are exactly four inequivalent such lattices in dimension 18 (cf. [5]). • n = 19: The highest attainable minimum Lee weight is 8 and there is a family of possible weight enumerators with three parameters. The highest attainable minimum Hamming weight is 5 and there is a family of possible weight enumerators with three parameters. The highest attainable minimum Euclidean weight is 8. Self-dual codes of length 19 are constructed by Theorem 3.16. When v = (2, 0, 0, . . . , 0, 0) and C is the code denoted by [15,7]-d14c in [9], the minimum Euclidean weights of C0 , C2 and S are 8, 8 and 4, respectively. Thus the extended self-dual code C19 of length 19 has dL = 4, dE = 8 and dH = 3. Therefore the highest Euclidean weight is just 8. The code also produces a unimodular lattice with no vectors of norm 1 by Construction A4 . There are exactly three inequivalent such lattices in dimension 19 (cf. [5]). • n = 20: The highest possible minimum Lee and Euclidean weights are 8. There is a family of possible weight enumerators with minimum Lee weight with six parameters. The highest attainable minimum Hamming weight is 6 and there is a family of possible weight enumerators with four parameters. The minimum Euclidean weight of the shadow of C16 is 8. The extended code C20 of length 20 constructed from C16 by Theorem 3.14 has minimum Euclidean weight 8 and minimum Lee weight 4. Therefore the highest minimum Euclidean weight is just 8. • n = 21: 19

The highest possible minimum Lee and Euclidean weights are 8. There is a family of possible weight enumerators with minimum Lee weight with six parameters. The highest attainable minimum Hamming weight is 6 and there is a family of possible weight enumerators with four parameters. C21,1 in [16] has minimum Euclidean weight 8 and minimum Lee weight 6. Thus the highest minimum Euclidean weight is just 8 and the highest minimum Lee weight is 6 or 8. • n = 22: The highest possible minimum Lee and Euclidean weights are 8. There is a family of possible weight enumerators with minimum Lee weight with six parameters. The highest attainable minimum Hamming weight is 6 and there is a family of possible weight enumerators with five parameters. Let us consider a direct sum construction C1 ⊕ C2 = {(c1 , c2 ) | c1 ∈ C1 and c2 ∈ C2 }. The minimum Euclidean weight of C1 ⊕ C2 is min{dE (C1 ), dE (C2 )} where dE (C) denotes the minimum Euclidean weight of C. There are self-dual codes with dE = 8 for lengths 8 and 14. Thus the code C8 ⊕ C14 has minimum Euclidean weight 8 where C8 is a Type II code of length 8 and C14 is a Type I code with dE = 8 of length 14 in [9]. Therefore the highest minimum Euclidean weight is just 8. C˜21,1 and C˜21,2 in [16] also have minimum Euclidean weight 8. C˜21,1 in [16] also has minimum Lee weight 8. Therefore the highest minimum Lee and Euclidean weights are just 8. • n = 23: The highest possible minimum Euclidean and Lee weights are 12 and 10, respectively. There is a unique possible weight enumerator with minimum Lee weight 10: a23 + 85008a11 b8 c4 + 40480a8 b8 c7 + 141680a2 b12 c9 + · · · . The highest attainable minimum Hamming weight is 7 and there is a family of possible weight enumerators with five parameters. The quadratic residue code of length 23 is known. Its minimum Euclidean and Lee weights are 12 and 10, respectively [2]. Thus the highest minimum Euclidean and Lee weights are just 12 and 10, respectively. Corollary 5.3 Any extended code of length 24 constructed from a Type I code of length 23 with dE = 12 by Theorem 3.15 is a Euclidean-optimal Type II code, that is dE = 16. Proof. Consider the possible weight enumerators of length 23 obtained by Theorem 3.4. Since dE = 12, we have α00 = 1, α01 = −46, α02 = 0, α10 = 23, α11 = −368, α20 = 184. 20

Then the terms of the possible weight enumerators of the shadow corresponding to the Euclidean weight 7 are only α12 16 7 α12 19 3 − a b + a b c. 2 2 Since any coefficient must be a non-negative integer, α12 is 0. Thus if dE = 12 then the shadow contains no codeword of Euclidean weight 7. 2 Thus even if a code with dE = 12 is not equivalent to the quadratic residue code, the extended code is Euclidean-optimal. If an Euclidean-optimal Type I code of length 23 exists then the extended code determines the Leech lattice via Construction A4 . Moreover no self-dual codes of lengths 20, 21 and 22 can be extended to a Euclideanoptimal Type II code of length 24 by the methods. • n = 24: The highest possible minimum Euclidean, Lee and Hamming weights are 12, 12 and 8, respectively. There is a family of possible weight enumerators with minimum Lee weight with 4 parameters and a family of possible weight enumerators of minimum Hamming weight with 5 parameters. C24,1 and C24,2 in [10] are Type I codes and their minimum Euclidean, Lee and Hamming weights are 12, 10 and 8, respectively. Thus the highest minimum Euclidean and Hamming weights are just 12 and 8, respectively. Rains [17] showed that self-dual codes of length 24 with minimum Lee weight 12 have the same symmetrized weight enumerators as the lifted Golay code. Thus highest minimum Lee weight of Type I codes is 10. If C is a self-dual code of length n then the set of codewords in C which are either 0 or 2 on a particular coordinate is a subcode of index 2. Removing that coordinate from the subcode gives a self-dual code D of length n − 1. The code D is called the shortened code of C on that coordinate in [9]. Here we take the first coordinate as that one. If d is the minimum Hamming weight of C then the minimum Hamming weight of D is d − 1 or d. Considering the shortened codes of C24,1 in [10], self-dual codes H23 , H22 , H21 and H20 are constructed of length 23, 22, 21 and 20, respectively. Then dH (H23 ) is 7 or 8 where dH (C) denotes the minimum Hamming weight of C. However the highest possible minimum Hamming weight is 7, thus dH (H23 ) is 7, which gives that the highest minimum Hamming weight of length 23 is just 7. Similarly dH (H22 ) is 6, which gives that the highest minimum Hamming weight of length 22 is just 6. We have verified by computer that dH (H21 ) is 5 and dH (H20 ) is 4. The highest attainable minimum Lee weight for a Type I code of length 35 is 12. Hence a [70, 35, 14] Type I binary code is not the image under the Gray map of a Type I code over 21

Z4 .

5.3

Summary

As a summary, we list the highest minimum Lee, Euclidean and Hamming weights of Type I codes in Tables 5, 6 and 7, respectively, for length n ≤ 24. In each table, the first column denotes the length n and the second column gives the highest minimum weight. In addition, the third and fourth columns give the number and references of the known codes having the indicated minimum weight, respectively. In Table 5 the code D4+n is the code C1 ⊕ C2 where C1 is the code with dL = 4 of length 4 and C2 is a code with dL ≥ 4 of length n. Since the highest minimum Euclidean weights of Type II codes of lengths 8, 16 and 24 are 8, 8 and 16, Table 6 gives the following: Proposition 5.4 The highest Euclidean weight of any self-dual code of length n ≤ 24 is known.

Postscript. After a manuscript of this paper was first circulated, recently Rains [17] has improved an earlier version of Tables 5, 6 and 7. The current versions of these tables contain the results in his paper.

Acknowledgments. The authors would like to thank Philippe Gaborit for his helpful information on Tables 5, 6 and 7 of lengths 10 to 15. The authors would also like to thank Eric Rains, Neil Sloane and the referees for their useful comments.

References [1] E. Bannai, S.T. Dougherty, M. Harada and M. Oura, Type II codes, even unimodular lattices and invariant rings, (submitted). [2] A. Bonnecaze, P. Sol´e and A.R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory 41 (1995), 366–377. [3] A. Bonnecaze, P. Sol´e, C. Bachoc and B. Mourrain, Type II codes over Z4 , IEEE Trans. Inform. Theory 43 (1997), 969–976. [4] R.A. Brualdi and V. Pless, Weight enumerators of self-dual codes, IEEE Trans. Inform. Theory 37 (1991), 1222–1225. 22

Table 5: The Highest Minimum Lee Weights of Type I Codes n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

dL 2 2 2 4 2 4 4 4 2 4 4 4 4 6 6 8 6 8 6 8 8 8 10 10

number of codes 1 1 1 1 2 1 1 3 11 5 3 39 8 1 15 ≥5 ≥ 17 7 ≥1 ≥1 384 ≥ 19367 30 ≥1

references [7] [7] [7] [7] [7] [7] [7] [7] [7] [9] [9] [9] [9] [9] [9] C16 [17] [17] [17] [17] [17] ˜ C21,1 in [16], [17] [3] [10]

[5] J.H. Conway and N.J.A. Sloane, On the enumeration of lattices of determinant one, J. Number Theory 15 (1982), 83–94. [6] J.H. Conway and N.J.A. Sloane, A new upper bound on the minimum distance of self-dual codes, IEEE Trans. Inform. Theory 36 (1990), 1319–1333. [7] J.H. Conway and N.J.A. Sloane, Self-dual codes over the integers modulo 4, J. Combin. Theory Ser. A 62 (1993), 30–45. [8] S.T. Dougherty, Shadow codes and weight enumerators, IEEE Trans. Inform. Theory 41 (1995), 762–768.

23

Table 6: The Highest Minimum Euclidean Weights of Type I Codes n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

dE 4 4 4 4 4 4 4 4 4 4 4 8 4 8 8 8 8 8 8 8 8 8 12 12

number of codes 1 1 1 2 2 3 4 6 11 16 19 19 66 35 28 ≥5 ≥ 17 ≥ 39 ≥1 ≥1 ≥ 2384 ≥ 19367 ≥ 30 ≥5

C21,1

references [7] [7] [7] [7] [7] [7] [7] [7] [7] [9] [9] [9] [9] [9] [9] C16 C17 , [17] C18 , [17] C19 C20 and C21,2 in [16], [17] C8 ⊕ C14 , [17] [3], [17] [10]

[9] J. Fields, P. Gaborit, J. Leon and V. Pless, All self-dual Z4 codes of length 15 or less are known, IEEE Trans. Inform. Theory 44 (1998), 311–322. [10] T.A. Gulliver and M. Harada, Certain self-dual codes over Z4 and the odd Leech lattice, Lecture Notes in Computer Sciences Vol. 1225 (1997), 130–137. [11] A.R. Hammons, Jr., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane and P. Sol´e, A linear construction for certain Kerdock and Preparata codes, Bull. Amer. Math. Soc. 29 (1993), 218–222. [12] M. Harada, Self-Dual Codes, Designs and Unimodular Lattices, Doctoral thesis, Okayama University, Japan, 1997.

24

Table 7: The Highest Minimum Hamming Weights of Type I Codes n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

dH 1 1 1 2 1 2 3 4 1 2 2 2 2 3 3 4 4 4 3 4 5 6 7 8

number of codes 1 1 1 1 2 1 1 1 11 5 3 39 8 4 47 ≥1 62 66 ≥1 ≥1 384 ≥ 19367 ≥ 1.72 × 106 ≥2

references [7] [7] [7] [7] [7] [7] [7] [7] [7] [9] [9] [9] [9] [9] [9] C16 [17] [17] C19 H20 H21 , [17] H22 , [17] H23 , [17] [10]

[13] M. Harada, P. Gaborit and P. Sol´e, Self-dual codes over Z4 and unimodular lattices: a survey, (submitted). ¨ [14] M. Klemm, Uber die Ident¨at von MacWilliams f¨ ur die Gewichtsfunktion von Codes. Archiv. Math 49 (1987), 400–406. [15] C. Mallows and N.J.A. Sloane, An upper bound for self-dual codes, Inform. and Control 22 (1973), 188–200. [16] V. Pless, P. Sol´e and Z. Qian, Cyclic self-dual Z4 -codes, Finite Fields and Their Appl. 3 (1997), 48–69. [17] E.M. Rains, Optimal self-dual codes over Z4 , (submitted). 25

[18] E.M. Rains and N.J.A. Sloane, Self-dual Codes, a chapter in Handbook of Coding Theory, V. Pless, R. Brualdi, C. Huffman eds, Elsevier (1998) to appear. [19] H.-T. Tsai, Existence of some extremal self-dual codes, IEEE Trans. Inform. Theory 38 (1992), 1829–1833.

26