Type II Codes over F2 + uF2 Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510, USA Email: [email protected],

Philippe Gaborit Laboratoire A2X Universit´e Bordeaux I 33400 Talence, France 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

1

Running head: Type II codes over F2 + uF2 Name: Steven T. Dougherty, Philippe Gaborit, Masaaki Harada and Patrick Sol´e Contact Author: Steven T. Dougherty Address: Department of Mathematics University of Scranton Scranton, PA 18510, USA Telephone: 717-941-6104 Fax: 717-941-6369 E-mail: [email protected]

2

Abstract The alphabet F2 + uF2 is viewed here as a quotient of the Gaussian integers by the ideal (2). Self-dual F2 + uF2 codes with Lee weights a multiple of 4 are called Type II. They give even unimodular Gaussian lattices by construction A, while Type I codes yield unimodular Gaussian lattices. Construction B makes it possible to realize the Leech lattice as a Gaussian lattice. There is a Gray map which maps Type II codes into Type II binary codes with a fixed point free involution in their automorphism group. Combinatorial constructions use weighing matrices and strongly regular graphs. Gleason type theorems for the symmetrized weight enumerators of Type II codes are derived. All self-dual codes are classified for length up to 8. The shadow of Type I codes yields bounds on the highest minimum Hamming and Lee weights. Index Terms: Codes over Rings, Gray Map, Automorphism Groups, Lattices and Shadows.

3

1

Introduction

Recent years witnessed a burst of activity in codes over Z4 with applications to (nonlinear) binary codes [18] and unimodular lattices [3] and [22]. Another important alphabet of size 4 besides Z4 is F2 + uF2 introduced in [1] to construct lattices. The philosophy of this article is to treat the latter as a function field analogue of the former, emphasizing analogies and differences. Like Z4 the ring contains a nilpotent element of order 2 and (Lee) weight 2, namely u, and an almost identical-looking Gray map. The main difference between the two alphabets is that the Lee weight over F2 + uF2 plays both the role of the Lee weight over Z4 as far as binary codes are concerned and the role of the Euclidean weight in lattice constructions. Type II codes over Z4 were introduced in [3] to construct Type II lattices, similarly Type II codes are introduced here to construct even Gaussian lattices. Even Gaussian lattices are known to exist in dimensions a multiple of 4 [12, p. 55]. Construction B and density doubling realize the Leech lattice as a Gaussian lattice. Besides Gaussian lattices another application of the alphabet F2 + uF2 lies in the construction via the Gray map of self-dual binary codes with a fixed point free involution in their automorphism groups. Binary self-dual codes with such an involution are investigated in [5] and [6]. All binary extremal double circulant self-dual codes are classified in [21] and [17] for length up to 72. Some of the double circulant codes have such an involution in their automorphism groups. The material is arranged as follows. Section 2 collects the required notations and definitions. Section 3 lays down foundations: the Gray map, lattice constructions, and shadows. Section 4 contains the structure theorems needed for a mass formula, which, in turn, makes the short length classification of Section 6 possible. All self-dual codes are classified in Section 6 for length up to 8. Sections 5 and 9 contain several constructions. Section 8 develops the invariant theory used in the study of weight enumerators. Section 10 builds up the shadow theory needed for the bounds of Section 11. Sections 5, 7 and 9 contain constructions.

2

Notations and Definitions

In this section, we give the required notations and definitions. In particular, we introduce Type II codes which are a remarkable class of self-dual codes. In the quotient Z[i]/(2) or equivalently Z[X]/(2, (X + 1)2 ), we let u denote the residue class of X + 1, thus introducing the ring R := F2 + uF2 where u satisfies u2 = 0. Throughout this paper, R denotes the ring F2 + uF2 . Roughly speaking, u plays the same role over the ring F2 + uF2 that 2 plays over Z4 . 4

A code C of length n over R is an R-submodule of Rn . An element of C is called a P codeword. Duality for codes is understood with respect to the form i xi yi . C is said to be self-dual if C = C ⊥ . We say that two codes are equivalent if one can be obtained from the other by permuting the coordinates and exchanging 1 and 1 + u in certain coordinates. The Lee composition of a vector x = (x1 , . . . , xn ) ∈ Rn is defined as (n0 (x), n1 (x), n2 (x)) where n0 (x) is the number of xi = 0, n2 (x) the number of xi = u and n1 (x) = n − n0 (x) − n2 (x) where n is the length. The Lee weight wL (x) of x is then defined as n1 (x) + 2n2 (x). The symmetrized weight enumerator (swe for short) is defined as sweC (a, b, c) =

X

an0 (x) bn1 (x) cn2 (x) .

x∈C

There is a natural (linear!) Gray map φ which is a F2 -linear isometry from (Rn , Lee distance) onto (F2n 2 , Hamming distance) where the Lee distance of two codewords x and y is the Lee weight of x − y (cf [1]). We let φ(x + uy) = (y, x + y), where x, y ∈ Fn2 . A self-dual code over R is said to be Type II if the Lee weight of every codeword is a multiple of 4 and Type I otherwise. Any code is permutation-equivalent to a code C with generator matrix: 



Ik A B1 + uB2  G= 1 0 uIk2 uD where A, B1 , B2 and D are matrices over F2 . We associate two binary codes: the residue code C(1) and the torsion code C(2) as follows: C(1) = {x ∈ Fn2 | ∃y ∈ Fn2 | x + uy ∈ C} and C(2) = {x ∈ Fn2 | ux ∈ C}. A generator matrix of C(1) is: G1 =



Ik1 A B1



and a generator matrix of C(2) is: 



Ik A B1  G2 =  1 . 0 Ik2 D We have |C| = |C(1) | · |C(2) | = 2k1 2k1 +k2 = 22k1 +k2 . A Gaussian lattice is a discrete Z[i]-module of Cn . An even Gaussian lattice contains only vectors of even (squared, hermitian) norm. Duality for Gaussian lattices is understood P with respect to the sesquilinear form i xi yi , the bar denoting complex conjugation. 5

3 3.1

First Properties Gray Map

First, we connect the properties of C to those of its Gray image. Proposition 3.1 If C is self-orthogonal so is φ(C). φ(C) is a Type II code iff the code C is Type II. The minimum Lee weight of C is the same as the minimum Hamming weight of φ(C). Proof. Let z = x + uy (resp. z 0 = x0 + uy 0 ). If zz 0 = 0 in R then zz 0 = xx0 + u(xy 0 + x0 y) entails both xx0 = 0 and x0 y + xy 0 = 0. Since the Gray map φ is an isometry from 2 (Rn , Lee distance) to (F2n 2 , Hamming distance), the last two assertions hold. The above proposition gives a restriction on the lengths and upper bounds on the minimum Lee weight. Corollary 3.2 There exists a Type II code of length n iff n ≡ 0

(mod 4).

Proof. It is known that if a binary Type II code exists for length n then n ≡ 0 (mod 8) (cf. [26]). Thus if a Type II code of length n exists then n must be divisible by four. K4 is a Type II code of length 4. Note that Kn will be introduced in Section 5. 2 Remark. The above restriction may be obtained using invariants for the symmetrized weight enumerators of Type II codes (see Theorem 8.2). Corollary 3.3 Let dL (II, n) and dL (I, n) be the highest minimum Lee weights of a Type II code and a Type I code, respectively, of length n. Then n + 4, dL (II, n) ≤ 4  12h i  4 n + 4, i dL (I, n) ≤  h 24 n 4 24 + 6, 



if n 6≡ 22

(mod 24),

otherwise.

Proof. The Gray map is an isometry from (Rn , Lee distance) to (F2n 2 , Hamming distance). An upper bound on the minimum Hamming weight of a binary Type II code (resp. a Type I code) is given in [26] (resp. [31]). 2

Lemma 3.4 Let C and C 0 be equivalent self-dual codes over R then φ(C) and φ(C 0 ) are equivalent.

6

Proof. Follows from the definition of the Gray map.

2

The converse assertion is not true in general. For example, consider the binary self-dual code B4 = {(0, 0, 0, 0), (1, 1, 0, 0), (0, 0, 1, 1), (1, 1, 1, 1)}. The inverse map φ−1 (B4 ) of the Gray map of B4 is {(0, 0), (1, 1), (1 + u, 1 + u), (u, u)}, however φ−1 (B4σ ) where σ = (23) is {(0, 0), (u, 0), (0, u), (u, u)}. We shall study the condition that the converse assertion is ture in Section 9. Proposition 3.5 If C is a self-dual code then C contains the all u vector. Proof. vector.

A self-orthogonal vector v has n1 (v) even and hence is orthogonal to the all u 2

The above proposition corresponds to the result that φ(C) contains the all-one vector.

3.2 3.2.1

Lattices Construction A

With every code C over R we attach the lattice √ A(C) = L(C)/ 2. where we have set L(C) := {x ∈ Z[i]n | x (mod 2) ∈ C}. This is the Gaussian analogue of the construction A2 (Cb ) of [12] given for a binary linear code Cb by √ 2A2 (Cb ) := {x ∈ Zn | x (mod 2) ∈ Cb }. We connect the properties of C to those of its attached lattice. We call the projection p(z) of a complex number z the pair (<(z), =(z)) and extend it component-wise to get a 1 − 1 map from Z[i]n onto Z2n . Proposition 3.6 If C is Type II then A(C) is even unimodular, and unimodular if C is Type I. Further, p(A(C)) = A2 (φ(C)). Proof. The first assertion follows from the fact that the Lee weight of a codeword is half the weight of its representative in A(C). The second follows by observing that, for a, b in Fn2 and x, y ∈ Zn we have p(a + ub + 2(x + iy)) = (a + b + 2x| b + 2y). 2 Remark. This gives a new proof of Corollary 3.2, since Gaussian unimodular even lattices only exist in length multiple of 4 [12, p. 55]. 7

3.2.2

Construction B

We now proceed to define the Gaussian analogue of the classical construction B hereby called B2 . Recall that this construction attaches to a doubly-even binary code Cb the lattice √ X 2B2 (Cb ) := {x ∈ Zn | x (mod 2) ∈ Cb and xi = 0 (mod 4)}. i n

Let Gn denote any code of R whose Gray image is the parity-check code of length n. For instance we may take the following code. Let Pn be the parity-check code and e any vector of the form (1, 0n−1 ). Define Gn as the R-span of Pn and ue. With every code C over R we attach the sphere packing: √ B(C) = L0 (C)/ 2, where we have set L0 (C) := C + 2Gn + 4Z[i]n . The following proposition is now immediate. Proposition 3.7 If C is Type II then B(C) is an integral lattice, and p(B(C)) = B2 (φ(C)). Using construction B applied to the inverse Gray image of the extended Golay we can realize the half-Leech lattice hΛ24 of [12, p. 191] as a Gaussian lattice. After density doubling and suitable rescaling one can obtain the Leech lattice itself as a Gaussian lattice [12, p. 149]. From Proposition 3.6 and 3.7 it transpires that the interest of these constructions does not lie in finding new lattices but in making apparent the Gaussian structure of lattices obtained from binary codes by constructions A and B.

3.3

Shadows

We define the even-weight subcode C0 of a Type I code C as the subcode consisting of all codewords of C with Lee weights a multiple of 4. Clearly C0 is of index 2 in C and C of index S S S S 2 in C0⊥ . We write C0⊥ = C0 C1 C2 C3 . We let C = C0 C2 and define the shadow S S of C as S = C1 C3 . Eventually we note that the Gray image of the shadow is the shadow of the Gray image. Proposition 3.8 If C is Type I then φ(Cj ) = φ(C)j for j = 0, 1, 2, 3, that is,

coset

C   y

Cj

−−−−−→

φ(C)   ycoset −−−−−→ φ(Cj ) = φ(C)j φ

φ

Corollary 3.9 If C is a Type I code of length n then φ(S) is the shadow of φ(C). Hence if the shadow of C has minimum Lee weight dS then φ(C) is a binary [2n, n] code whose shadow has minimum Hamming weight dS . 8

4

Structure of Type II Codes

In this section we give a characterization of the structure of Type II codes, as a function of the structure of their residue and torsion codes. If C is a code over R of length n and x and y are two vectors of C, we denote by ni,j (x, y) the number of occurrences of the couples (i, j) and (j, i), where i and j are elements of R, in the columns of the (2, n) matrix obtained from the juxtaposition of the vectors x and y. Since there is no ambiguity, in the following we shall simply denote ni,j (x, y) by ni,j . There is a lot of similarity between the structure of codes over R and the structure of codes over Z4 . In [15], self-dual codes over R are examined, considering the effective parallelism between “2” in Z4 and “u” in R, we deduce from [14]: Proposition 4.1 The set of self-dual codes over R is the set of codes over R, which are permutation-equivalent to a code C with a generator matrix of the form: 



I + uB A   k1 , 0 uD where A, B and D are matrices over F2 , satisfying: (1) B is symmetric, ⊥ (2) A and D are such that C(1) = C(2) and C(1) is even.

We now give a result on the residue code of a Type II code: Proposition 4.2 If C is a Type II code then the residue code C(1) of C contains the all-ones vector 1. Proof. If C is Type II then the Lee weights of all the codewords are multiple of 4, and so ⊥ the number of u components in a codeword only composed of u and 0 is even, so C(1) = C(2) contains 1. 2

Proposition 4.3 If C is a self-dual code over R and if x and y are two codewords of C such that wL (x) ≡ wL (y) ≡ 0 (mod 4) then wL (x + y) ≡ 0 (mod 4). Proof. We can write x = x1 + ux2 and y = y1 + uy2 , where xi , yi ∈ Fn2 . From the definition of the Lee weight we have: wL (x + y) = wL (x) + wL (y) − 4nu,u − 2n1,u − 2n1+u,u − 2n1,1 − 2n1+u,1+u . We know that C is self-dual so: x · y = (x1 + ux2 ) · (y1 + uy2 ) = x1 · y1 + u(x1 · y2 + x2 · y1 ) = 0, 9

so that: x1 · y1 = n1,1 + n1+u,1+u + n1,1+u ≡ 0

(mod 2),

and x1 · y2 + x2 · y1 = n1,u + n1+u,u + 2n1,1+u + 2n1+u,1+u ≡ n1,u + n1+u,u + n1,1+u ≡ 0

(mod 2).

Hence n1,u + n1+u,u + n1,1 + n1+u,1+u ≡ 2n1,1+u ≡ 0 Therefore wL (x + y) ≡ 0

(mod 2) 2

(mod 4).

Theorem 4.4 The Type II codes over R are the codes over R which are permutationequivalent to a code C with a generator matrix of the form: 



I + uB A   k , 0 uD with: (1) The residue code C(1) is even and contains 1, ⊥ (2) C(2) = C(1) ,

(3) B is symmetric and the Lee weights of the k first rows of the generator matrix are a multiple of 4. Proof. This result is a direct consequence of Propositions 4.1, 4.2 and 4.3, the only thing to see is the parity condition. 2 We deduce from the preceding theorem a mass formula for Type II codes: Theorem 4.5 Let NdII (n) be the number of distinct Type II codes of length n and let σ1 (n, k) be the number of distinct even binary codes of length n and dimension k containing 1, then NdII (n) =

X

σ1 (n, k) · 21+

k(k−1) 2

.

k≤ n 2

Proof. Applying Proposition 4.1, for a given even binary code containing 1, there are k(k−1) 21+ 2 possibilities for the matrix B which corresponds to the 1 + k(k−1) liberty degrees 2 for the components of B, namely the k components of the diagonal plus the below-diagonal components except the first row which has a Lee weight a multiple of 4 on the k first rows (except the first one, of course), of the generator matrix. The upper components of the matrix B are obtained by symmetry and we know that the first row has a Lee weight a 10

multiple of 4, since the sum of the first k rows is equal to 1 weight multiple of 4.

(mod u) and so has a Lee 2

Remark. Let us denote by σ0 (n, k) the number of distinct even codes of length n and . The quantity σ1 (n, k), dimension k not containing 1, then from [15]: σ1 (n, k + 1) = σ0 (n,k) 2k is then obtained by induction , knowing that σ1 (n, k) + σ0 (n, k) = σ(n, k) the number of distinct even codes of length n and dimension k in [27]. This proof, although very similar to the proof of the mass formula for Type II codes over Z4 in [15], differs because of the structure of the matrix B and because of the possible residual codes.

5

First Examples

In this section, we give several families of self-dual codes over R. First we introduce the Klemm-like codes defined for every integer n a multiple of 2 as Kn := Rn + uPn where Rn is the repetition code and Pn its dual the parity-check code. Let fn denote the swe of Kn . Clearly fn = 2n−1 bn + ((a + c)n + (a − c)n )/2. K2m is a family of self-dual codes with dL = 4, in particular if m is even then K2m is Type II. K2m is a special case of the following multilevel construction. Proposition 5.1 Let C be a binary self-orthogonal code containing the all-one vector and with weights a multiple of 4. Then C + uC ⊥ is a self-dual Type II code. Proof. If x + uy is the typical codeword then the Gray map tells us that wL (x + uy) = w(y) + w(x + y) = w(x) + 2w(y) + 2w(x ∗ y), with w(y), w(x ∗ y) even and w(x) a multiple of 4.

2

Corollary 5.2 If C is a Type II binary code then C +uC is Type II and φ(C +uC) = C ⊕C. c We now generalize the Klemm-like construction. Let m ≥ 2 be an integer and r ≤ b m−1 2 m another integer. Denote by RM (r, m) the Reed-Muller code of order r and length 2 . Let Cm,r := RM (r, m) + uRM (m − r − 1, m). Corollary 5.3 For m ≥ 2 and r ≤ b m−1 c the code Cm,r is Type II and if m is even and 2 m−2 r = 2 we have φ(Cm,r ) = RM (r + 1, m + 1).

11

Proof. It is well-known that the dual of RM (r, m) is RM (m − 1 − r, m). The congruence condition follows from McEliece Theorem: the largest power of 2 that divides the weight of m−1 a codeword of RM (r, m) is 2b r c [25, p. 447]. The Gray image is a Reed-Muller code when m − r − 1 = r + 1, by the (u| u + v) construction of Reed-Muller codes. 2 Remark. Cm,0 = K2m . We present a class of double circulant self-dual codes. Let D2m be a code of length 2m with generator matrix of the form ( I , uJ + (1 + u)I ) where J is the m by m all-one matrix. Proposition 5.4 D2m is a self-dual code of length 2m with dL = 4 and dH = 2. Proof. The self-duality follows from the form of the generator matrix G. It is not difficult to see that dL = 4 and dH = 2. 2

6

Classification

6.1

Type II Codes of Length up to 8

In this subsection we give a classification for Type II codes of lengths 4 and 8. The method used here is similar to the method used for codes over Z4 in [14], it consists in first considering all the possible residual binary codes and then construct from these binary codes all the associated Type II codes. Then we compute the automorphism group of each code with the same method used for Z4 in [14] adapted in to the case of codes over R. We know the classification is complete when it checks the mass formula given in Theorem 4.5. Table 1 gives the list of the codes together with the order of their automorphism groups and the symmetrized weight enumerators for Lee weights inferior to 8. The generator matrices are given below. The codes in the table are given names according to the binary code generated by the set of the codewords of weight 4 and weight 2 of their residual codes, the letters “a”, “b” and “c” are used to differentiate codes with the same residual code. For instance the label [8, 3] d4 + 2d2 b denotes a code of length 8 whose subcode generated by words of Hamming weight 2 and 4 is “d4 + 2d2 ”. In the classification the code indicated by “∗” means a direct sum A ⊕ B where A ⊕ B = {(a, b) | a ∈ A, b ∈ B}. The notations follow those in [30]. For length 4 the highest minimum Lee weight is 2 and the highest minimum Hamming weight is 4, for length 8 these weights are 4 in both cases. • n = 4:

12

Table 1: Type II Codes of Lengths 4 and 8 Code

Symmetrized Weight Enumerators

Group Order b4 8 4 0 0 16 4 12 24 24 8 12 28

26 · 3 25 214 · 32 · 5 · 7 211 · 32 · 5 213 · 32 211 · 3 211 · 3 211 · 3 211 · 3 211 29 · 3 27 · 3 · 7

[4, 1] d4 (K4 ) [4, 2] 2d2 (D4 ) K8 [8, 2] d2 a ∗[8, 2] 2d4 [8, 3] d4 + 2d2 a [8, 3] d4 + 2d2 b [8, 3] d8 a [8, 4] 4d2 a ∗[8, 4] 4d2 b [8, 4] 4d2 c [8, 4] e8 a

b2 c 0 8 0 12 0 16 8 0 0 16 12 0



c2 6 2 28 16 12 8 8 4 4 4 4 0

b8 0 0 128 64 64 32 32 32 16 16 16 16

b6 c 0 0 0 64 0 64 64 0 64 64 64 0



1 1 1 1    [4, 1] d4 =  0 u 0 u   0 0 u u

b4 c2 0 0 0 0 96 56 40 144 48 80 80 168

b2 c3 0 0 0 40 0 32 48 0 64 32 32 0



c4 1 1 70 30 38 14 14 22 6 6 6 14



1 0 1 u  [4, 2] 2d2 =  0 1 u 1

• n = 8: 

K8 =

             

1 u u u u u u



[8, 2] 2d4 =

          

1 u 0 0 0 0 0 1 0 u u 0 0

1 0 u 0 0 0 0 1 0 u 0 0 0

1 0 0 u 0 0 0 1 0 0 u 0 0

1 0 0 0 u 0 0 1 0 0 0 0 0

0 1 0 0 u u

1 0 0 0 0 u 0 0 1 0 0 u 0

1 0 0 0 0 0 u 0 1 0 0 0 u

1 0 0 0 0 0 0

          [8, 2]      

0 1 0 0 0 0

d2 a

     =     

1 0 u u u u

       [8, 3]     

13

1 0 u 0 0 0

1 0 0 u 0 0



d4 + 2d2 a

    =    

1 0 u u u

1 0 0 0 u 0

1 0 0 u 0

1 0 0 0 0 u

1 u 0 0 u

1 u 0 0 0 0

0 1 0 0 0 0

u 1 0 0 0 0

1 0 0 0 0

0 1 0 0 0

u 1 0 0 0

           

0 0 1 0 0

u 0 1 0 0

         



[8, 3] d4 +2d2 b =



[8, 4] 4d2 a =

     



[8, 4] 4d2 c =

6.2

     

        

1 0 0 u u

1 0 0 u 0

1 0 0 0 u

1 0 0 0 0

0 1 0 0 0

0 1 u 0 0

0 u 1 0 0

1 0 0 u

1 u u 0

u 1 0 u

0 1 u 0

0 u 1 0

u 0 1 u

0 0 u 1

u u 0 1



0 0 1 0

0 u 0 1

0 0 0 1



1 0 0 0

1 u 0 0

u 1 0 0

0 1 u u

0 u 1 0

0 0 1 0 0





     [8, 3]    

    [8, 4]  

d8 a

    =     

   4d2 b =   

    [8, 4] e8 a  

    =  

1 0 0 u u

1 0 0 u 0

1 0 0 0

1 u 0 0

1 0 0 1

1 0 0 0

1 0 1 0 u u 1 0 0

1 1 0 1

1 0 1 0 0 0 1 0 0

1 1 0 0

0 1 0 0 u 0 0 1 0

0 1 1 1

0 1 0 0 0 0 0 1 u

0 1 1 0

0 1 1 0 u 0 0 u 1

0 0 1 1

0 1 1 0 0 0 0 0 1

0 0 1 0

         

      

      

Type I Codes of Length up to 8

In this subsection we classify Type I codes of length up to 8. We recall that Type I codes are self-dual codes which have at least one codeword of Lee weight ≡ 2 (mod 4), so that the lists of Type I and Type II codes, joined together give the list of all self-dual codes of length n ≤ 8. We give all Type I codes which contain no self-dual code A1 with generator matrix (u) as a subcode of length 1. Here again the method used is mostly similar to the method used for self-dual codes over Z4 in [14]. In the following we call admissible, the codes over F2 which do not have a column identically equal to zero and which are self-orthogonal. We first begin by the following lemma. Lemma 6.1 Let C be a self-dual code. If the residual code C(1) of C contains a column identically equal to zero then C contains a subcode which is equivalent to the code A1 and is decomposable. Proof. Without loss of generality we can suppose that the last column of the residual code is zero, then we deduce that the vector (0, 0, . . . , 0, 1) is in the dual of C(1) and therefore (0, 0, . . . , 0, u) is in C, which proves the lemma. 2 Therefore we only have to find all the admissible codes of length n ≤ 8. In order to do this, we list by the mass formula of [27], all the possible binary self-orthogonal codes of length n ≤ 8, from which we deduce in Table 2 all the admissible codes. Binary codes are 14

labeled according to the usual notations of [30], the label 1n stands for the all-one code of length n and the special label f1 stands for the code: 



1 1 1 1 1 1 0 0   . 0 0 1 1 1 1 1 1

Table 2: The Admissible Codes of Length n ≤ 7 Code d2 d4 2d2 16 d4 ⊕ d2 d6 3d2 e7 18 d2 d4 ⊕ d4 f1 d6 ⊕ d2 4d2 e8

Length 2 4 4 6 6 6 6 7 8 8 8 8 8 8 8

Group Order 2 4! 23 6! 24 · 3 24 · 3 24 · 3 23 · 3 · 7 8! 5 2 · 32 · 5 27 · 32 26 · 3 25 · 3 27 · 3 26 · 3 · 7

Now we construct from the possible admissible codes, all the Type I codes associated to a given residual code using a method similar to the Z4 case. Using the list of Type II codes of the preceding subsection and the mass formula in [15] we verify that all the codes have been found. Tables 3 and 4 list all the inequivalent Type I codes over R of length n ≤ 8 without trivial subcode A1 . We give generator matrices of the codes in Table 3. 



1 1 0 0  K2 = (11) 2K2 =  0 0 1 1 

[6, 2] d4 d2 a =

     

1 0 u u

1 0 u 0

1 0 0 u

1 0 0 0

0 1 0 0

0 1 0 0

     [6, 2]  

15

    d4 d2 b =   

1 u u u

1 0 u 0

1 0 0 u

1 0 0 0

0 1 0 0

u 1 0 0

      

Table 3: Type I Codes of Length n ≤ 7 without Trivial Component Code K2 ∗2K2 K6 ∗[6, 2] d4 d2 a [6, 2] d4 d2 b [6, 2] d6 ∗[6, 3] 3d2 a ∗[6, 3] 3d2 b [6, 3] 3d2 c [6, 3] 3d2 d [7, 3] e7



K6 =

Symmetrized Weight Enumerators

Group Order

        

b2 2 4 0 2 0 0 6 2 0 0 0

4 32 5 2 · 6! 28 3 28 3 27 3 27 3 27 27 27 3 24 3 · 7

1 u u u u

1 u 0 0 0

1 0 u 0 0

1 0 0 u 0

1 0 0 0 u

1 0 0 0 0

c 0 0 0 0 0 0 0 0 0 0 0

b4 0 4 0 8 0 12 12 4 4 12 14

b2 c 1 0 0 0 8 0 0 8 8 0 0

c2 0 2 15 7 7 3 3 3 3 3 0

b6 0 0 32 16 16 0 8 8 8 8 0

b4 c 0 0 0 0 16 24 0 16 16 0 42

b2 c2 0 4 0 12 0 0 12 4 8 24 0

c3 0 0 0 0 0 3 0 0 0 0 7

       [6, 2]    



   d6 =   



1 u u u

1 0 u 0

1 1 0 u



1 1 0 0

0 1 0 u

u 1 0 0

      



1 1 0 0 0 0 1 1 0 u 0 0        [6, 3] d2 a =   0 0 1 1 0 0  [6, 3] d2 b =  u 0 1 1 0 0  0 0 0 0 1 1 0 0 0 0 1 1 







1 1 0 u 0 0 1 1 u 0 u 0        [6, 3] 3d2 c =  u 0 1 1 0 u  [6, 3] 3d2 d =  0 u 1 1 u 0   0 0 0 u 1 1 u 0 u 0 1 1 

[7, 3] e7 =

     

1 0 1 0

1 0 0 0

16

1 1 1 0

1 1 0 0

0 1 1 u

0 1 0 u

0 0 1 u

      

Table 4 lists all the codes of length 8 without trivial components. To save space, we only give the first terms of their symmetrized weight enumerator of Lee weight ≤ 6. We remark that there is only one code with both Lee and Hamming weight 4, namely [8, 4] e8 b. The generator matrix is   1 1 1 1 0 u 0 0    0 u 1 1 1 1 0 u    . [8, 4] e8 b =   0 0 0 u 1 1 1 1    1 0 1 0 1 0 1 0

Table 4: Type I Codes of Length 8 without Trivial Component Code [8, 2] d2b ∗[8, 2] 2d4 b [8, 2] f1 ∗[8, 3] d4 + 2d2 c ∗[8, 3] d4 + 2d2 d [8, 3] d4 + 2d2 e [8, 3] d4 + 2d2 f ∗[8, 3] d6 + d2 a [8, 3] d6 + d2 b ∗[8, 3] d6 + d2 c [8, 3] d8 b ∗[8, 4] 4d2 d ∗[8, 4] 4d2 e ∗[8, 4] 4d2 f ∗[8, 4] 4d2 g [8, 4] 4d2 h [8, 4] 4d2 i [8, 4] 4d2 j [8, 4] 4d2 k [8, 4] e8 b

7

Symmetrized Weight Enumerators

Group Order 211 32 5 213 32 211 3 211 3 210 3 211 3 210 3 29 3 28 3 29 3 211 211 3 210 29 29 3 210 29 29 211 27 3

b2 2 0 0 4 2 0 0 2 0 2 0 8 4 2 2 0 0 0 0 0

c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

b4 0 0 4 12 0 4 0 12 4 12 8 24 8 4 12 8 0 4 8 12

b2 c 0 0 0 0 8 0 4 0 4 0 0 0 8 8 0 0 8 4 0 0

c2 16 12 8 8 8 8 8 4 4 4 4 4 4 4 4 4 4 4 4 0

b6 32 0 32 32 16 32 16 24 24 24 0 32 16 16 32 16 16 16 32 0

b4 c 0 64 16 0 32 0 32 24 24 24 64 0 32 32 0 32 32 32 0 64

b2 c2 30 0 0 28 14 32 16 6 8 6 0 24 12 14 30 16 16 16 32 0

c3 0 0 16 0 0 0 0 8 8 8 0 0 0 0 0 0 0 0 0 0

Construction of Type II Binary Codes

In this section, we investigate if all Type II binary codes of length 8n can be constructed from length 4n Type II codes over R, via the Gray-map. For every binary Type II code C 17

of length up to 24, we give a Type II code over R whose Gray map image is C. We show the following result by constructing the desired Type II codes. Proposition 7.1 All Type II binary codes of length 8n can be constructed from length 4n Type II codes over R, via the Gray-map for n ≤ 3. All the notations for Type II binary codes used in this section follow [28] and [30]. For lengths 8 and 24 all the indecomposable Type II binary codes are characterized by their Hamming weight distributions. The methods consist in finding Type II codes over R, which have the same Lee weight distribution as the Type II binary codes. For length 16, there are two Type II binary codes, namely E8 ⊕ E8 and E16 , which have the same weight distribution. The Type II codes over R corresponding are distinguished by the code generated by the codewords of weight 4 of their Gray image, which are respectively E8 ⊕ E8 and E16 . Of course there may be several distinct Type II codes over R corresponding to one given Type II binary code. Table 5 gives a correspondence between Type II codes over R and over F2 . We give generator matrices of inequivalent indecomposable Type II codes over R, which are labeled according to their residual codes. • Length 4: K4 • Length 8: 

[8, 4] 4d2 c =

     

1 0 0 u

1 0 0 0

0 1 0 u

0 1 0 0

0 0 1 u

0 0 1 0

0 0 0 1

u u u 1

      

• Length 12:            

           

[12, 6] b12 a = 1 u 0 0 0 u 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 [12, 6] b12 c = 1 u 0 u 0 u 1 0 0 0 0 0 1 u 0 u 0 u 1 0 0 0 0 0 1 u 0 u 0 0

0 0 0 0 0 1

u 0 u 0 0 1

0 1 1 1 1 1

0 1 1 1 1 1

1 0 1 1 1 1

1 0 1 1 1 1

1 1 0 0 0 1

1 1 0 0 0 1

1 1 0 0 1 0

1 1 0 0 1 0

1 1 0 1 0 0

1 1 0 1 0 0

1 1 1 0 0 0





          

          

1 1 1 0 0 0





          

          

18

[12, 6] b12 b = 1 0 u 0 0 0 1 0 u 0 u 0 1 u 0 0 u u 1 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 1

0 1 1 1 1 1

1 0 1 1 1 1

1 1 0 0 0 1

1 1 0 0 1 0

1 1 0 1 0 0

1 1 1 0 0 0



[12, 6] b12 d = 1 0 0 u 0 0 1 0 0 u 0 0 1 u 0 u 0 u 1 0 0 u 0 0 1 0 0 u 0 u

0 0 u 0 u 1

0 1 1 1 1 1

1 0 1 1 1 1

1 1 0 0 0 1

1 1 0 0 1 0

1 1 0 1 0 0

1 1 1 0 0 0



          

          

[12, 6]  1   0    0    u   0  0

(e8 + 2d2 )a = 0 0 u 0 0 1 0 u 0 0 0 1 0 0 0 u 0 1 0 0 0 0 0 1 0 0 0 0 0 1

0 0 0 1 1 1

0 0 1 0 1 1

0 0 1 1 0 1 

K12 =

                        

1 u u u u u u u u u u

0 0 1 1 1 0 1 u 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0

1 0 0 0 0 0 1 0 u 0 0 0 0 0 0 0 0

[12, 6] (e8 + 2d2 )b =  1 u u u 0 0   u 1 0 0 0 0    u 0 1 u 0 0    u 0 u 1 0 0   0 0 0 0 1 0  0 0 0 0 0 1

           

1 0 0 u 0 0 0 0 0 0 0

1 0 0 0 u 0 0 0 0 0 0

1 0 0 0 0 u 0 0 0 0 0

1 0 0 0 0 0 u 0 0 0 0

1 0 0 0 0 0 0 u 0 0 0

1 0 0 0 0 0 0 0 u 0 0

1 0 0 0 0 0 0 0 0 u 0

1 0 0 0 0 0 0 0 0 0 u

1 0 0 0 0 0 0 0 0 0 0

                         

Table 5: Correspondence Table Type II Codes over R Length Code 4 K4 8 K4 ⊕ K 4 8 K8 12 [12, 6] b12 a [12, 6] (e8 + 2d2 )a 12 12 [12, 6] (e8 + 2d2 )b 12 [12, 6] b12 b 12 K12 12 [12, 6] b12 c 12 [12, 6] b12 d 12 3K4 12 K4 ⊕ K 8

19

Binary Type II Codes Length Code 8 E8 16 E8 ⊕ E8 16 E16 24 A24 24 B24 24 C24 24 D24 24 E24 24 F24 24 Golay 24 3E8 24 E8 ⊕ E16

0 0 0 1 1 1

0 0 1 0 1 1

0 0 1 1 0 1

0 0 1 1 1 0

0 1 0 0 0 0

1 0 0 0 0 0

           

8

Invariants

We let G2 denote the matrix group of order 96 with generators 



1 2 1  1  M =  1 0 −1  , 2 1 −2 1 and





1 0 0   ML =  0  .  0 i 0 0 −1 Theorem 8.1 The symmetrized weight enumerator of a Type II code over R is left invariant by the group G2 of order 96. Proof. The two generators M, ML are respectively the matrix of the MacWilliams relations and the congruence of n1 (x) + 2n2 (x) modulo 4 for a codeword x. The result follows by definition of Type II codes. 2 The group G2 has for its Molien series M4 (t) :=

1 + t8 . (1 − t4 )2 (1 − t12 )

This means a secondary invariant of degree 8 and three primary invariants: two of degree 4 and one of degree 12. Magma computations give as secondary invariant s8 := a8 + 28 a6 c2 + 70 a4 c4 + 28 a2 c6 + 128 b8 + c8 , as primary invariants of degree 4 p41 := (b2 − ac)2 and p42 := 5 a4 + 6 a2 c2 + 48 ab2 c + 16 b4 + 5 c4 , and as the primary invariant of degree 12 p12 := 495 a8 c4 + 7920 a8 b4 + 924 a6 c6 + 126720 a4 b8 + 495 a4 c8 +66 a2 c10 + 126720 b8 c4 + 7920 b4 c8 + 66 a10 c2 + 760320 a2 b8 c2 +1182720 a3 b6 c3 + 66528 a5 b2 c5 + 554400 a4 b4 c4 + 354816 ab6 c5 + 2640 a9 b2 c +31680 a7 b2 c3 + 2640 ab2 c9 + 31680 a3 b2 c7 + 135168 ab10 c + 354816 a5 b6 c +221760 a2 b4 c6 + 221760 a6 b4 c2 + 1025 a12 + 1025 c12 + 4096 b12 . A Taylor series up to order 32 is 1 + 2 t4 + 4 t8 + 7 t12 + 10 t16 + 14 t20 + 19 t24 + 24 t28 + 30 t32 + O(t36 ). 20

Theorem 8.2 The swe of a Type II code over R belongs to the ring R ⊕ s8R where R is either C[p41, p42, p12] or C[f4 , (b2 − ac)2 , f12 ]. Proof. The first statement should be attributed to Magma. To check the algebraic independence of f4 , (b2 − ac)2 , f12 one verifies (in Maple) that the Jacobian of f4 , (b2 − ac)2 , f12 does not vanish identically and concludes by the implicit function theorem. 2 Remark. The degrees of the basic polynomials in the above rings are divisible by four. This gives an alternative proof of Corollary 3.2.

9

Advanced Examples

We shall need a characterization of those binary codes that are Gray images of codes over R. For an arbitrary splitting of coordinate positions into two equal parts define the swap map σ as σ((x|y)) = (y|x). A swap map shall be said to be attached to a Gray map if they both correspond to the same partition of coordinate places. Theorem 9.1 Fix a Gray map. A binary code C of length 2n is the image of an R-code of length n by that Gray map iff it is left wholly invariant by the swap map attached to the Gray map. More generally C is the image of some R-code by some Gray map if it admits a fixed point free involution in its automorphism group. Proof. This is necessary as the R-code has to be left invariant by the map z 7→ (1 + u)z and φ((1 + u)(x + uy)) = σ((x|y)). This is also sufficient since being left invariant by s and therefore by 1 + s is equivalent for a binary code to be an F2 [u] module. The second assertion follows. 2 The following corollary allows us to use results and constructions from [5]. This shows for instance the existence of a Type II R-code of length 32 and minimum Lee weight 12. Note that C 0 and C 00 in [5, Theorem 1] are the residue and torsion codes of the inverse Gray map of C. The code C in [5, Theorem 2] is the Gray image of C 0 + uC 00 . The second code in [5, Section 3] is the Klemm-like code of order k. Corollary 9.2 For a given Gray map the binary code C is the Gray map image of a selfdual code over R iff C is a self-dual code and invariant by the swap map attached to the Gray map. More generally the self-dual code C is the image of some self-dual R-code by some Gray map iff it admits a fixed point free involution in its automorphism group. Proof. The necessity follows from Proposition 3.1 and Theorem 9.1.

21

To prove sufficiency consider a pair of vectors of Rn , z = x + uy and z 0 = x0 + uy 0 and the dot product of their Gray images φ(z)φ(z 0 ) = xx0 + (x0 y + xy 0 ). Letting y = y 0 = 0 yields xx0 = 0 which, in turn, entails xy 0 + x0 y = 0. From the last two equations we get zz 0 = xx0 + u(x0 y + y 0 x) = 0. 2 The above corollary is a useful method for construction of self-dual codes over R with the highest minimum Lee weight (see Tables 10 and 11). Binary extremal self-dual codes in [5] and [6] have such involutions. These codes yield self-dual codes over R with the highest minimum Lee weight (see Section 11). We introduce a class of Type II codes over R using the above corollary, namely Rquadratic residue codes. Let us consider binary extended quadratic residue codes QRp+1 of length p + 1 where p ≡ −1 (mod 8). The codes QRp+1 are binary Type II codes when p ≡ −1 (mod 8). It is known that the group P SL(2, p) acts on QRp+1 and P SL(2, p) has an element x → −1/x, so P SL(2, p) contains a fixed point free involution when p ≡ −1 (mod 8). Corollary 9.3 The inverse Gray map of QRp+1 is a Type II code over R of length p ≡ −1 (mod 8).

p+1 2

when

Such codes over R are called R-quadratic residue codes RQRp+1/2 . For example, QR24 , QR48 and QR80 are binary extremal Type II codes, thus RQR12 , RQR24 and RQR40 are Type II codes over R with the highest minimum Lee weights for that length. Now let us consider double circulant codes. Double circulant codes are divided into two classes, namely pure double circulant and bordered double circulant (cf. [25]). A code with generator matrix ( I , R ) where R is a circulant matrix is called pure double circulant. Corollary 9.4 The inverse Gray map of a binary pure double circulant self-dual code of length 4n is a self-dual code over R. Proof. It is sufficient to show that a binary pure double circulant self-dual code D of length 4n has a fixed point free involution. It follows from the form of the generator matrix that the automorphism group of D contains an element σ = (1, 2, . . . , 2n)(2n + 1, 2n + 2, . . . , 4n). Then σ n is a fixed point free involution. 2 All binary extremal double circulant self-dual codes have been classified in [17] and [21] for length up to 72. P48 , P52,1 , P60 in [21] have a fixed point free involution. Moreover we 22

checked that some double circulant codes have fixed point free involutions, namely C56,1 , P34 , B36 , P38 , P46 , P50,1 in [21], and C66,1 in [17]. The above codes yield self-dual codes with the highest minimum Lee weight. A binary code is said to be t-quasicyclic if it is invariant by the shift operator iterated t times. So a 1-quasicyclic code is a cyclic code. Corollary 9.5 C is the Gray map image of a cyclic code iff C is 2-quasicyclic and invariant by the swap map attached to the Gray map. For instance a class B in [25, p. 507] is a particular class of 2-quasicyclic codes and invariant under the swap map. This class contains Type II codes with parameters [24, 12, 8], [40, 20, 8], [88, 44, 16] and so on. Therefore these codes determine the existence of Type II codes B12 and B20 of lengths 12 and 20 with minimum Lee weight 8. It is a long-standing open question if there is a binary extremal Type II [72, 36, 16] code. As another consequence, we relate some Type II codes over R with the above binary code satisfying a certain condition. Corollary 9.6 Suppose that there is no R-Type II code with dL = 16 of length 36. Then there is no binary extremal Type II [72, 36, 16] code with a fixed point free involution. Binary extremal Type II [72, 36, 16] codes with an automorphism of odd order have been widely studied (cf., e.g. [9], [24] and [29]). The next construction uses combinatorial matrices. Recall that W is a weighing matrix of weight k and order n iff W is a (1, −1, 0)-matrix with W W T = kIn . Corollary 9.7 Let A be an n by n (1, 0)-symmetric matrix satisfying A2 = In . The row R-span of In + (1 + u)A determines a self-dual R-code of length n. In particular if A is a weighing matrix W = (wij ) of weight congruent to −1 (mod 4) then the row R-span of 0 0 = 1 if ) and wij In + (1 + u)W 0 determines a Type II R-code of length n, where W 0 = (wij 0 wij 6= 0 and wij = 0 if wij = 0. Proof. We claim that the row span of ( In , A ) is a [2n, n] self-dual binary code which Q admits the involution ni=1 (i, i + n) as an automorphism. Indeed its generator matrix is ( In , A ) and AT = A shows self-duality while left multiplication by A along with A2 = I shows the existence of the said automorphism. The result follows by Corollary 9.2. 2 There are many symmetric weighing matrices of odd weight. For an infinite family of such matrices of order 4t+2 and weight 5, see [8, p. 304]. The well-known properties of the Jacobsthal matrix yield the codes of Gray image class B yet again. As a small example the

23

matrix



A=

                

1 1 1 1 1 0 0 0

1 1 1 1 0 1 0 0

1 1 1 1 0 0 1 0

1 1 1 1 0 0 0 1

1 0 0 0 1 1 1 1

0 1 0 0 1 1 1 1

0 0 1 0 1 1 1 1

0 0 0 1 1 1 1 1

                 

determines a [16, 8, 4] Type I binary code. This code yields a Type I code with dL = 4. For the definition of a strongly regular graph with parameters (v, k, λ, µ) we shall take the following property of its adjacency matrix A A2 = (k − µ)Iv + (λ − µ)A + µJ, where J is the all-one matrix. Thus ( Iv , A ) generates a self-dual code if and only if k is odd, and both λ and µ are even. This code is Type II iff k = −1 (mod 4). Construction K1 of [23] gives an infinite family of such matrices. The following matrix B is an adjacency matrix of a unique (16, 5, 2, 2) strongly regular graph: 

B=

                                      

0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0

1 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0

1 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0

1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1

1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1

0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1

0 1 0 1 0 0 0 0 0 0 0 1 1 0 0 1

0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0

0 1 0 0 0 1 0 0 0 0 1 1 0 1 0 0

0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1

0 0 1 0 1 0 0 1 0 1 0 0 0 0 1 0

0 0 1 0 0 1 0 1 1 0 0 0 0 1 0 0

0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 0

0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0

0 0 0 0 1 1 1 1 0 0 1 0 0 0 0 0

                    .                   

The matrix ( I16 , B ) generates a Type I [32, 16, 6] code. This binary code yields a Type I code of length 16 with dL = 6. 24

10 10.1

Shadows Symmetrized Weight Enumerators

In [1] the following theorem was proved for Type I codes. Theorem 10.1 The symmetrized weight enumerator of a Type I code is an isobaric polynomial in q1 = a + c,

(1)

q2 = ac − b2 and

(2)

q3 = b2 (a − c)2 ,

(3)

that is X

aj,k q1n−2j−4k q2j q3k .

j,k

Proof. The swe is invariant under a dihedral group of order 8 [2] (and not 6 as in [1, Theorem 4.4]) with generators 



1 2 1  1  M =  1 0 −1  , 2 1 −2 1 and





1 0 0    M4 =  0 −1 0  , 0 0 1 and Molien series

2

1 . (1−t)(1−t2 )(1−t4 )

We now proceed to study the swe of the shadow. We begin with an easy lemma. Lemma 10.2 If C is a Type I code, then the swe of C0 is 1 sweC0 (a, b, c) = (sweC (a, b, c) + sweC (a, ib, −c)). 2 We can now give a simple expression for the swe of the shadow as a function of the swe of the code. Theorem 10.3 The swe of S is related to the swe of C by the relation sweS (a, b, c) = sweC (b + (a + c)/2, i(a − c)/2, b − (a + c)/2)

25

Proof. We proceed as in [11, p. 1323] by computing first by the preceding lemma, the swe of its even weight subcode, then by the MacWilliams relation sweC ⊥ (a, b, c) =

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

the swe of C0⊥ , then, and finally the swe of the shadow by subtracting the swe of C from the swe of C0⊥ . 2 We now give a Gleason-like Theorem for the shadow. Theorem 10.4 If the swe of a Type I code C can be expressed as X

αjk (a + c)n−2j−4k (ac − b2 )j (b2 (a − c)2 )k ,

j,k

then the swe of its shadow is X

2k

αjk (−1)k 2n−2j−6k (b)n−2j−4k (b2 − ac)j (a2 − c2 ) .

j,k

In particular the Lee weight enumerator WL (y) of the shadow is X

α0k (−1)k 2n−6k y n−4k (1 − y 4 )2k .

k

The Lee weights of codewords of S are congruent to n modulo 4. Proof. The first assertion follows from the preceding lemma. The second follows from the first by letting a = 1, c = b2 . The third assertion follows from the second or, alternatively, by the fact that the Gray map of the shadow is the shadow of the Gray image of the code. 2

10.2

Cosets and Extension of Type I Codes

Let s and t be vectors such that C = hC0 , ti and C0⊥ = hC, si. Then C2 = (C0 + t), C1 = (C0 + s) and C3 = (C0 + s + t). Notice that C0⊥ /C0 is the Klein 4-group for all n. Let α = s · s and β = s · t. It is easy to determine the orthogonality relations among Ci ’s if α and β are known, where the result is given in Table 6. The Lee weight of s is n (mod 4), hence 2n2 (s) + n1 (s) ≡ n (mod 4). If n is even this implies that n1 (s) is even and hence α = 0. If n is odd the n1 (s) is odd and there are no self-orthogonal vectors in the shadow, moreover α = 1. 26

Table 6: Orthogonality Relations

C0 C1 C2 C3

C0 0 0 0 0

C1 0 α β α+β

C2 0 β 0 β

C3 0 α+β β α

We note that β cannot be 0 by construction. Notice also that ut is in C0 . Since 2n2 (t) + n1 (t) ≡ 2

(mod 4),

n1 (t) is even and ut has doubly-even Lee weight. We have that ut·s = 0 and then u(t·s) = 0 thus t · s = u. Therefore we have the following lemma. Lemma 10.5 Suppose that C is a self-dual code of length n. Then Table 7 holds where n is odd and Table 8 holds where n is even with the value in position (i, j) is x · y for any vector x ∈ Ci and any vector y ∈ Cj .

Table 7: Orthogonality Relations for n odd

C0 C1 C2 C3

C0 0 0 0 0

C1 0 1 u 1+u

C2 0 u 0 u

C3 0 1+u u 1

Table 8: Orthogonality Relations for n even

C0 C1 C2 C3

C0 0 0 0 0

C1 0 0 u u

C2 0 u 0 u

C3 0 u u 0

Now we give a method of an extension of Type I codes using shadow codes. 27

Theorem 10.6 Suppose that C is a Type I code of odd length n. Let C ∗ be the code of length n + 1 obtained by extending C0⊥ as follows: (0, C0 ), (1, C1 ), (u, C2 ), (1 + u, C3 ). Then C ∗ is a self-dual code of length n + 1. 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)). Proof. It is easy to show that C ∗ is a linear code. The self-orthogonality of the code follows from Lemma 10.5. Since |C ∗ | = 2n+1 , the code is self-dual. A direct calculation determines the swe. 2 Similarly we have the following: Theorem 10.7 Suppose that C is a Type I code of even length n. Let C ∗ be the code of length n + 2 obtained by extending C0⊥ as follows: (0, 0, C0 ), (u, u, C0 ), (u, 0, C2 ), (0, u, C2 ), (1, 1, C1 ), (1+u, 1+u, C1 ), (1, 1+u, C3 ), (1+u, 1, C3 ). Then C ∗ is a self-dual code of length n + 2. 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)). Remark. Some construction techniques of binary self-dual codes using shadows were given in [7]. Theorems 1 and 2 in [7] may be also obtained from Theorems 10.6 and 10.7, respectively, by Proposition 3.8. Thus, similarly to Theorem 2 in [7], Theorem 10.7 can not construct codes with high minimum weight.

10.3

Other Shadows

In Section 2, 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. However, there are other ways to define the shadow so that the shadow is non-trivial for Type II codes. For example, C0 can be chosen so that it consists of the vectors in C that are orthogonal to the vector (u, 0, 0, . . . , 0). Then C0 is of index 2 and the shadow can be defined as before, see [13] for details. The difficulty with this shadow is that its swe cannot be determined from the swe of the code. Another possibility for the shadow would be to let C0 be the subcode of vectors whose Euclidean weights were a multiple of 8, and continue as before. In this case the swe can be determined and is non-trivial unless the code has all Euclidean weights a multiple of 8. This case would mirror the theory for Z4 , [13]. 28

11

Highest Minimum Weights of Small Lengths

In this section, using Theorem 10.4 we proceed as in [11] and compute the possible symmetrized weight enumerators of Type I codes and their shadows. By considering the possible symmetrized weight enumerators and Corollary 3.3, the highest possible minimum weight is determined. We give examples of self-dual codes with the highest minimum Lee weight for that length. These codes determine the exact highest minimum Lee weights.

11.1

Highest Possible Minimum Weights

Most of the methods to determine the possible symmetrized weight enumerators for the highest possible minimum weight are the same as in [11], however a difference is to use the following easy lemma: Lemma 11.1 The coefficient of an−i−j bi cj in the possible symmetrized weight enumerator must be even for i 6= 0. Proof. When i 6= 0 both vectors v and (1 + u)v have identical Lee compositions with v 6= (1 + u)v. 2 As an example, we consider the highest minimum weights of length 7. We denote the possible symmetrized weight enumerators of Type I codes and their shadows by W and S, respectively. The highest possible minimum Lee weight is 4 and there is a family of symmetrized weight enumerators with 3 parameters: W = (β − 2 α) a3 b4 + (β − 2 α) c3 b4 + a7 + c7 + (3 γ + 3 β + 2 α) a2 cb4 + (3 γ + 3 β + 2 α) ac2 b4 + (−6 β + 84 − 3 γ − 2 α) a3 c2 b2 + (−6 β + 84 − 3 γ − 2 α) a2 c3 b2 + (−2 β + 2 α + 28) ac4 b2 + (−2 β + 2 α + 28) a4 cb2 −γ ab6 − γ cb6 + (β − 14) a5 c2 + (−35 + 3 β + γ) a4 c3 + (−35 + 3 β + γ) a3 c4 + (β − 14) a2 c5 S = (224 − 6 γ − 16 β) b5 ac + (−96 + 2 γ + 8 β) b7 + (14 − α) b3 a4 + (14 − α) b3 c4 + (6 γ + 2 α + 8 β − 28) b3 a2 c2 + (−2 γ − 2 α) c3 a3 b + α ba5 c + α bac5 , where |γ| must be even. If the highest possible minimum Hamming weight is 4 then there is a unique symmetrized weight enumerator W, S with α = 0, β = 14 and γ = −7. However −γ must be even by Lemma 11.1, thus the highest possible minimum Hamming weight is 3. We often invoke 29

Lemma 11.1 to eliminate cases. In Section 6, Type I codes of length up to 8 were classified. The code [7, 3] e7 in Table 3 is a unique codes with dL = 4 and dH = 3. The swe is α = γ = 0 and β = 14 in W . In Table 9 we list the highest possible minimum Lee and Hamming weights dL (n) and dH (n), for Type I codes of length 9 ≤ n ≤ 24. We also list the numbers Nn (dL ) and Nn (dH ) of undetermined parameters αjk in the possible symmetrized weight enumerators for the given dL and dH . Table 9: The Highest Possible Minimum Weights of Type I Codes n 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

11.2

dL (n) 4 4 6 6 6 6 6 8 8 8 8 8 8 8 10 12

Nn (dL ) 6 9 6 10 10 14 14 15 15 20 20 26 26 32 27 28

dH (n) 3 3 4 5 5 6 5 7 7 7 8 8 8 8 9 10

Nn (dH ) 5 8 6 7 7 8 11 9 9 14 10 16 16 22 17 19

Self-Dual Codes with the Highest Minimum Lee Weights

In Table 10 (resp. Table 11), we list the exact highest minimum Lee weight dL of Type I codes (resp. Type II codes) for length up to 36 (resp. 40). In the tables, the second column lists the highest minimum Lee weight and the third column gives the codes with the indicated minimum Lee weight for length n. Almost all the self-dual codes with highest minimum Lee weight in the tables are obtained from extremal binary self-dual codes using Corollary 9.2. In particular, recently the first binary extremal Type I [62, 31, 12] code of length 62 has been constructed in [20]. Its automorphism group has been found in [20] and the group contains a fixed point free involution. It is not known if binary Type I codes with parameters [56, 28, 12], [70, 35, 14] and [72, 36, 14] exist. One Type I [70, 35, 12] code is known (cf. [19]) and we checked that its 30

automorphism group is trivial. It is easy to see that the Type I [70, 35, 10] code obtained from the quadratic residue Type II [72, 36, 12] code by subtracting has a fixed point free involution. Thus the highest minimum Lee weight for length 70 is 10, 12 or 14. Question. Construct or prove the non-existence of Type I codes with dL = 12 of length 28 and Type I codes with dL = 14 of lengths 35 and 36. Acknowledgment. The authors would like to thank Akihiro Munemasa for helpful comments.

References [1] C. Bachoc, “Application of coding theory to the construction of modular lattices,” J. Combin. Theory Ser. A vol. 78, (1997), pp. 92–119. [2] C. Bachoc, e-mail to P. Sol´e, June 9, 1997. [3] A. Bonnecaze, P. Sol´e and A.R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices,” IEEE Trans. Inform. Theory vol. IT-41, (1995), pp. 366–377. [4] A. Bonnecaze and P. Udaya, “Cyclic codes and self-dual codes over F2 +uF2 ,” (preprint). [5] S. Buyuklieva, “On the binary self-dual codes with an automorphism of order 2,” Des. Codes and Cryptogr. vol. 12, (1997), pp. 39–48. [6] S. Buyuklieva and I. Boukliev, “Extremal self-dual codes with an automorphism or order 2,” IEEE Trans. Inform. Theory vol. IT-44, (1998), pp. 323–328. [7] R.A. Brualdi and V. Pless, “Weight enumerators of self-dual codes,” IEEE Trans. Inform. Theory vol. IT-37, (1991), pp. 1222–1225. [8] H.C. Chan, C.A. Rodger and J. Seberry, “On inequivalent weighing matrices,” Ars Combin. vol. 21, (1986), pp. 299–333. [9] J.H. Conway and V. Pless, “On primes dividing the group order of a doubly-even (72, 36, 16) code and the group order of a quaternary (24, 12, 10) code,” Discrete Math. vol. 38, (1982), pp. 143–156. [10] J.H. Conway, V. Pless and N.J.A. Sloane, “The binary self-dual codes of length up to 32: a revised enumeration,” J. Combin. Theory Ser. A vol. 60, (1992), pp. 183–195. [11] J.H. Conway, N.J.A. Sloane, “An upper bound on the minimum distance of self-dual codes,” IEEE Trans. Inform. Theory vol. IT-36, (1990), pp. 1319–1333. 31

Table 10: 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 25 26 27 28 29 30 31 32 33 34 35 36

dL 2 2 2 2 2 4 4 4 4 4 6 6 6 6 6 8 6 8 8 8 8 8 10 10 10 10 10 ≤ 12 10 12 12 12 12 12 10, 12 or 14 ≤ 14

References every code every code every code every code every code K6 C7 Section 9 Corollary 9.2, H18 in [28] K10 Corollary 9.2, the twice shortened Golay code Corollary 9.2, Z24 in [30] Corollary 9.2, A26 in [10] Corollary 9.2, A28 in [10] Corollary 9.2, the twice shortened RM (2, 5) [16] Corollary 9.2, P34 in [21] Corollary 9.2, B36 in [21] Corollary 9.2, P38 in [21] Corollary 9.2, [5] Corollary 9.2, [5] Corollary 9.2, [5] Corollary 9.2, P46 in [21] Corollary 9.2, P48 in [21] Corollary 9.2, P50,1 in [21] Corollary 9.2, P52,1 in [21] Corollary 9.2, [6] Corollary 9.2, [6] Corollary 9.2, P60 in [21] Corollary 9.2, C62 in [20] Corollary 9.2, [5] Corollary 9.2, C66,1 in [17] Corollary 9.2, [6]

32

Table 11: The Highest Minimum Lee Weights of Type II codes n 4 8 12 16 20 24 28 32 36 40

dL 4 4 8 8 8 12 12 12 12 or 16 16

References Section 6 Section 6 B12 , Section 9 C4,1 B20 , Section 9 RQR24 Corollary 9.2, C56,1 in [21] Corollary 9.2, [1] RQR36 (dL = 12) RQR40

[12] J.H. Conway and N.J.A. Sloane, “Sphere Packing, Lattices and Groups (2nd ed.),” Springer-Verlag, New York, 1993. [13] S.T. Dougherty, M. Harada, and P. Sol´e, “Shadow codes over Z4 ,” (submitted). [14] J. Fields, P. Gaborit, J. Leon, V. Pless, “All self-dual Z4 codes of length 15 or less are known,” IEEE Trans. Inform. Theory vol. IT-44, (1998), pp. 311–323. [15] P. Gaborit, “Mass formula for self-dual codes over Z4 and Fq + uFq rings,” IEEE Trans. Inform. Theory vol. IT-42, (1996), pp. 1222–1228. [16] T.A. Gulliver, e-mail to M. Harada, November 7, 1997. [17] T.A. Gulliver and M. Harada, “Classification of extremal double circulant self-dual codes of lengths 64 to 72,” Des. Codes and Cryptogr. vol. 13, (1998), pp. 257–269. [18] 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. vol. 29, (1993), pp. 218–222. [19] M. Harada, “The existence of a self-dual [70, 35, 12] code and formally self-dual codes,” Finite Fields Their Appl. vol. 3, (1997), pp. 131–139. [20] M. Harada, “Construction of an extremal self-dual code of length 62,” (submitted). [21] M. Harada, T.A. Gulliver and H. Kaneta, “Classification of extremal double circulant self-dual codes of length up to 62,” Discrete Math., (to appear).

33

[22] M. Harada, P. Sol´e and P. Gaborit, “Self-dual codes over Z4 and unimodular lattices: a survey,” (submitted). [23] X. Hubaut, “Strongly regular graphs,” Discrete Math. vol. 13, (1975), pp. 357–381. [24] W.C. Huffman and V.Y.Yorgov, “A [72, 36, 16] doubly even code dose not have an automorphism of order 7,” IEEE Trans. Inform. Theory vol. IT-33, (1987),pp. 749–752. [25] F.J. MacWilliams and N.J.A. Sloane, “The Theory of Error-Correcting Codes,” NorthHolland, Amsterdam 1977. [26] C.L. Mallows and N.J.A. Sloane, “An upper bound for self-dual codes,” Inform. Control vol. 22, (1973), pp. 188–200. [27] V. Pless, “The number of isotropic subspaces in a finite geometry,” Atti. Accad. Naz. Lincei Rendic vol. 39, (1965), pp. 418–421. [28] V. Pless, “A classification of self-orthogonal codes over GF (2),” Discrete Math. vol. 3, (1972), pp. 209–246. [29] V. Pless, “23 does not divide the order of the group of a (72, 36, 16) doubly even code,” IEEE Trans. Inform. Theory vol. IT-28, (1981), pp. 113–117. [30] V. Pless and N.J.A. Sloane, “On the classification and enumeration of self-dual codes,” J. Combin. Theory Ser. A vol. 18, (1975), pp. 313–335. [31] E.M. Rains, “Shadow bounds for self-dual codes,” IEEE Trans. Inform. Theory vol. IT-44, (1998), pp. 134–139.

34