Higher Weights for Ternary and Quaternary Self-Dual Codes ∗ Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510 USA T. Aaron Gulliver Department of Electrical and Computer Engineering University of Victoria P.O. Box 3055, STN CSC Victoria, BC, Canada V8W 3P6 [email protected] and Manabu Oura Division of Mathematics Sapporo Medical University Sapporo 060 - 8556 Japan [email protected] June 22, 2011

Abstract We study higher weights applied to ternary and quaternary self-dual codes. We give lower bounds on the second higher weight and compute the second higher weights for optimal codes of length less than 24. We relate the joint weight enumerator with the higher weight enumerator and use this relationship to produce Gleason theorems. Graded rings of the higher weight enumerators are also determined. ∗

This work was supported in part by Northern Advancement Center for Science & Technology and the Natural Sciences and Engineering Research Council of Canada.

1

1

Introduction

In this paper we consider the theory of higher weights applied to ternary and quaternary self-dual codes, denoted as Type III and Type IV codes, respectively [13]. Higher weights are generalizations of Hamming weights and are also referred to as generalized Hamming weights and Wei weights ([7], [15], [16]). Type III codes are self-dual codes over F3 , and have the property that the Hamming weight of each vector is divisible by 3. Type IV codes are self-dual codes over F4 with the property that the Hamming weight of each vector is even. For a complete description of self-dual codes see [13]. We shall investigate the natural weight enumerators corresponding to the Hamming weights for these codes and also give bounds on the second higher weight. The higher weights of Type I and Type II codes were studied in [4]. Just as with the general theory of these codes, the study of the Hamming and complete weight enumerators for non-binary codes is significantly different from the binary case. In the binary case one can use the fact that the support of the sum of two vectors is simply those coordinates where one vector has a 1 and the other has a 0, since if the coordinates agree then their sum is 0. This makes the binary case much easier to handle since addition is the exclusive whereas this is no longer true for addition in larger fields. Specifically, it becomes harder to establish bounds on the higher weights for a code.

1.1

Notations and Definitions

Let Fq be a finite field with q elements. A code over Fq is a subset of Fnq . A code is linear if it P is a subspace of Fnq . To the space Fn3 we attach the standard inner-product: [v, w] = vi wi , and for a ternary code C we define C ⊥ = {v ∈ Fn3 | [v, w] = 0 ∀w ∈ C}. We denote the elements of F4 by F4 = {0, 1, ω, ω 2 }. The field F4 has the usual involution, x, with 0 = 0, 1 = 1, ω = ω 2 and ω 2 = ω. To the space Fn4 we attach the Hermitian P inner-product, v · w = vi wi , and for a code C define C ⊥ = {v ∈ Fn4 | v · w = 0 ∀w ∈ C}. If a code C has C ⊆ C ⊥ then we say that C is self-orthogonal, and if C = C ⊥ then C is self-dual. We shall describe the notion of higher weights following the notation in [15]. See this paper (and also [7] and [16]), for a complete description of higher weights. Let D ⊆ Fnq be a linear subspace, then ||D|| = |Supp(D)|, where Supp(D) = {i | ∃v ∈ D, vi 6= 0}. 2

For a linear code C define dr (C) = min{||D|| | D ⊆ C, dim(D) = r}. Notice that the minimum Hamming weight of a code C is d1 (C). It also follows that dk = |Supp(C)| where k is the dimension of the code. Moreover, di < dj when i < j (Proposition 3.1 in [15]). The higher weight spectrum is defined as Ari = |{D ⊆ C | dim(D) = r, ||D|| = i}|. This gives the following higher weight enumerators X (r) (r) WC = WC (y) = Ari y i , i

or in a homogeneous form (r)

(r)

WC = WC (x, y) =

X

Ari xn−i y i .

i

Then for each r ≤ dim(C) " we # have a "weight # enumerator. If C is a code with dimension k k −1)(q k −q)...(q k −q r−1 ) k k (r) over Fq then WC (1) = , where = (q , which is the number of (q r −1)(q r −q)...(q r −q r−1 ) r r subspaces of dimension r in a k dimensional space. Qr−1 s (q − q j ), and for simplicity, we shall sometimes write We adopt the notation [s]r = j=0 0 instead of (0, . . . , 0).

2

Joint Weight Enumerators and Higher Weights (r)

(0)

(1)

(r)

Since WC ⊥ (x, y) involves all of WC (x, y), WC (x, y), . . . , WC (x, y), a straightforward application of invariant theory is not possible. However, we shall use the complete joint enumerator to produce a Gleason’s theorem for the higher weight enumerator. Related techniques were used in [4] to produce a Gleason’s theorem for the binary case. As in the usual Gleason’s theorem the situation becomes much more complicated when the size of the field increases. We begin with some definitions. The g fold complete weight enumerator is defined as X Y xana (v1 ,...,vg ) , JCg (xa ) = JCg (xa : a ∈ Fgq ) = v1 ,...,vg ∈C a∈Fgq

where na (v1 , . . . , vg ) denotes the number of coordinates i such that the vector formed from the i-th position of the g vectors, v1 , v2 , . . . , vg , is the vector a, i.e. a = (v1i , . . . , vgi ). Writing JCg (x, y) instead of JCg (x0 = x, xa = y(a 6= 0)), two sets of homogeneous (r) polynomials WC (x, y)’s and JCg (x, y)’s are related as follows (a similar, but independent, theorem is proven in [4] for the binary case). 3

Theorem 2.1 For C a code over Fq g X (r) [g]r WC (x, y). =

JCg (x, y)

(1)

r=0

Proof. We have X

JCg (x, y) =

xn0 (v1 ,...,vg ) y n−n0 (v1 ,...,vg )

v1 ,...,vg ∈C

=

g X

X

r=0

xn0 (v1 ,...,vg ) y n−n0 (v1 ,...,vg )

v1 ,...,vg ∈C

dimhv1 ,...,vg i=r

=

g X X r=0

]{(v1 , . . . , vg ) ∈ C g ; hv1 , . . . , vg i = D}

D⊂C

dim D=r n0 (v1 ,...,vg ) n−n0 (v1 ,...,vg )

×x

y

.

As in [4], we can show that the number ]{(v1 , . . . , vg ) ∈ C g ; hv1 , . . . , vg i = D} is equal to [g]r . Hence we have JCg (x, y)

=

g X X r=0

=

g X

D⊂C

dim D=r

=

X

[g]r

r=0 g X

[g]r xn0 (v1 ,...,vg ) y n−n0 (v1 ,...,vg )

xn−kDk y kDk

D⊂C

dim D=r (r)

[g]r WC (x, y).

r=0

2

This completes the proof of Theorem 2.1.

The MacWilliams type identities for higher weights, proven in [4] is a direct corollary of this theorem. The MacWilliams relations are given with a different proof in [7], [15]. Let F be a subspace of Fgq of dimension g − 1. Define the map ψ as follows. For a ∈ Fgq , if a = 0 then ψ(xa ) = x, if a ∈ F then ψ(xa ) = y and if a ∈ / F then ψ(xa ) = 0. For Xα = (xα1 , xα2 , . . . , xαqg ) we let ψ(Xa ) denote replacing each variable xα with ψ(xα ). For F a subspace of Fgq define the following characteristic function ΨF (v1 , ..., vg ): ( ΨF (v1 , ..., vg ) =

1, 0,

if (v1i , ..., vgi ) ∈ F , for any i otherwise.

4

Theorem 2.2 Let C be a code over Fq of length n. The following identity holds [g]g WCg (x, y) = JCg (x, y) −

JCg (ψ(Xa )) +

X g F ⊂Fq

dim F =g−1





X

X

ΨF (v1 , ..., vg ) − 1 xn0 (v1 ,...vg ) y n−n0 (v1 ,...vg ) .

 v1 ,...vg ∈C

F ⊂Fgq : dim F =g−1

Proof. Theorem 2.2 follows from (1) and g−1 X

(2)

X

(r)

[g]r WC (x, y) =

JCg (ψ(Xa ))−

g F ⊂Fq

r=0

dim F =g−1





X

X

ΨF (v1 , ..., vg ) − 1 xn0 (v1 ,...vg ) y n−n0 (v1 ,...vg ) .

 v1 ,...vg ∈C

F ⊂Fgq : dim F =g−1

We shall prove (2). We have LHS of (2) =

g−1 X

X

r=0

xn0 (v1 ,...,vg ) y n−n0 (v1 ,...,vg )

v1 ,...,vg ∈C

dimhv1 ,...,vg i=r

X

=

xn0 (v1 ,...,vg ) y n−n0 (v1 ,...,vg ) .

v1 ,...,vg ∈C

0≤dimhv1 ,...,vg i≤g−1

and on the other hand X

X

g F ⊂Fq

v1 ,...,vg ∈C

RHS of (2) =

xn0 (v1 ,...,vg ) y n−n0 (v1 ,...,vg )

dim F =g−1 PF (v1 ,...,vg )

 −

X

 X

 v1 ,...vg ∈C

ΨF (v1 , ..., vg ) − 1 xn0 (v1 ,...vg ) y n−n0 (v1 ,...vg ) .

F ⊂Fgq : dim F =g−1

Using these, we check the summation. The element (v1 , . . . , vg ) ∈ C g with dimhv1 , . . . , vg i ≤ g − 1 is counted once and only once in the summation of the RHS of (2). Hence we have the equality RHS of (2) = LHS of (2). 2

This completes the proof of (2).

In the case when g = 2, the last sum of Theorem 2.2 is qxn and we have X (2) 1 (2) (2) (JC (x, y) − JC (ψ(Xa )) + qxn ). WC (x, y) = [2]2 2 F ⊂Fq

dim F =1

5

Note that [2]2 = (q 2 − 1)(q 2 − q)   if q = 2, 6  = 48 if q = 3,    180 if q = 4. Order the elements of F2q lexicographically, for example in F23 the elements are ordered 00, 01, 02, 10, 11, 12, 20, 21, 22. For ternary codes this gives: Corollary 2.3 Let C be a ternary code then (2)

1 (2) (2) (J (1, y, y, y, y, y, y, y, y) − JC (1, 0, 0, y, 0, 0, y, 0, 0) 48 C (2) (2) − JC (1, y, y, 0, 0, 0, 0, 0, 0) − JC (1, 0, 0, 0, y, 0, 0, 0, y)

WC (y) =

(2)

− JC (1, 0, 0, 0, 0, y, 0, y, 0) + 3). For quaternary codes this gives: Corollary 2.4 Let C be a quaternary code then (2)

1 (2) (J (1, y, y, y, y, y, y, y, y, y, y, y, y, y, y, y) 180 C (2) − JC (1, y, y, y, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

WC (y) =

(2)

− JC (1, 0, 0, 0, y, 0, 0, 0, y, 0, 0, 0, y, 0, 0, 0) (2)

− JC (1, 0, 0, 0, 0, y, 0, 0, 0, 0, y, 0, 0, 0, 0, y) (2)

− JC (1, 0, 0, 0, 0, 0, y, 0, 0, 0, 0, y, 0, y, 0, 0) (2)

− JC (1, 0, 0, 0, 0, 0, 0, y, 0, y, 0, 0, 0, 0, y, 0) + 4).

3

Type III and Type IV Codes

In this section, we consider the higher weights of ternary and quaternary self-dual codes. (0) (1) Example 1: Let C be the [8, 4, 3] ternary code t24 [13]. Then WC = 1, WC = 8y 3 +32y 6 , (2) (3) (4) WC = 2y 4 + 16y 6 + 64y 7 + 48y 8 , WC = 8y 7 + 32y 8 and WC = y 8 . Table 3 gives d2 and d3 for ternary self-dual codes up to length 24. In particular, values are given for all ternary self-dual codes up to length 16, and known maximal length codes up to length 24. See [1], [11], and [12] for notation and a description of the codes. From Corollary 1 in [6] we have

6

Theorem 3.1 d2 ≥

q+1 d1 . q

Thus if C is a ternary self-dual code with d1 = 3g, then d2 ≥ 34 d = 4g. In the special case where wt(v) = wt(w) = wt(v + w) = 3g = wt(2v + w) = d1 we have d2 = 4g. Very often when there are a significant number of minimum weight vectors this occurs (see Table 3). However, if there are few minimum weight vectors, the bound can be exceeded. For example the code (e3 p13 )+ (which has only q − 1 = 2 minimum weight codewords) has d1 = 3 and d2 = 7. (0) (1) Example 2: Let C be the [8, 4, 4] quaternary code e8 [13]. Then WC = 1, WC = (2) (3) (4) 14y 4 + 56y 6 + 15y 8 , WC = 28y 6 + 112y 7 + 217y 8 , WC = 8y 7 + 77y 8 and WC = y 8 . Table 4 gives d2 for quaternary self-dual codes up to length 24. In particular, values are given for all maximal quaternary self-dual codes up to length 18, and several known maximal codes up to length 24. See [1], [5] and [9] for notation and a description of the codes. From the previous theorem, if C is a quaternary self-dual code, then d2 ≥ 45 d. This result is not as clean as that for ternary codes because for the bound to be met d must be divisible by 4.

4

Invariants and Gleason Type Theorems

Let ξ be a complex cubic root of unity and define   1 1 1   T3 =  1 ξ ξ 2  , 1 ξ2 ξ and



 1 1 1 1 1 1 −1 −1   . 1 −1 1 −1  1 −1 −1 1

  T4 =  

The following MacWilliams relations are well known [10]. Let C1 , C2 , . . . , Cg be codes in Fqn and let C˜ denote either C or C ⊥ . Then JC˜1 ,C˜2 ,...,C˜g (Xa ) = Qg

i=1

where

( δC˜ =

1 δC˜

|Ci | 0 1

· (⊗gi=1 T δC˜ )JC1 ,...,Cg (Xa ),

i

if if

C˜ = C, C˜ = C ⊥ ,

T is T3 if q = 3 and T is T4 if q = 4, and ⊗ represents the tensor product. 7

Let S3 = √13 T3 , S4 = 12 T4 and I` denote the ` by ` identity matrix. If C is a ternary self-dual code then JCg (Xa ) is held invariant by the 3g by 3g matrix δ (i)

MA3 = ⊗gi=1 S3A , where

( δA (i) =

1 0

if i ∈ A . if i ∈ /A

For each A contained in {1, 2, . . . , g} we have a matrix that holds the weight enumerator invariant. Denote the set of all 2g such matrices MA3 , A ⊆ {1, 2, . . . , g}, by Ω31 . In addition, √ JCg (Xa ) is held invariant by iI3g since the length of the code is divisible by 4, where i = −1. Since the weight of every vector in a ternary self-dual code is divisible by 3, JCg (Xa ) is held invariant by the diagonal matrix KB3 where ( ξ if i ∈ B 3 (KB )ii = , 0 if i ∈ /B and (KB )ij = 0 if i 6= j, where B runs over all subsets of {1, 2, . . . , g} that give the weight of a vector, that is J(χB (1), . . . , χB (3g − 1)) = wt(v) for some vector in C. Denote the set of all such matrices KB3 by Ω32 . If C is a self-dual code over F4 then JCg (Xa ) is held invariant by δ (i)

MA4 = ⊗gi=1 S4A , where

( δA (i) =

1 0

if i ∈ A , if i ∈ /A

and A runs over all subsets of {1, 2, . . . , g}. Denote the set of all such matrices MA4 by Ω41 . In addition, JCg (Xa ) is held invariant by −I4g since the length of the code is divisible by 2. Since the weight of every vector in a Type IV code is divisible by 2, JCg (Xa ) is held invariant by the diagonal matrix KB4 where ( −1 if i ∈ B 4 (KB )ii = , 0 if i ∈ /B and (KB )ij = 0 if i 6= j, where B runs over all subsets of {1, 2, . . . , g} that give the weight of a vector, that is J(χB (1), . . . , χB (4g − 1)) = wt(v) for some vector in C. Denote the set of all such matrices KB4 by Ω42 . We define the following groups by giving their generators: Gg3 = hΩ31 , Ω32 , iI3g i 8

and Gg4 = hΩ41 , Ω42 , iI4g i. Let R3g be the ring of invariants for Gg3 , that is R3g is the ring of polynomials consisting of all polynomials held invariant by each matrix in Gg3 , and R4g be the ring of invariants for Gg4 , that is R4g is the ring of polynomials consisting of all polynomials held invariant by each matrix in Gg4 . Then we have the following theorems. Theorem 4.1 Let C be a ternary self-dual code. Then WC2 (y) is of the form 1 48

(JC2 (1, y, y, y, y, y, y, y, y) − JC2 (1, 0, 0, y, 0, 0, y, 0, 0)

− JC2 (1, y, y, 0, 0, 0, 0, 0, 0) − JC2 (1, 0, 0, 0, y, 0, 0, 0, y) − JC2 (1, 0, 0, 0, 0, y, 0, y, 0) + 3), where JC2 is an element of R32 . 2

Proof. Follows from Theorem 2.3.

(2)

Theorem 4.2 Let C be a quaternary self-dual code. Then WC (y) is of the form 1 180

(JC2 (1, y, y, y, y, y, y, y, y, y, y, y, y, y, y, y)

−

JC2 (1, y, y, y, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

−

JC2 (1, 0, 0, 0, y, 0, 0, 0, y, 0, 0, 0, y, 0, 0, 0)

−

JC2 (1, 0, 0, 0, 0, y, 0, 0, 0, 0, y, 0, 0, 0, 0, y)

−

JC2 (1, 0, 0, 0, 0, 0, y, 0, 0, 0, 0, y, 0, y, 0, 0)

−

JC2 (1, 0, 0, 0, 0, 0, 0, y, 0, y, 0, 0, 0, 0, y, 0) + 4)

where JC2 is an element of R42 . 2

Proof. Follows from Theorem 2.4. Let 1 Tqg = √ t q

1 qg − 1 1 −1

! .

The fact that this matrix holds JCg (x, y) invariant for a self-dual code follows directly by collapsing variables of the MacWilliams relation for the complete joint weight enumerator. If C is a ternary self-dual code then JCg (x, y) is held invariant by T3g and by iI2 since the length is divisible by 4. If C is a quaternary self-dual code then JCg (x, y) is held invariant by T4g and by −I2 since the length is divisible by 2. Define the following groups: 9

G3g = hT3g , iI2 i, and G4g = hT4g , −I2 i. Let Rg3 be the ring of invariants for G3g and Rg4 be the ring of invariants for G4g . Given P (r) the relationship stated in Theorem 2.1, namely that JCg (x, y) = gr=0 [g]r WC (x, y), we have the following theorem. Theorem 4.3 Let C be a code over Fq with q = 3 or 4, then g X (r) [g]r WC (x, y) ∈ Rkg , r=0

where k = 3 for ternary codes and k = 4 for quaternary codes.

5

Rings of the Higher Weight Enumerators (r)

Let Wg be a graded ring generated by the higher weight enumerators WC (x, y) (0 ≤ r ≤ g) Fq of self-dual codes C over Fq . In this section, we determine the structures of Wg for q = 3 Fq and 4. Theorem 5.1 We have W

(0)

W

(1)

F3 F3

= C[x4 ],
= C[x4 , y 12 ] 1 ⊕ xy 3 ⊕ (xy 3 )2 ⊕ (xy 3 )3 . (0)

Proof. The first assertion follows from the fact that WC (x, y) = xn for any code C and that a self-dual ternary code exists if and only if n ≡ 0 (mod 4). (1) We now prove the second assertion. For any ternary self-dual C, JC (x, y) can be written as a polynomial of x4 + 8xy 3 and x12 + 264x6 y 6 + 440x3 y 9 + 24y 12 (Gleason, see Theorem 28 (1) in [2], p. 202). From this fact, we know that W is a finitely generated ring by x4 , xy 3 , y 12 F3 over C. Using the Gr¨obner basis (cf. [3], [14]), we have W

(1)

F3

= C[x4 , xy 3 , y 12 ]
= C[x4 , y 12 ] 1 ⊕ xy 3 ⊕ (xy 3 )2 ⊕ (xy 3 )3 .

This completes the proof of Theorem 5.1. In order to determine the ring Wg , g ≥ 2, we need two lemmas. F3 10

2

Lemma 5.2 For any g ≥ 1 and for any ternary self-dual code C, we have Ag1 = Ag2 = Ag5 = 0. Proof. Since the weight of any element of a ternary self-dual code is divisible by three, we have Ag1 (C) = Ag2 (C) = A15 (C) = 0 for any g ≥ 1. We now show that Ag5 (C) = 0 for g ≥ 2. For g = 2, the weight of two linearly independent elements is four or five and this cannot occur. For g ≥ 3, we take three linearly independent elements which have the following forms: v1 = (∗, ∗, ∗, 0, 0, . . .), v2 = (0, ∗, ∗, ∗, 0, . . .), v3 = (0, ∗, ∗, 0, ∗, . . .), where ∗ denotes a nonzero element of F3 . Since the weight of the sum v1 + v2 is divisible by three, we may put v1 = (∗, a, b, 0, 0, . . .), v2 = (0, −a, c, ∗, 0, . . .), where a 6= 0, b + c 6= 0. The same argument applies to the sum v2 + v3 and v1 + v3 and we have three cases: Case 1: v3 = (0, a, −b, 0, ∗, . . .). The weight of v1 + v2 + v3 = (∗, a, c, ∗, ∗, . . .) is five. This cannot occur since the weight is divisible by three because of self-duality. Case 2: v3 = (0, −a, −c, 0, ∗, . . .). The weight of v1 + v2 + v3 = (∗, −a, b, ∗, ∗, . . .) is five. This cannot occur. Case 3: v3 = (0, d, −b, 0, ∗, . . .), a + d 6= 0, b = c. The weight of v1 + v2 + v3 = (∗, d, b, ∗, ∗, . . .) is five. This cannot occur. This completes the proof of Lemma 5.2. 2

Lemma 5.3 Any element of the set {0, 3, 4, l ∈ Z≥6 } can be written once and only once in the form 4a + 3b, a ∈ Z≥0 , b ∈ {0, 1, 2, 3}. 2

Proof. Trivial.

Theorem 5.4 For g ≥ 2, we have
Wg = C[x4 , y 4 ] 1 ⊕ xy 3 ⊕ (xy 3 )2 ⊕ (xy 3 )3 . F3

11

(1)

(1)

Proof. The ternary tetracode t4 [13] has Wt4 (x, y) = y 4 . Hence W contains the ring F3 C[x4 , y 4 , xy 3 ] which can be written as (3)


C[x4 , y 4 , xy 3 ] = C[x4 , y 4 ] 1 ⊕ xy 3 ⊕ (xy 3 )2 ⊕ (xy 3 )3 .

In order to prove the equality of the two rings Wg and C[x4 , y 4 , xy 3 ], it suffices to show that, F3 for any ternary self-dual code C of length n = 4c, any term xn−i y i whose coefficient Agi does not vanish is contained in the right hand side of (3). By Lemma 5.2, we have i 6= 1, 2, 5. In this case, there exists unique a ∈ Z≥0 and b ∈ {0, 1, 2, 3} such that i = 4a + 3b. Hence xn−i y i = x4c−(4a+3b) y 4a+3b = (x4 )c−a−b (y 4 )a (xy 3 )b , and this is contained in the right hand side of (3). This completes the proof of Theorem 5.4. 2 Similar results hold for Wg . F4 Theorem 5.5 We have (0)

W = C[x2 ], F4 (1) W = C[x2 , y 2 ]. F4 For g ≥ 2, we have Wg = C[x2 , y 2 ](1 ⊕ xy 5 ). F4 (0)

Proof. The first assertion follows from the fact that WC (x, y) = xn for any code C and that a self-dual quaternary code exists if and only if n ≡ 0 (mod 2). (1) We now prove the second assertion. For any quaternary self-dual C, JC (x, y) can be written as a polynomial of x2 + 3y 2 and x6 + 45x2 y 4 + 18y 6 (Theorem 30 in [2], p. 203). (1) From this fact, we know that W is a finitely generated ring in x2 , y 2 over C and the result F4 follows. (2) Finally, we show the case for g ≥ 2. The hexacode h6 [13] has Wh6 (x, y) = 6xy 5 + 15y 6 . Hence Wg contains C[x2 , y 2 , xy 5 ], which can be written as F4 (4)

C[x2 , y 2 , xy 5 ] = C[x2 , y 2 ](1 ⊕ xy 5 ).

In order to prove the equality of the two rings Wg and C[x2 , y 2 , xy 5 ], we need two claims. F4 Claim 1: Ag1 = Ag3 = 0 for any g ≥ 1 and for any quaternary self-dual code. 12

Claim 2: Any element of the set {0, 2, l ∈ Z≥4 } can be written once and only once in the form 2a + 5b, a ∈ Z≥0 , b ∈ {0, 1}. These claims correspond to Lemmas 5.2, 5.3 in the F3 case, respectively, and so we omit the proofs. Now for any quaternary self-dual code C of length n = 2c, take any term xn−i y i whose coefficient Agi does not vanish. By Claim 1, we have that i 6= 1, 3. In this case, there exists unique a ∈ Z≥0 and b ∈ {0, 1} such that i = 2a + 5b. Hence we have xn−i y i = x2c−(2a+5b) y 2a+5b = (x2 )c−a−3b (y 2 )a (xy 5 )b , and this is contained in the right hand side of (4). This completes the proof of Theorem 5.5.

2

Acknowledgment: We are grateful to the anonymous referee for their careful reading and helpful suggestions.

References [1] J.H. Conway, V. Pless and N.J.A. Sloane, “Self-dual codes over GF(3) and GF(4) of length not exceeding 16,” IEEE Trans. Inform. Theory, vol. 25, pp. 312–322, 1979. [2] J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, 2nd Ed., Springer, New York, 1988. [3] D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd Ed., Springer, New York, 1997. [4] S.T. Dougherty, T.A. Gulliver and M. Oura, “Higher weights and graded rings for binary self-dual codes,” Discrete Appl. Math., (to appear). [5] T.A. Gulliver, “Optimal double circulant self-dual codes over GF(4),” IEEE Trans. Inform. Theory, vol. 46, pp. 271–274, Jan. 2000. [6] T. Helleseth, T. Kløve, V.I. Levenshtein and Ø. Ytrehus, “Bounds on the minimum support weights,” IEEE Trans. Inform. Theory, vol. 41, pp. 432–440, Mar. 1995. [7] T. Kløve, “Support weight distributions of linear codes,” Discrete Math., vol. 106/107, pp. 311–316, 1992.

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[8] F.J. MacWilliams, C.L. Mallows and N.J.A. Sloane, “Generalizations of Gleason’s theorem on weight enumerators of self-dual codes,” IEEE Trans. Inform. Theory, vol. 18, pp. 794–805, 1972. [9] F.J. MacWilliams, A.M. Odlyzko and N.J.A. Sloane, “Self-dual codes over GF(4),” J. Combin. Theory Ser. A, vol. 25, pp. 288–318, 1978. [10] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-correcting Codes, NorthHolland, Amsterdam, 1977. [11] C.L. Mallows, V. Pless, and N.J.A. Sloane, “Self-dual codes over GF (3),” SIAM J. Appl. Math, vol. 31, pp. 649–666, 1976. [12] V. Pless, N.J.A. Sloane, and H.N. Ward, “Ternary codes of minimum weight 6, and the classification of the self-dual codes of length 20,” IEEE Trans. Inform. Theory, vol. 26, pp. 305–316, May 1980. [13] E.M. Rains and N.J.A. Sloane, Self-Dual Codes, in V. S. Pless and W. C. Huffman (Eds.), Handbook of Coding Theory, Elsevier, Amsterdam, pp. 177–294, 1998. [14] B. Sturmfels, Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, New York, 1993. [15] M.A. Tsfasman and S.G. Vladut, “Geometric approach to higher weights,” IEEE Trans. Inform. Theory, vol. 41, pp. 1564–1588, 1995. [16] V.K. Wei, “Generalized Hamming weights for linear codes,” IEEE Trans. Inform. Theory, vol. 37, pp. 1412-1418, 1991.

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Table 1: Higher Weight Enumerators for the [12, 6, 6] Golay Code W1 132 220

12

W2

W3

W4

W5

W6

495 880 2970 3960 2706

220 1980 9900 21780

66 1320 9625

12 352

1

weight i 6 8 9 10 11 12

Table 2: The Genus 6 Weight Enumerator for a the [12, 6, 6] Golay Code coefficient of y i 1 96096 261621360 84184100000 18386225938080 2433667533897600 147642497091345984

15

weight i 0 6 8 9 10 11 12

Table 3: Ternary Self-Dual Codes with n ≤ 24 n Code 4 t4 8 t24 12 e4+ 3 12 e34 12 g12 16 e44 16 (e43 e4 )+ 16 g12 e4 16 (e43 f4 )+ 16 (e23 g10 )+ 16 (e3 p13 )+ 16 f82+ 20 10f2 20 4f4 + 2f2 20 5f4 20 4f5 20 2g9 + f2 20 2f10 24 XQ23 24 S(24)

16

d1 3 3 3 3 6 3 3 3 3 3 3 6 6 6 6 6 6 6 9 9

d2 d3 4 4 7 6 8 4 7 8 9 4 7 4 7 4 9 6 8 6 9 7 9 8 10 8 11 8 10 8 10 8 11 8 9 8 9 12 14 12 14

Table 4: Quaternary Self-Dual Codes with n ≤ 24 n Code 4 i22 6 h6 8 e8 10 d+ 10 10 e2+ 5 12 d+ 12 12 (e7 e5 )+ 12 d2+ 6 12 d3+ 4 14 d+ 14 2+ 14 e7 14 (d8 e5 f1 )+ 14 (e25 f1 )+ 14 (d8 d6 )+ 14 (d26 f2 )+ 14 (d6 d24 )+ 14 (d34 f2 )+ 14 (d24 16 )+ 14 q14 16 f82+ 16 (f52 16 )+ 16 1+ 16 4+ 16 f4 18 S18 20 C20 22 C22 24 C24,1 24 C24,2 24 C24,3 24 C24,4

17

d1 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 8 8 8

d2 4 5 6 5 6 6 5 6 7 6 6 5 5 6 6 6 7 7 8 8 8 8 8 10 10 10 12 10 11 10