All Self-Dual Z_{4} Codes of Length 15 or Less Are Known

Joe Fields†, Philippe Gaborit‡, Jeffrey S. Leon, Vera Pless

Abstract

We classify the self-dual ℤ_{4} codes of all lengths from 10 through 15, extending the previous classification through length 9 [J.H. Conway, N.J.A. Sloane, Self-Dual Codes over the Integers Modulo 4, J. Comb. Theory A 62 (1993), 31-45].

Keywords

Coding theory, Self-dual codes, Codes over rings, Classification.

I. Introduction

In [2] Conway and Sloane began the classification of self-dual ℤ_{4} codes. They classified all the codes up to length 9 without a “mass-formula”. This formula was found by Gaborit [4] for both self-dual and Type II codes. Using this and an automorphism group program for ℤ_{4} codes, Pless, Leon and Fields [9] classified all of the Type II codes of length 16 (the next possible length after 8 where these codes exist). At the same time, Gaborit was pushing forward with the classification begun by Conway and Sloane. An unexpected consequence of the classification of the length 16 Type II codes was the determination of all of the length 15 self-dual codes by shortening the length 16 codes. Furthermore, all of the codes of lengths ≤ 14 appear as direct summands in the list of all length 15 codes.

In this paper we present both approaches to the classification of ℤ_{4} self-dual codes. In section 3, “The Bottom-Up Approach”, the process of finding all of the self-dual ℤ_{4} codes by actually constructing them, is described. In section 4, “The Top-Down Approach”, we discuss how the codes can be found by shortening the length 16 codes. Both methods are verified by orbit counting using the mass formula of [4]. The agreement of these two distinct approaches serves to confirm our results.

Appendix I lists all of the indecomposable codes of length 10 through 15, and gives the orders of their groups and partial symmetrized weight enumerators. The highest minimum Lee weight of a self-dual code of length 15 is 6. We give generator matrices for all such codes. Also, generator matrices for the indecomposable codes of lengths 10 and 11 are given. The highest minimum Lee weight attained in lengths 10 through 13 (and, with one exception, at length 14) is 4. Generator matrices for all of the codes are available on the world wide web at http://www.math.uic.edu/~pless.

II. Preliminaries

It follows from the discussion in [9] that a self-dual ℤ_{4} code C of length n has a generator matrix of the form:

G = \left[{A (I_{k} + 2B) \atop 2D}\right]  (1)

where A, B and D are binary matrices. Further,

G_{1} = [ A I_{k} ]

†Fields, Leon and Pless are with the University of Illinois at Chicago

‡Gaborit is with Université Bordeaux I, he thanks Christine Bachoc for many helpful suggestions and Gabriele Nebe for her advice in programming

generates a doubly-even binary [n, k] code C_{1} and

G_{2} = \left[{A I_{k} \atop D}\right]

generates C_{2} = C_{1}^{⊥}.

In this situation C is said to have type 4^{k}2^{n−2k} and indeed contains this many codewords.

The symmetrized weight enumerator of a ℤ_{4} code is a polynomial in two variables b and c such that the coefficient of b^{i}c^{j} is the number of vectors in the code having exactly i 1’s and 3’s and j 2’s.

Two ℤ_{4} codes are equivalent if one can be obtained from the other by a permutation of the coordinates possibly followed by multipying some coordinates by 3 (sign changes). The group G(C) of a ℤ_{4} code C is the set of all coordinate permutations and sign changes that send the code onto itself. We denote by M_{n}, the monomial group over ℤ_{4}, which is the set of all sign-change permutations of n coordinates; it has order n!2^{n}. The index |M_{n}|/|G(C)| gives the number of codes equivalent to C.

We call a doubly-even binary code admissible if it contains the all-ones vector. From a given admissible code, one can construct ℤ_{4} codes for which the admissible code plays the role of C_{1}.

We label each binary doubly-even code with its length, dimension and a description of the subcodes generated by weight 4 vectors, using the standard notation d_{2m}, e_{7} and e_{8}. Additionally, we use the label f in situations where C_{1} does not contain any weight 4 vectors. To avoid confusing the binary and ℤ_{4} codes, we use a typographic convention: the labels of ℤ_{4} codes are rendered in bold face.

Note that it is possible for doubly-even codes of different dimensions to have similar labels since the labels make no mention of ’glue’ vectors. Thus a length 15 doubly-even code with the label [15, 4]−2d_{6} has a generator matrix equivalent to the following:

[ \begin{matrix}1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \end{matrix} ]

Whereas the doubly-even code with the label [15, 5] − 2d_{6} has the following generator matrix:

[ \begin{matrix}1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 \end{matrix} ]

These labels completely distinguish the inequivalent binary doubly-even codes of lengths 10 through 16, however there are often inequivalent ℤ_{4} self-dual codes arising from the same C_{1}. These are further distinguished by appending a, b, … to their labels.

If we have a length n ℤ_{4} code C, then the set of vectors in C which are either 0 or 2 on a particular coordinate is a subcode C′. Removing that coordinate from C′ gives a code of length n − 1 which is called the shortened code of C on that coordinate.

Lemma 1: If C is a Type II ℤ_{4} code of type 4^{k}2^{n−2k} then shortening on any coordinate gives a self-dual code of type 4^{k−1}2^{n−2k+1}.

Proof: If we are shortening C on the jth coordinate, denote the shortened code by S_{j}(C). Let us write h̃ for a vector consisting of all ones and threes. Since C is a Type II code, it contains a vector h̃. Thus for every coordinate position of C there are vectors that have 1’s in that coordinate. Choose a basis for C such that exactly one basis vector has a 1 in the jth column, and all other basis vectors have 0’s or 2’s there. Without loss of generality, this vector is h̃.

In some sense, the subcode C′ (the subcode which has only 0’s and 2’s in the jth coordinate) is of “co-dimension” 1. All of the other vectors (besides h̃) in the basis for C are in C′. Since we are considering a spanning set for C the vector 2h̃ (i.e. the all-twos vector) is not in their span. To say that C has type 4^{k}2^{n−2k}, is to assert that every basis for C contains at least k vectors containing 1’s and 3’s. Clearly, we will have k − 1 vectors having 1’s and 3’s in them in our basis for S_{j}(C). Furthermore, by construction, this code is self-orthogonal and of length n − 1. There were n − 2k vectors consisting solely of 2’s and 0’s in the basis for C, and we need to add the all-twos vector to get a complete basis for S_{j}(C), so there are n − 2k + 1 vectors having only 2’s and 0’s in the basis for S_{j}(C).

A self-dual code of length n − 1 having at least k − 1 vectors with 1’s and/or 3’s in any basis has type 4^{k−1}2^{(n−1)−2(k−1)}. The shortened code we just constructed is self-orthogonal and has the right number of vectors, so it must be self-dual.

III. The Bottom-Up Approach

In this part we deal with the bottom up approach up to length 13, which consists in checking the mass formula of [4] by building the indecomposable self-dual ℤ_{4} codes of a given length n and then in completing them with decomposable codes obtained by direct sums of self-dual codes of length n − 1 or less.

Theorem 5 of [4] shows us that in order to build self-dual codes over ℤ_{4}, we first have to find all the inequivalent doubly even binary codes of a given length n. But self-dual ℤ_{4} codes whose projection onto one of the coordinate is not all of ℤ_{4} are decomposable [2], for they contain a subcode isomorphic to A_{1} (the length one code composed of the two words ’0’ and ’2’). Therefore, we only need to find the inequivalent doubly even binary codes of length 10 to 13 with no coordinate position identically zero.

Using mass formulas of [4], we build the classification of doubly even binary codes for length 10 ≤ n ≤ 13, from which we extract the codes we are interested in. There are 21 such codes which are listed in Table I.

In order to classify doubly even binary codes, we first search for possible sub-codes generated by vectors of weight 4, which are obtained as direct sums of the standard codes d_{2m},e_{7},e_{8} and we eventually add ’glue’ words of weight 8 to describe all the different codes. We know we have all the inequivalent codes, when the mass formula checks.

Code  Auto Order
[10,2]-d4  2^{7}3^{2}5
[10,3]-d6d4  2^{7}3^{2}
[10,4]-d10  2^{8}3.5
[11,3]-d6  2^{6}3^{2}5
[11,4]-e7d4  2^{6}3^{2}7
[12,1]-f  2^{10}3^{3}5.7
[12,2]-d4  2^{10}3^{3}5.7
[12,2]-f  2^{10}3^{4}
[12,3]-d6  2^{8}3^{3}5
[12,3]-2d4  2^{10}3

[12,3]-3d4  2^{10}3^{4}
[12,4]-e7  2^{6}3^{2}5.7
[12,4]-d8  2^{9}3^{2}
[12,4]-2d6  2^{9}3^{2}
[12,4]-d8d4 2^{10}3^{2}
[12,5]-d12 2^{10}3^{2}5
[12,5]-e8d4 2^{9}3^{2}7
[13,3]-d4  2^{6}3^{4}
[13,4]-d6d4  2^{7}3^{2}
[13,5]-d10  2^{8}3^{2}5
[13,5]-e7d6 2^{7}3^{2}7

Table I. List of the possible codes C_{1}, for lengths 10 ≤ n ≤ 13.

In [4] we explained how to obtain all the self-dual Z_{4} codes which have the same C_{1}.

Let C be a ℤ_{4} self-dual code with its associated binary codes C_{1} and C_{2} and let (e_{1}, .., e_{k}) be a basis of C_{1} which we complete to a basis (e_{1}, .., e_{n}) of 𝔽_{2}^{n}. Let us consider a k-set of vectors (e_{1}^{∗}, .., e_{k}^{∗}), verifying e_{i}.e_{j}^{∗} = δ_{ij}, the Kronecker symbol, 1 ≤ i, j ≤ k. We know it is possible because of the existence of a dual basis of (e_{1}, .., e_{n}).

We can associate to any self-dual ℤ_{4} code C a map f :

f : C_{1} ⟼ (\overline{e_{1}^{∗}, .., e_{k}^{∗}})

where (\overline{e_{1}^{∗}, .., e_{k}^{∗}}) is the subspace generated by the vectors (e_{1}^{∗}, .., e_{k}^{∗}). Although f is not linear, it is determined by the image of a basis of C_{1}.

Let M(m_{ij})_{1≤i,j≤k} be the matrix associated to f, f(e_{i}) = ∑_{j=1}^{j=k_{1}} m_{ij}e_{j}^{∗}.

In the particular case of the generator matrix (1), where:

G_{1} = [ A I_{k} ] ,

we can choose for the set (e_{1}^{∗}, .., e_{k}^{∗}) the rows of the matrix:

Q = [ 0 I_{k} ]

which leads to M=B, with B the k × k matrix of the generator matrix of the self-dual code C (1).

In [4] we showed that the general way to build indecomposable Z_{4} self-dual codes is to choose a C_{1} in Table I, compute C_{2} = C_{1}^{⊥} and describe all the possible matrices M such that e_{i}.e_{j} ≡ 2[m_{ij} + m_{ji}] mod 4 , ∀ 1 ≤ i, j ≤ k. There are 2^{\frac{k(k+1)}{2}} such matrices, which represent the self-dual ℤ_{4} codes associated to a given C_{1}. Then we check if the code is really indecomposable by comparison with direct sums of self-dual codes of lower length.

In order to check the mass formula we have to compute the order of the automorphism groups of all the inequivalent self-dual codes we know for a given length. We are sure that our list is complete when it matches the mass formula.

The method described here differs from Leon’s algorithm [6], which computes directly the automorphism groups. We have used a program computing the orbit Orb_{(G)}C and the stabilizer Aut_{(G)}C of a code C under the action of a given matrix group G. The advantage of such a program is that it can easily be carried out, but if we use it directly with G = M_{n} and a ℤ_{4} code C, in order to compute |Aut_{(M_{n})}C| = \frac{|M_{n}|}{|Orb_{(Mn)}C|}, we quickly get orbits with big size, which is too expensive in computation time.

To deal with this problem we define a matrix group m_{1} = {diag(a_{1}, a_{2}, …, a_{n}) |a_{i} ∈ {∓1}}, and we notice that :

Aut_{(M_{n})}C = Aut_{(m_{1}⋊Aut_{(Sn)}C_{1})}C,

which leads to :

|Aut_{(M_{n})}C| = \frac{|Aut_{(S_{n})}C_{1}| × 2^{n}}{|Orb_{(m_{1}⋊Aut_{(Sn)}C_{1})}C|},

so that we only need to compute Aut_{(S_{n})}C_{1} and Orb_{(m_{1}⋊Aut_{(Sn)}C_{1})}C. We compute Aut_{(S_{n})}C_{1} once and when we change the matrix M, we compute and store the different elements of Orb_{(m_{1}⋊Aut_{(Sn)}C_{1})}C, in order to be sure that the next new matrix M is not associated to a code already found.

For example let us take C_{1} = [13,5]-e7d6, with k = 5, given by a generator matrix G_{1}:

G_{1} = [ \begin{matrix}1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \end{matrix} ]

We compute the orbit and a set of generators of the automorphisms of C_{1} under the action of S_{13}. We get 772200 equivalent codes and hence \frac{13!}{772200} automorphisms. The code C_{2} is given by C_{2} = C_{1}^{⊥} and a k_{1}-set (e_{1}^{∗}, .., e_{k}^{∗}) associated with a matrix Q is deduced from the basis of C_{1} :

Q = [ \begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \end{matrix} ]

We choose a matrix M among the 2^{\frac{5.6}{2}} possible matrices :

M = [ \begin{matrix}0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ]

The matrix M, which characterizes the map F allows us to build a new self-dual code over ℤ_{4} : [13,5]-e7d6a, with generator matrix obtained by G_{1} + 2MQ for the first k rows and by two times a supplement of C_{1} in C_{2} for the last n − 2k rows ( or the 2-part). When the matrix M changes, only the first k rows of the generator matrix change.

[ \begin{matrix}1 & 3 & 1 & 3 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 3 & 1 & 3 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 2 & 0 \end{matrix} ]

We compute the orbit of [13,5]-e7d6a under the action of m_{1} ⋊ Aut_{(S_{n})}C_{1} and obtain 10752 equivalents from which we deduce that the order of the automorphism group of the ℤ_{4} code [13,5]-e7d6a under the action of M_{n} is

|Aut_{(M_{n})}[13, 5] − e7d6a| = \frac{2^{13}.13!}{772200.10752} = 2^{11}.3

IV. The Top-Down Approach

A. The Length 15 Codes

Recall that the Euclidean weight of a ℤ_{4} vector v is an integer that counts a 2 component of v as 4, and a 1 or 3 component of v as 1. A Type II code is a self-dual code containing an h̃ vector in which all vectors have Euclidean weight divisible by 8. Type II codes exist only at lengths n ≡ 0 (mod 8).

Note that any self-dual code of length 15 is a shortened code of a Type II length 16 code. By Lemma 1, shortening a Type II length 16 code does give a self-dual length 15 code. Conversely, given a generator matrix G′ of a length 15 self-dual code, we can find a generator matrix G for a Type II length 16 code. Add a column to G′, and fill in this column with 0’s and 2’s so that the Euclidean weights of all rows become 0 (mod 8). Add another row to G′ consisting of a vector h̃ where the 1’s and 3’s in h̃ are adjusted so as to make it orthogonal to the extended G′.

To see how this works, consider the following generator matrix for [16, 4] − d_{6}.

G = [ \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 3 & 3 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 1 \\ 2 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \end{matrix} ]

Shortening on the 12th coordinate we obtain

S_{12}(G) = [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 2 & 2 & 1 \\ 2 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ]

a generator matrix for [15, 3]−d_{6}. It is clear how to add a column and a row equivalent to the all-ones vector to S_{12}(G) and get a generator matrix for a Type II code. No matter how we do this, we get a generator matrix for a code equivalent to the one we started with.

Theorem 1: There are 290 inequivalent self-dual ℤ_{4} codes of length 15, 111 of them are indecomposable. The highest minimum Lee weight is 6, and there are 14 codes attaining this minimum Lee weight. The number of codes of each type 4^{k}2^{15−2k} are as follows:

\begin{matrix}k & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ number & 1 & 3 & 9 & 26 & 58 & 86 & 77 & 30 \end{matrix}

These codes were found by shortening each of the 133 length 16 Type II codes on coordinates in different orbits of the groups of these codes. That these 290 codes represent all of the inequivalent length 15 self-dual codes has been verified using the mass formula from [4].

In Appendix I we give the labels of the indecomposable self-dual codes of length 15, their group orders (the group orders are factored as (order of the sign change group) × (order of the permutation group)), and partial information about their symmetrized weight enumerators.

The highest minimum Lee weight is 6, and there are 14 codes with this minimum weight. Five of them are shortened from the five Type II codes having minimum Lee weight 8, the rest are shortened from some of the Type II codes of minimum Lee weight 6. We give the generator matrices of the codes of minimum Lee weight 6 in Appendix II.

The Gray map of [15, 4] − f gives the linear singly-even [30,15,6] code, a child of the doubly-even code R(1, 5).

There is a relation between the number of Type II codes of length 16 and the number of self-dual codes of length 15. We end this section by clarifying this relation.

Theorem 2: Let n be divisible by 8. Let τ(n, k) denote the number of admissible binary codes of length n and dimension k, and σ(n − 1, k − 1) the number of doubly-even [n − 1, k − 1] codes. Then

τ(n, k) = σ(n − 1, k − 1).

Further, the number of Type II codes of length n and type 4^{k}2^{n−2k} is twice the number of ℤ_{4} self-dual codes of length n − 1 and type 4^{k−1}2^{n−2k+2}.

Proof: The statement for the binary codes can be demonstrated in two ways, one way is by a combinatorial proof and the other uses formulas.

Given an [n − 1, k − 1] doubly-even code C′, adjoining a new coordinate, say the last one, on which C′ is zero, and adding the all-ones vector produces an admissible [n, k] code. Conversely, shortening an admissible [n, k] code on, say, its last coordinate produces a doubly-even [n − 1, k − 1] code.

For the other proof, note that τ(n, k) satisfies the following recursion [7]

τ(n, 1) = 1

and

τ(n, k) = τ(n, k − 1)\frac{2^{n−2k+1} + 2^{n/2−k} − 1}{2^{k−1} − 1}.

Also, σ(n − 1, k − 1) satisfies the same recursion and σ(n − 1, 0) = 1.

Now the number of Type II codes of length n and type 4^{k}2^{n−2k} [4] is τ(n, k)2^{k(k−1)/2+1}. From the same reference, the number of self-dual codes of length n − 1 ≡ 7 (mod 8) and type 4^{k−1}2^{n−2k+1} is

σ(n − 1, k − 1)2^{k(k−1)/2}.

This demonstrates the second statement in the theorem.

This can also be demonstrated combinatorially by noting that there are two ways to extend a selfdual ℤ_{4} code of length n − 1 and type 4^{k−1}2^{n−2k+1} to a Type II code of length n and type 4^{k}2^{n−2k} which correspond to the two possibilities (1 or 3) for the new column in the new vector h̃.

B. Lengths 10,11,12,13 and 14

Lemma 2: If C contains a self-dual ℤ_{4} code of length n whose symmetrized weight enumerator contains the term rc, the code obtained by puncturing the r columns of a generator matrix for C that consist entirely of 0’s and 2’s gives a generator matrix of a self-dual ℤ_{4} code C′ of length n − r.

Further C′ is either indecomposable or is the direct sum of indecomposable codes each of which is length 4 or more.

Proof: From the data in [2], 4 is the smallest length of an indecomposable code whose length is greater than 1.

From the lemma, it is clear that all the information about indecomposable codes of lengths 10,11,12,13 and 14 is contained in the data on all of the length 15 self-dual codes and the data in [2] about indecomposable codes of smaller lengths (4,6,7 and 8).

Theorem 3: The number of indecomposable codes of length 10 ≤ n ≤ 14 and type 4^{k}2^{n−2k} are:

n\k 1 2 3 4 5 6
10 0 1 1 2
11 0 0 1 1
12 1 2 6 11 11
13 0 0 1 2 4
14 0 2 6 20 32 32

We punctured all of the length 15 codes which had 5c in their symmetrized weight enumerators obtaining all of the length 10 indecomposable codes and one direct sum [4, 1] − d4 ⊕ [6, 2] − d6. We checked that we had indeed found all the codes by finding their group orders, finding all direct sums involving indecomposable summands of lengths 4,6,7 and 8 and their group orders and then using the mass-formula.

We did similar computations for length 11 using length 15 self-dual codes whose symmetrized weight enumerators contained the term 4c. We found all direct sums of codes of shorter lengths from the data in [2] and the indecomposable codes of length 10. All group orders were computed and the mass formula checked. Similar computations were done in turn for lengths 12, 13 and 14, using length 15 self-dual codes whose symmetrized weight enumerators contained the terms 3c, 2c and 1c respectively. Again, all direct sums of the appropriate lengths were constructed using the data from [2] and our newfound knowledge of the indecomposable codes of lengths between 10 and the length in question.

We give the labels, group orders, and partial information about symmetrized weight enumerators of the self-dual codes having no trivial component of length n, for 10 ≤ n ≤ 14, in Appendix I.

We give the generator matrices of the indecomposable codes of lengths 10 and 11 in Appendix II. There are too many indecomposable codes of lengths 12 ≤ n ≤ 15 for inclusion here, so we give generator matrices only for those indecomposable codes (one of length 14 and fifteen of length 15) having minimum Lee weight 6.

V. Appendix I: Data For All Self-Dual Codes (without Trivial Components) of Lengths 10,11,12,13,14 & 15

Most of the codes below have minimum Lee weight 4,the few exceptions to this rule (which have minimum Lee weight 6) are marked with an asterisk.

A. LENGTH 10

name group order orbit swe coefficients
(sign change)perm. lengths c^{2} c^{3} c^{4} b^{4} b^{4}c^{1} b^{4}c^{2} b^{8}
[10,2]-d4  (2^{7})2^{7}3^{2}5  4,6 17 24 46 4 24 60 128
[10,3]-d4⊕d6  (2^{6})2^{7}3^{2}  4,6  9 8 22 20 24 108 96
[10,3]-d4d6  (2^{6})2^{7}3  2,4,4 9 8 22 4 56 124 96
[10,4]-d10a  (2^{5})2^{6}5  10  5 0 10 20 40 140 80
[10,4]-d10b  (2^{5})2^{6}3  4,6  5 0 10 4 72 156 80

B. LENGTH 11

name group order orbit swe coefficients
(sign change)perm. lengths c^{2} c^{3} c^{4} b^{4} b^{4}c^{1} b^{4}c^{2} b^{8}
[11,3]-d6  (2^{5})2^{6}3^{2}5  5,6 10 19 28 6 42 126 128
[11,4]-d4⊕e7  (2^{4})2^{6}3^{2}7  4,7  6 7 8 22 42 126 112
[11,4]-d4e7  (2^{4})2^{6}3^{2}  1,4,6 6 7 8 6 58 174 112

C. LENGTH 12

name group order orbit swe coefficients
(sign change)perm. lengths c^{2} c^{3} c^{4} b^{4} b^{4}c^{1} b^{4}c^{2} b^{8}
[12,1]-f  (2^{11})2^{10}3^{5}5^{2}7 · 11  12  66 0 495 0 0  0  0
[12,2]-d4⊕f(8)  (2^{10})2^{10}3^{3}5 · 7  4,8 34 0 239 8 0 224 128
[12,2]-d4  (2^{10})2^{10}3^{3}5 · 7  4,8 34 0 239 0 64 0  0
[12,2]-f  (2^{9})2^{10}3^{4}  12  18 64 111 0 0  0 192
[12,3]-2d4a  (2^{7})2^{10}3  4,8 10 32 47 8 32 96 160
[12,3]-2d4b  (2^{7})2^{10}3  4,8 10 32 47 0 32 128 160
[12,3]-d4⊕d4⊕d4  (2^{9})2^{10}3^{4}  12  18 0 111 24 0 288 192
[12,3]-3d4a  (2^{9})2^{10}3^{3}  4,8 18 0 111 0 64 128 64
[12,3]-3d4b  (2^{9})2^{10}3^{4}  12  18 0 111 0 0 384 192
[12,3]-d4⊕2d4  (2^{9})2^{10}3^{3}  4,8 18 0 111 8 64 96 64
[12,3]-d6a  (2^{8})2^{8}3^{3}5  6,6 18 0 111 0 96 0  0
[12,3]-d6b  (2^{8})2^{8}3^{2}5  2,4,6 18 0 111 8 32 224 128
[12,4]-d4⊕d8II  (2^{7})2^{10}3^{2}  4,8 10 0 47 8 96 32 32
[12,4]-d4⊕d8I  (2^{7})2^{10}3  4,8 10 0 47 24 32 224 160
[12,4]-d4d8a  (2^{7})2^{10}3  4,8 10 0 47 0 96 64 32
[12,4]-d4d8b  (2^{7})2^{8}3  2,2,4,4 10 0 47 8 64 160 96
[12,4]-d4d8c  (2^{7})2^{9}3  4,8 10 0 47 8 32 288 160
[12,4]-d4d8e  (2^{7})2^{8}3^{2}  2,4,6 10 0 47 0 64 192 96
[12,4]-d4d8f  (2^{7})2^{10}3  4,8 10 0 47 0 32 320 160
[12,4]-d6⊕d6  (2^{6})2^{9}3^{2}  12  6 16 15 24 48 96 144
[12,4]-d6d6a  (2^{6})2^{9}  4,8  6 16 15 8 48 160 144
[12,4]-d6d6b  (2^{6})2^{8}3  12  6 16 15 0 48 192 144
[12,4]-d8a  (2^{4})2^{9}3  4,8  6 16 15 8 48 160 144

name  group order  orbit  swe coefficients
(sign change)perm. lengths c^{2} c^{3} c^{4} b^{4} b^{4}c^{1} b^{4}c^{2} b^{8}
[12,4]-d8b  (2^{4})2^{9}3^{2}  4,8  6 16 15 0 48 192 144
[12,4]-e7a  (2^{5})2^{6}3^{2}5 · 7  5,7 10 0 47 0 112 0  0
[12,4]-e7b  (2^{5})2^{6}3^{2}5  1,5,6 10 0 47 8 48 224 128
[12,5]-d12a  (2^{6})2^{8}3  12  6 0 15 24 48 192 144
[12,5]-d12b  (2^{6})2^{7}3  2,4,6 6 0 15 8 80 128 80
[12,5]-d12c  (2^{6})2^{8}  4,8  6 0 15 8 48 256 144
[12,5]-d12d  (2^{6})2^{7}5  2,10 6 0 15 0 80 160 80
[12,5]-d12e  (2^{6})2^{8}3^{2}  12  6 0 15 0 48 288 144
[12,5]-d12f  (2^{6})2^{8}3  12  6 0 15 0 48 288 144
[12,5]-d4⊕e8II  (2^{4})2^{9}3^{2}7  4,8  6 0 15 8 112 0 16
[12,5]-d4⊕e8I  (2^{4})2^{9}3^{2}  4,8  6 0 15 24 48 192 144
[12,5]-d4e8a  (2^{4})2^{9}3^{2}  4,8  6 0 15 0 112 32 16
[12,5]-d4e8b  (2^{4})2^{6}3^{2}  1,1,4,6 6 0 15 8 64 192 112
[12,5]-d4e8c  (2^{4})2^{8}3  4,8  6 0 15 8 48 256 144
[12,5]-d4e8e  (2^{4})2^{6}3^{2}7  1,4,7 6 0 15 0 64 224 112
[12,5]-d4e8f  (2^{4})2^{9}3^{2}  4,8  6 0 15 0 48 288 144

D. LENGTH 13

name group order orbit swe coefficients
(sign change)perm. lengths c^{2} c^{3} c^{4} b^{4} b^{4}c^{1} b^{4}c^{2} b^{8}
[13,3]-d4  (2^{7})2^{6}3^{4}  4,9  9 45 82 2 18 72 192
[13,4]-d4⊕d6a  (2^{5})2^{7}3^{2}  3,4,6 5 25 34 10 42 120 176
[13,4]-d4⊕d6b  (2^{5})2^{7}3  2,3,4,4 5 25 34 2 34 152 176
[13,5]-d10a  (2^{3})2^{6}3 · 5  3,10 3 15 10 10 50 160 168
[13,5]-d10b  (2^{3})2^{6}3^{2}  3,4,6 3 15 10 2 42 192 168
[13,5]-d6⊕e7  (2^{4})2^{7}3^{2}7  6,7  3 15 10 26 66 96 168
[13,5]-d6e7b  (2^{4})2^{7}3  1,2,4,6 3 15 10 10 50 160 168
[13,5]-d6e7c  (2^{4})2^{6}3  3,4,6 3 15 10 2 42 192 168

E. LENGTH 14

name group order orbit swe coefficients
(sign change)perm. lengths c^{2} c^{3} c^{4} b^{4} b^{4}c^{1} b^{4}c^{2} b^{8}
[14,2]-d4  (2^{11})2^{11}3^{4}5^{2}7  4,10 47 40 301 4 40 180 0
[14,2]-f  (2^{11})2^{10}3^{4}5^{2}  2,12 31 72 285 0 0  0 128
[14,3]-d4⊕d4(10)  (2^{10})2^{10}3^{3}5  4,4,6 23 24 149 12 24 220 160
[14,3]-d4a  (2^{9})2^{11}3^{2}  2,4,8 15 40 141 0 32 64 128
[14,3]-d4b  (2^{9})2^{11}3^{2}  2,4,8 15 40 141 4 8 116 192
[14,3]-d4+d4(10)a  (2^{10})2^{10}3^{3}5  4,4,6 23 24 149 4 72 116 32
[14,3]-d4+d4(10)b  (2^{10})2^{9}3^{3}5  2,2,4,6 23 24 149 0 32 192 96
[14,3]-d6⊕f8  (2^{10})2^{11}3^{3}5 · 7  6,8 31 8 157 12 24 348 128
[14,3]-d6a  (2^{10})2^{11}3^{2}5 · 7  2,4,8 31 8 157 4 72 244 0
[14,3]-f  (2^{8})2^{10}3 · 7  14  7 56 133 0 0  0 224
[14,4]-d4⊕d4⊕d6  (2^{9})2^{11}3^{3}  6,8 15 8 77 28 24 300 256
[14,4]-2d4a  (2^{5})2^{9}3  6,8  3 32 65 4 24 84 208

name  group order  orbit  swe coefficients
(sign change)perm. lengths c^{2} c^{3} c^{4} b^{4} b^{4}c^{1} b^{4}c^{2} b^{8}
[14,4]-2d4b  (2^{5})2^{9}3  6,8  3 32 65 0 16 96 208
[14,4]-2d4d6a  (2^{9})2^{10}3^{2}  2,4,4,4 15 8 77 4 72 180 64
[14,4]-2d4d6b  (2^{9})2^{10}3^{2}  2,4,8  15 8 77 0 64 192 64
[14,4]-2d4d6c  (2^{9})2^{10}3^{2}  2,4,8  15 8 77 0 32 256 128
[14,4]-2d4d6d  (2^{9})2^{11}3^{2}  2,4,8  15 8 77 4 40 244 128
[14,4]-2d4d6e  (2^{9})2^{11}3^{2}  2,4,8  15 8 77 4 8 308 192
[14,4]-3d4a  (2^{7})2^{11}  2,4,8  7 24 69 0 32 128 160
[14,4]-3d4b  (2^{7})2^{11}3  2,12  7 24 69 12 24 156 224
[14,4]-3d4c  (2^{7})2^{11}  2,4,8  7 24 69 4 40 116 160
[14,4]-3d4d  (2^{7})2^{11}3  2,12  7 24 69 0 0 192 224
[14,4]-d4⊕d4d6  (2^{9})2^{10}3^{2}  2,4,4,4 15 8 77 12 56 220 128
[14,4]-d4d6a  (2^{8})2^{10}3^{2}  4,4,6  11 16 73 4 88 84 48
[14,4]-d4d6b  (2^{8})2^{9}3  2,2,2,4,4 11 16 73 0 48 160 112
[14,4]-d4d6c  (2^{8})2^{10}3  2,4,4,4 11 16 73 12 40 188 176
[14,4]-d4d6d  (2^{8})2^{9}3  2,4,4,4 11 16 73 4 24 212 176
[14,4]-d6⊕2d4  (2^{9})2^{11}3^{3}  6,8  15 8 77 12 88 156 64
[14,4]-d6a  (2^{6})2^{7}3^{3}  2,6,6  7 24 69 0 48 96 128
[14,4]-d6b  (2^{6})2^{7}3^{2}  2,2,4,6  7 24 69 4 24 148 192
[14,4]-d8a  (2^{9})2^{9}3^{2}5  4,4,6  19 0 81 4 88 212 16
[14,4]-d8b  (2^{9})2^{8}3^{3}5  2,6,6  19 0 81 0 48 288 80
[14,4]-d8c  (2^{9})2^{9}3^{2}5  4,4,6  19 0 81 12 40 316 144
[14,5]-2d6a  (2^{3})2^{8}3^{2}  2,12  1 20 31 12 48 108 200
[14,5]-2d6b  (2^{3})2^{8}  2,4,8  1 20 31 4 32 132 200
[14,5]-2d6c  (2^{3})2^{7}3  2,12  1 20 31 0 24 144 200
[14,5]-d4⊕d10a  (2^{8})2^{9}3 · 5  4,10  11 0 41 28 40 300 240
[14,5]-d4⊕d10b  (2^{8})2^{9}3^{2}  4,4,6  11 0 41 12 72 220 112
[14,5]-d4d10a  (2^{8})2^{9}3  2,4,4,4 11 0 41 4 88 180 48
[14,5]-d4d10b  (2^{8})2^{8}3^{2}  2,2,4,6 11 0 41 0 80 192 48
[14,5]-d4d10c  (2^{8})2^{9}3  2,4,4,4 11 0 41 12 72 220 112
[14,5]-d4d10d  (2^{8})2^{8}3  2,2,2,2,2,4 11 0 41 0 48 256 112
[14,5]-d4d10e  (2^{8})2^{9}3  2,4,4,4 11 0 41 12 40 284 176
[14,5]-d4d10f  (2^{8})2^{9}3  2,4,4,4 11 0 41 4 56 244 112
[14,5]-d4d10g  (2^{8})2^{9}3^{2}  4,4,6  11 0 41 4 24 308 176
[14,5]-d4d10h  (2^{8})2^{9}3  2,4,4,4 11 0 41 4 24 308 176
[14,5]-d4d8a  (2^{4})2^{10}  2,4,8  3 16 33 0 48 128 144
[14,5]-d4d8b  (2^{4})2^{10}  2,4,8  3 16 33 12 40 156 208
[14,5]-d4d8c  (2^{4})2^{8}  2,2,2,4,4 3 16 33 4 40 148 176
[14,5]-d4d8d  (2^{4})2^{9}  2,4,8  3 16 33 4 24 180 208
[14,5]-d4d8e  (2^{4})2^{10}3  2,4,8  3 16 33 4 56 116 144
[14,5]-d4d8f  (2^{4})2^{8}3  2,2,4,6  3 16 33 0 32 160 176
[14,5]-d4d8g  (2^{4})2^{10}  2,4,8  3 16 33 0 16 192 208
[14,5]-d4e7a  (2^{5})2^{7}3^{2}7  3,4,7  5 12 35 4 96 68 56
[14,5]-d4e7b  (2^{5})2^{6}3^{2}  1,2,2,3,6 5 12 35 0 56 144 120

name  group order  orbit  swe coefficients
(sign change)perm. lengths c^{2} c^{3} c^{4} b^{4} b^{4}c^{1} b^{4}c^{2} b^{8}
[14,5]-d4e7c  (2^{5})2^{7}3^{2}  1,3,4,6 5 12 35 12 48 172 184
[14,5]-d4e7d  (2^{5})2^{6}3  1,2,3,4,4 5 12 35 4 32 196 184
[14,5]-d6⊕d8I  (2^{7})2^{11}3  6,8  7 8 37 28 56 204 224
[14,5]-d6⊕d8II  (2^{7})2^{11}3^{2}  6,8  7 8 37 12 120 60 32
[14,5]-d6d8a  (2^{7})2^{9}3  2,2,4,6 7 8 37 4 72 148 96
[14,5]-d6d8b  (2^{7})2^{11}  2,4,8 7 8 37 4 72 148 96
[14,5]-d6d8c  (2^{7})2^{9}  2,4,4,4 7 8 37 0 64 160 96
[14,5]-d6d8d  (2^{7})2^{9}  2,2,2,4,4 7 8 37 12 56 188 160
[14,5]-d6d8e  (2^{7})2^{9}  2,4,4,4 7 8 37 0 32 224 160
[14,5]-d6d8f  (2^{7})2^{10}  2,4,8 7 8 37 0 32 224 160
[14,5]-d6d8g  (2^{7})2^{10}  2,4,8 7 8 37 12 24 252 224
[14,5]-d6d8h  (2^{7})2^{8}  2,4,4,4 7 8 37 4 40 212 160
[14,5]-d6d8i  (2^{7})2^{11}  2,4,8 7 8 37 4 40 212 160
[14,5]-d6d8j  (2^{7})2^{10}3  6,8  7 8 37 0 0 288 224
[14,6]-e7⊕e7  (2^{2})2^{7}3^{2}7^{2}  14  0 14 14 28 84 84 196
[14,6]-2e7a  (2^{2})2^{7}3^{2}  2,12  0 14 14 12 52 132 196
[14,6]-2e7b  (2^{2})2^{6}3  6,8  0 14 14 4 36 156 196
∗ [14,6]-2e7c  (2^{2})2^{4}3 · 7  14  0 14 14 0 28 168 196
[14,6]-d12a  (2^{2})2^{8}3  2,12  1 12 15 12 48 156 200
[14,6]-d12b  (2^{2})2^{7}3  2,2,4,6 1 12 15 4 48 148 168
[14,6]-d12c  (2^{2})2^{8}  2,4,8 1 12 15 4 32 180 200
[14,6]-d12d  (2^{2})2^{7}5  2,2,10 1 12 15 0 40 160 168
[14,6]-d12e  (2^{2})2^{8}3^{2}  2,12  1 12 15 0 24 192 200
[14,6]-d12f  (2^{2})2^{8}3  2,12  1 12 15 0 24 192 200
[14,6]-d14a  (2^{7})2^{8}  2,4,4,4 7 0 21 4 72 196 96
[14,6]-d14b  (2^{7})2^{8}7  14  7 0 21 28 56 252 224
[14,6]-d14c  (2^{7})2^{8}3^{2}  2,12  7 0 21 0 96 144 32
[14,6]-d14d  (2^{7})2^{8}3  4,4,6 7 0 21 12 88 172 96
[14,6]-d14e  (2^{7})2^{8}  2,4,4,4 7 0 21 0 64 208 96
[14,6]-d14f  (2^{7})2^{8}  2,4,4,4 7 0 21 12 56 236 160
[14,6]-d14g  (2^{7})2^{8}  2,4,4,4 7 0 21 0 32 272 160
[14,6]-d14h  (2^{7})2^{8}3  2,12  7 0 21 12 24 300 224
[14,6]-d14i  (2^{7})2^{8}3  4,4,6 7 0 21 4 40 260 160
[14,6]-d14j  (2^{7})2^{8}5  4,10  7 0 21 4 40 260 160
[14,6]-d14k  (2^{7})2^{8}  2,4,4,4 7 0 21 4 40 260 160
[14,6]-d14l  (2^{7})2^{8}3 · 7  14  7 0 21 0 0 336 224
[14,6]-d6⊕e8I  (2^{4})2^{10}3^{2}  6,8  3 8 17 28 72 156 208
[14,6]-d6⊕e8II  (2^{4})2^{10}3^{2}7  6,8  3 8 17 12 136 12 16
[14,6]-d6e8a  (2^{4})2^{10}3  2,4,8 3 8 17 4 88 100 80
[14,6]-d6e8b  (2^{4})2^{7}3  2,2,4,6 3 8 17 0 64 144 112
[14,6]-d6e8c  (2^{4})2^{7}3  1,1,2,4,6 3 8 17 12 56 172 176
[14,6]-d6e8d  (2^{4})2^{9}  2,4,8 3 8 17 0 48 176 144
[14,6]-d6e8e  (2^{4})2^{9}  2,4,8 3 8 17 12 40 204 208

name  group order  orbit  swe coefficients
(sign change)perm. lengths c^{2} c^{3} c^{4} b^{4} b^{4}c^{1} b^{4}c^{2} b^{8}
[14,6]-d6e8f  (2^{4})2^{6}  2,2,2,4,4 3 8 17 4 40 196 176
[14,6]-d6e8g  (2^{4})2^{7}3  6,8  3 8 17 4 24 228 208
[14,6]-d6e8h  (2^{4})2^{7}3 · 7  1,2,4,7 3 8 17 4 72 132 112
[14,6]-d6e8i  (2^{4})2^{10}3  2,4,8 3 8 17 4 56 164 144
[14,6]-d6e8j  (2^{4})2^{7}3  2,2,4,6 3 8 17 0 32 208 176
[14,6]-d6e8k  (2^{4})2^{9}3  6,8  3 8 17 0 16 240 208

F. LENGTH 15

name group order orbit swe coefficients
(sign change)perm. lengths c^{2} c^{3} c^{4} b^{4} b^{4}c^{1} b^{4}c^{2} b^{8}
[15,3]-d4  (2^{9})2^{9}3^{2}5^{2}  1,4,10 20 47 181 2 22 110 128
[15,3]-d6  (2^{9})2^{10}3^{5}5 · 7  6,9  36 31 165 6 66 330 0
[15,3]-f  (2^{9})2^{7}3^{5}  3,12  12 55 189 0 0  0 192
[15,4]-2d4a  (2^{7})2^{8}3^{2}  1,4,4,6  8 27 97 2 38 94 144
[15,4]-2d4b  (2^{7})2^{7}3^{2}  1,2,2,4,6 8 27 97 0 16 112 176
[15,4]-2d4c  (2^{7})2^{8}3^{2}  1,4,4,6  8 27 97 6 18 122 208
[15,4]-d4⊕d6(11)  (2^{8})2^{9}3^{3}5  4,5,6  16 19 89 14 42 242 176
[15,4]-d4a  (2^{6})2^{9}3  3,4,8  4 31 101 0 16 48 192
[15,4]-d4b  (2^{6})2^{9}3  3,4,8  4 31 101 2 6 62 224
[15,4]-d4d6a  (2^{8})2^{9}3^{3}5  4,5,6  16 19 89 6 82 186 48
[15,4]-d4d6b  (2^{8})2^{8}3^{2}5  1,1,4,4,5 16 19 89 2 38 222 112
[15,4]-d6a  (2^{7})2^{10}3^{2}  1,2,4,8 12 23 93 2 38 158 128
[15,4]-d6b  (2^{7})2^{10}3^{3}  1,6,8  12 23 93 6 18 186 192
[15,4]-e7  (2^{8})2^{10}3^{3}5 · 7  1,6,8  28 7 77 6 82 378 0
[15,4]-e7⊕2d4  (2^{7})2^{10}3^{3}7  7,8  12 7 45 14 106 210 64
[15,4]-e7⊕f8  (2^{8})2^{10}3^{3}5 · 7^{2}  7,8  28 7 77 14 42 434 128
∗ [15,4]-f  (2^{5})2^{6}3^{2}5 · 7  15  0 35 105 0 0  0 240
[15,5]-2d4d6a  (2^{5})2^{9}  1,2,4,4,4 4 15 53 2 38 126 160
[15,5]-2d4d6b  (2^{5})2^{9}  1,2,4,8  4 15 53 0 32 128 160
[15,5]-2d4d6c  (2^{5})2^{9}  1,2,4,4,4 4 15 53 6 34 138 192
[15,5]-2d4d6d  (2^{5})2^{9}  1,2,4,8  4 15 53 0 16 144 192
[15,5]-2d4d6e  (2^{5})2^{10}3  1,6,8  4 15 53 6 50 122 160
[15,5]-2d4d6f  (2^{5})2^{10}  1,2,4,8  4 15 53 2 22 142 192
[15,5]-2d4d6g  (2^{5})2^{10}3  1,6,8  4 15 53 14 26 162 256
[15,5]-2d4d6h  (2^{5})2^{10}  1,2,4,8  4 15 53 2 6 158 224
[15,5]-2d4e7a  (2^{7})2^{9}3^{3}  1,4,4,6 12 7 45 6 82 218 64
[15,5]-2d4e7b  (2^{7})2^{9}3^{2}  1,2,4,8 12 7 45 2 54 238 96
[15,5]-2d4e7c  (2^{7})2^{9}3^{2}  1,2,4,8 12 7 45 2 38 254 128
[15,5]-2d4e7d  (2^{7})2^{10}3^{3}  1,6,8  12 7 45 6 34 266 160
[15,5]-2d4e7e  (2^{7})2^{10}3^{3}  1,6,8  12 7 45 6 18 282 192
[15,5]-2d6a  (2^{6})2^{8}3^{3}  3,6,6  6 13 51 6 90 114 72
[15,5]-2d6b  (2^{6})2^{8}3^{2}  2,3,4,6  6 13 51 14 50 170 200
[15,5]-2d6c  (2^{6})2^{7}3  1,1,2,3,4,4 6 13 51 2 46 150 136
[15,5]-2d6d  (2^{6})2^{5}3^{2}  3,3,3,6  6 13 51 0 24 168 168

name  group order  orbit  swe coefficients
(sign change)perm.  lengths  c^{2} c^{3} c^{4} b^{4} b^{4}c^{1} b^{4}c^{2} b^{8}
[15,5]-2d6e  (2^{6})2^{7}3  2,2,3,4,4  6 13 51 6 26 178 200
∗ [15,5]-3d4a  (2^{3})2^{8}3  3,4,8  0 19 57 0 16 80 208
[15,5]-3d4b  (2^{3})2^{8}3^{2}  3,12  0 19 57 6 18 90 240
[15,5]-3d4c  (2^{3})2^{8}3  3,4,8  0 19 57 2 22 78 208
∗ [15,5]-3d4d  (2^{3})2^{8}3^{2}  3,12  0 19 57 0 0 96 240
[15,5]-d4⊕d4⊕e7  (2^{7})2^{10}3^{3}7  7,8  12 7 45 30 42 306 288
[15,5]-d4⊕d4e7(11)  (2^{7})2^{9}3^{3}  1,4,4,6  12 7 45 14 58 258 160
[15,5]-d4d6a  (2^{4})2^{8}3  1,4,4,6  2 17 55 2 46 86 152
[15,5]-d4d6b  (2^{4})2^{7}  1,2,2,2,4,4 2 17 55 0 24 104 184
[15,5]-d4d6c  (2^{4})2^{8}  1,2,4,4,4  2 17 55 6 26 114 216
[15,5]-d4d6d  (2^{4})2^{7}  1,2,4,4,4  2 17 55 2 14 118 216
[15,5]-d8a  (2^{5})2^{7}3^{2}  1,4,4,6  6 13 51 2 46 150 136
[15,5]-d8b  (2^{5})2^{6}3^{3}  1,2,6,6  6 13 51 0 24 168 168
[15,5]-d8c  (2^{5})2^{7}3^{2}  1,4,4,6  6 13 51 6 26 178 200
[15,6]-d4d10a  (2^{3})2^{8}  1,2,4,4,4  2 9 31 2 46 134 152
[15,6]-d4d10b  (2^{3})2^{7}3  1,2,2,4,6  2 9 31 0 40 136 152
[15,6]-d4d10c  (2^{3})2^{8}3  1,4,4,6  2 9 31 6 42 146 184
[15,6]-d4d10d  (2^{3})2^{8}  1,2,4,4,4  2 9 31 6 42 146 184
[15,6]-d4d10e  (2^{3})2^{7}  1,2,2,2,2,2,4 2 9 31 0 24 152 184
[15,6]-d4d10f  (2^{3})2^{8}  1,2,4,4,4  2 9 31 6 26 162 216
[15,6]-d4d10g  (2^{3})2^{8}  1,2,4,4,4  2 9 31 2 30 150 184
[15,6]-d4d10h  (2^{3})2^{8}5  1,4,10  2 9 31 14 34 170 248
[15,6]-d4d10i  (2^{3})2^{8}3  1,4,4,6  2 9 31 2 14 166 216
[15,6]-d4d10j  (2^{3})2^{8}  1,2,4,4,4  2 9 31 2 14 166 216
[15,6]-d6d8a  (2^{2})2^{7}3  1,2,2,4,6  0 11 33 2 38 110 176
[15,6]-d6d8b  (2^{2})2^{9}3  1,6,8  0 11 33 14 42 130 240
[15,6]-d6d8c  (2^{2})2^{9}  1,2,4,8  0 11 33 2 38 110 176
∗ [15,6]-d6d8d  (2^{2})2^{7}  1,2,4,4,4  0 11 33 0 32 112 176
[15,6]-d6d8e  (2^{2})2^{7}  1,2,2,2,4,4 0 11 33 6 34 122 208
∗ [15,6]-d6d8f  (2^{2})2^{7}  1,2,4,4,4  0 11 33 0 16 128 208
∗ [15,6]-d6d8g  (2^{2})2^{8}  1,2,4,8  0 11 33 0 16 128 208
[15,6]-d6d8h  (2^{2})2^{8}  1,2,4,8  0 11 33 6 18 138 240
[15,6]-d6d8i  (2^{2})2^{6}  1,2,4,4,4  0 11 33 2 22 126 208
[15,6]-d6d8j  (2^{2})2^{9}3^{2}  1,6,8  0 11 33 6 66 90 144
[15,6]-d6d8k  (2^{2})2^{9}  1,2,4,8  0 11 33 2 22 126 208
∗ [15,6]-d6d8l  (2^{2})2^{8}3  1,6,8  0 11 33 0 0 144 240
[15,6]-d6e7a  (2^{3})2^{7}3^{2}7  2,6,7  1 10 32 6 94 78 84
[15,6]-d6e7b  (2^{3})2^{6}3  1,1,1,2,4,6 1 10 32 2 50 114 148
[15,6]-d6e7c  (2^{3})2^{7}3^{2}  1,2,6,6  1 10 32 14 54 134 212
[15,6]-d6e7d  (2^{3})2^{6}  1,2,2,2,4,4 1 10 32 6 30 142 212
[15,6]-d6e7e  (2^{3})2^{4}3  2,3,3,3,4  1 10 32 0 28 132 180
[15,6]-d6e7f  (2^{3})2^{4}3  2,3,4,6  1 10 32 2 18 146 212
[15,6]-e7⊕d8I  (2^{5})2^{10}3 · 7  7,8  4 7 29 30 74 194 256
[15,6]-e7⊕d8II  (2^{5})2^{10}3^{2}7  7,8  4 7 29 14 138 98 32

name  group order  orbit  swe coefficients
(sign change)perm. lengths c^{2} c^{3} c^{4} b^{4} b^{4}c^{1} b^{4}c^{2} b^{8}
[15,6]-e7⊕e8I  (2^{2})2^{9}3^{2}7  7,8  0 7 21 30 90 138 240
[15,6]-e7⊕e8II  (2^{2})2^{9}3^{2}7^{2}  7,8  0 7 21 14 154 42 16
[15,6]-e7d8a  (2^{5})2^{8}3^{2}  1,2,6,6 4 7 29 6 82 138 96
[15,6]-e7d8b  (2^{5})2^{8}  1,2,4,4,4 4 7 29 2 54 158 128
[15,6]-e7d8c  (2^{5})2^{8}3  1,2,2,4,6 4 7 29 14 58 178 192
[15,6]-e7d8d  (2^{5})2^{8}  1,2,4,4,4 4 7 29 2 38 174 160
[15,6]-e7d8e  (2^{5})2^{7}  1,2,4,8 4 7 29 0 32 176 160
[15,6]-e7d8f  (2^{5})2^{7}  1,2,4,4,4 4 7 29 6 34 186 192
[15,6]-e7d8g  (2^{5})2^{7}3  1,6,8  4 7 29 0 16 192 192
[15,6]-e7d8h  (2^{5})2^{10}3  1,6,8  4 7 29 6 66 154 128
[15,6]-e7d8i  (2^{5})2^{10}3  1,6,8  4 7 29 6 50 170 160
[15,6]-e7d8j  (2^{5})2^{9}  1,2,4,8 4 7 29 2 22 190 192
[15,6]-e7d8k  (2^{5})2^{9}3  1,6,8  4 7 29 14 26 210 256
[15,6]-e7d8l  (2^{5})2^{9}3  3,4,8  4 7 29 2 6 206 224
[15,7]-d14a  (2^{2})2^{6}7  1,14  0 7 21 14 42 154 240
[15,7]-d14b  (2^{2})2^{6}  1,2,4,4,4 0 7 21 2 38 134 176
∗ [15,7]-d14c  (2^{2})2^{6}  1,2,4,4,4 0 7 21 0 32 136 176
[15,7]-d14d  (2^{2})2^{6}  1,2,4,4,4 0 7 21 6 34 146 208
[15,7]-d14e  (2^{2})2^{6}3  1,4,4,6 0 7 21 6 50 130 176
∗ [15,7]-d14f  (2^{2})2^{6}3^{2}  1,2,12  0 7 21 0 48 120 144
∗ [15,7]-d14g  (2^{2})2^{6}  1,2,4,4,4 0 7 21 0 16 152 208
[15,7]-d14h  (2^{2})2^{6}3  1,2,12  0 7 21 6 18 162 240
[15,7]-d14i  (2^{2})2^{6}3  1,4,4,6 0 7 21 2 22 150 208
[15,7]-d14j  (2^{2})2^{6}5  1,4,10  0 7 21 2 22 150 208
[15,7]-d14k  (2^{2})2^{6}  1,2,4,4,4 0 7 21 2 22 150 208
∗ [15,7]-d14l  (2^{2})2^{6}3 · 7  1,14  0 7 21 0 0 168 240
[15,7]-e7e8a  (2^{2})2^{6}3  1,2,2,4,6 0 7 21 2 54 118 144
[15,7]-e7e8b  (2^{2})2^{6}3^{2}  1,1,1,6,6 0 7 21 14 58 138 208
∗ [15,7]-e7e8c  (2^{2})2^{5}  1,2,4,4,4 0 7 21 0 32 136 176
[15,7]-e7e8d  (2^{2})2^{5}  1,2,2,2,4,4 0 7 21 6 34 146 208
[15,7]-e7e8e  (2^{2})2^{3}3  1,4,4,6 0 7 21 2 22 150 208
[15,7]-e7e8f  (2^{2})2^{9}3^{2}  1,6,8  0 7 21 6 66 114 144
[15,7]-e7e8g  (2^{2})2^{9}3^{2}  1,6,8  0 7 21 6 82 98 112
[15,7]-e7e8h  (2^{2})2^{6}3^{2}7  1,1,6,7 0 7 21 6 82 98 112
[15,7]-e7e8i  (2^{2})2^{8}  1,2,4,8 0 7 21 2 38 134 176
[15,7]-e7e8j  (2^{2})2^{8}3  1,6,8  0 7 21 14 42 154 240
[15,7]-e7e8k  (2^{2})2^{6}3  1,2,2,4,6 0 7 21 2 38 134 176
[15,7]-e7e8l  (2^{2})2^{8}3  3,4,8  0 7 21 2 22 150 208
∗ [15,7]-e7e8m  (2^{2})2^{6}3  3,4,8  0 7 21 0 16 152 208
[15,7]-e7e8n  (2^{2})2^{6}3  3,4,8  0 7 21 6 18 162 240
∗ [15,7]-e7e8o  (2^{2})2^{5}3  1,4,4,6 0 7 21 0 16 152 208
∗ [15,7]-e7e8p  (2^{2})2^{6}3 · 7  7,8  0 7 21 0 0 168 240

VI. Appendix II: Generator Matrices of the Indecomposable Self-Dual Codes of

Lengths 10 & 11 and Codes Having Minimum Lee Weight 6

[10,2]-d4 =  [10,3]-d4d6 =
[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\ 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \\ 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 2 & 0 & 0 \end{matrix} ]  [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 & 2 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 2 & 1 \\ 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 2 & 2 & 0 & 0 & 0 \end{matrix} ]
[10,4]-d10a =  [10,4]-d10b =
[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 2 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 2 & 2 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 2 & 2 & 2 & 1 \\ 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 \end{matrix} ]  [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 2 & 1 & 0 & 2 \\ 1 & 1 & 0 & 0 & 1 & 0 & 2 & 2 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 2 & 0 & 2 & 1 \\ 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 \end{matrix} ]
[11,3]-d6 =  [11,4]-d4e7 =
[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 1 \\ 2 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 2 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 2 & 2 & 1 \\ 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 \end{matrix} ]

[14,6]-2e7c =

[ \begin{matrix}0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 2 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 & 2 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 2 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 2 & 0 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 2 & 0 & 0 & 2 & 2 & 1 \\ 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ]

[15,4]-f =

[ \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 \\ 2 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 2 & 0 & 2 & 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 \\ 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 2 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 2 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 2 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ]

[15,5]-3d4a =  [15,5]-3d4d =

[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 2 & 2 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 2 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 2 & 0 & 1 & 2 & 2 \\ 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 2 & 0 & 0 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 2 & 2 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 2 & 2 & 2 & 1 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 2 & 2 & 2 & 0 & 1 \\ 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 2 & 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 2 & 0 & 0 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ]

[15,6]-d6d8d =  [15,6]-d6d8f =

[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 0 & 2 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 2 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 2 & 0 & 1 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 2 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 2 & 2 & 2 & 2 & 1 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 2 & 0 & 0 & 0 & 2 & 1 \\ 2 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 0 & 2 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 2 & 2 & 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 3 & 2 & 0 & 0 & 2 & 1 & 0 \\ 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 3 & 2 & 2 & 2 & 2 & 2 & 1 \\ 2 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ]

[15,6]-d6d8g =  [15,6]-d6d8l =

[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 2 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 2 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 2 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 2 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 2 & 2 & 2 & 1 & 0 \\ 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 1 \\ 2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 & 0 & 0 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 2 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 1 & 2 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 2 & 2 & 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 2 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 2 & 0 & 2 & 1 & 0 \\ 1 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 1 \\ 2 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 2 & 0 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ]

[15,7]-d14c =  [15,7]-d14f =

[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 & 0 & 2 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 2 & 2 & 1 & 0 & 2 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 2 & 2 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0 & 2 & 0 & 2 & 1 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 2 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 0 & 2 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 2 & 2 & 2 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 2 & 2 & 2 & 1 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ]

[15,7]-d14g =  [15,7]-d14l =

[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 & 2 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 1 & 0 & 0 & 2 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 2 & 2 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 0 & 2 & 2 & 1 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 0 & 2 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 & 2 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 1 & 0 & 0 & 2 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 2 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 0 & 2 & 2 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 0 & 2 & 2 & 1 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ]

[15,7]-e7e8c =  [15,7]-e7e8m =

[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 & 2 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 2 & 2 & 1 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 2 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 2 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 2 & 2 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 2 & 2 & 2 & 1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 2 & 0 & 2 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 2 & 2 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 2 & 2 & 1 & 2 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 2 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 2 & 2 & 0 & 2 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 2 & 2 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 0 & 0 & 2 & 0 & 2 & 1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ]

[15,7]-e7e8o =  [15,7]-e7e8p =

[ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 & 0 & 2 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 2 & 2 & 1 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 2 & 0 & 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 2 & 0 & 0 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 2 & 2 & 2 & 2 & 1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ] [ \begin{matrix}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 0 & 0 & 2 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 1 & 0 & 0 & 2 & 0 & 2 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 2 & 2 & 1 & 0 & 0 & 2 & 2 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 2 & 0 & 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 2 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 2 & 0 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \end{matrix} ]

References

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[2] J.H. Conway, N.J.A. Sloane, Self-Dual Codes over the Integers Modulo 4, J. Comb. Theory A 62 (1993), 31-45.

[3] Joe Fields, Vera Pless Methods Used in the Classification of ℤ_{4} Codes, Proceedings, Allerton Conference (1996), to appear.

[4] P. Gaborit, Mass Formulas for Self-Dual Codes Over ℤ_{4} and 𝔽 + u𝔽_{q} Rings, IEEE Trans. Inform. Theory IT42 n. 4 (1996), 1222-1228.

[5] Jeffrey S. Leon, Permutation Group Algorithms Based on Partitions, I: Theory and Algorithms, J. Symbolic Computation 12 (1991), 533-583.

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[7] V. Pless, A Classification of Self-Orthogonal Codes over GF(2), Discrete Math. 3 (1972), 209-246.

[8] V. Pless, Parents, Children, Neighbors and the Shadow, Contemporary Math. 168 (1994), 279-290.

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