Neighborhoods of binary self-dual codes

In this paper, we introduce and investigate the neighborhood of binary self-dual codes. We prove that there is no better Type I code than the best Type II code of the same length. Further, we give some new necessary conditions for the existence of a singly-even $(56,28,12)$-code and a doubly-even $(72,36,16)$-code.


Introduction
Neighbors of self-dual codes were first investigated in [1] and [2].Later, neighbors were used to find extremal (64,32,12) codes in [3] and to find new codes of length 68 in [4].Recently, the graph of neighboring codes was investigated [5].In the current paper, we introduce the definition of a neighborhood of binary self-dual codes.The paper is organized as follows: In Section 2, we mention the main definitions and preliminary results that were essential for our work.Section 3 contains some auxiliary results.In Section 4, we investigate the relationship between neighboring selfdual codes and introduce the neighborhood of self-dual codes.In Section 6, we show that equivalent codes can have different neighborhoods.Finally, in Section 7, we give some research problems.

Preliminaries
Let 2  denote the finite field of two elements, and let n be a positive integer.Then, a subspace of 2 n  is called a binary linear code.We denote a linear code by C.Then, its dual code C ⊥ is defined as where ⋅ denotes the usual scalar product of two vectors.A code is called It is a well-known fact that the dimension of a self-dual code of length n is , 2 n as for each linear code C, we have ( ) ( ) The weight of a codeword , c C ∈ denoted by w(c) is the number of its nonzero coordinates.The minimum weight (or minimum distance) of a code C is the smallest nonzero weight of its codewords.We denote the minimum distance of C by ( ).
Self-dual binary codes can be classified into Type I and Type II codes [6,7].A self-orthogonal binary code C is said to be doubly-even if all of its codewords have weight divisible by 4. If C has a codeword of weight not divisible by 4, then C is a singly-even code [8].Singly-even self-dual codes are called Type I codes and doubly-even self-dual codes are called Type II codes [7].It is well known that Type II codes only exist for length divisible by 8 [9].
We know that the minimum distance of a binary self-dual code is bounded by ( ) [1,9].If a code reaches equality in this bound, then the code is called extremal.Especially for larger codelengths it seems that equality cannot be reached in many cases.Codes with the highest possible minimum distance are called optimal codes.The search for extremal and optimal binary codes is a difficult task for many codelengths n and several researchers have contributed to this theory (see e.g., [10,11]).There are still many open problems about extremal and optimal binary codes [7].

Auxiliary results
Let a and b be two codewords (i.e., binary vectors) of the same length.We denote the numbers of coordinates, which are 1 in both codewords by μ(a, b), i.e., The weight of the sum of two vectors is the following:

Neighbors and neighborhoods
The definition of neighbors among self-dual codes was introduced as follows: Definition 2. Two self-dual codes of length n are called neighbors, provided their intersection is a code of dimension 1.

n −
It is well known that a Type I code has a maximal doubly-even subcode of codimension 1 [1,2].Throughout the paper, we will denote this maximal doubly-even subcode by max , C further we denote the all-1 codeword by 1.

c ⋅ =
We know by [1] and [2] that if C is Type I, then it has two Type II neighbors.
Investigating the neighbors of self-dual codes leads us to some interesting facts about the relationship between neighboring codes.We find out that codes with the same maximal subcode have a special relationship with each other.Therefore, we come up with the definition of a neighborhood for binary self-dual codes.
We extend the definition of neighbors to a set of codes, which are pairwise neighbors.We will call this set of codes a neighborhood.Definition 3. Let C max be a self-orthogonal code of length n, and dimension 1. 2 n − Then, there exist three selfdual codes C 1 , C 2 , and C 3 , which contain C max as a maximal subcode of codimension 1, i.e., they are pairwise neighbors.Then, we say that {C 1 , C 2 , C 3 } are a neighborhood of codes.We will denote this set by  .
Remark 4. The neighborhood  of a doubly-even self-dual code of length divisible by 8 always consists of three codes, one of Type I and two of Type II.
Proof.Let { } ( ) we have by ( ) ( ) w c w c γ is doubly-even, then among the other two codewords exactly one is doubly-even and one is singly-even.
In the following, we investigate the minimum distances of three members of a neighborhood  and their relations.First, it turns out that if the Type I member of a neighborhood  has a minimum distance 2, then the minimum distance of the two Type II members coincides.
Theorem 5. Let  be a neighborhood of binary self-dual codes of length divisible by 8.If the singly-even member of  has a minimum distance 2, then the minimum distance of the doubly-even members coincides.
Proof.We denote the members of  by C 1 , C 2 , and C 3 .Their coinciding doubly-even subcode will be denoted by Furthermore, we assume that ( ) ( ) γ cannot be orthogonal, we have ( ) and ( ) There is no better Type I code than the best possible Type II code of the same length.This answers a question of Rains and Sloane [6] and Conway and Sloane ( [1], p.1321, open question 1) who asked what is the smallest n such that a Type I code of length n is better than the best Type II code of the same length.Using the neighborhood approach, we can see that such a code cannot exist.This shows that the neighborhood approach enables us to see binary self-dual codes in a different way and thus to see relations and properties which were not obvious before.

The existence of certain self-dual codes
The existence of a Type I (56,28,12)-code is an open question [12].It is known that Type II (56,28,12)-code exists [13][14][15].Considering our previous results, we come to the following fact: Corollary 8.If there exists a Type I (56,28,12)-code, then it is a neighbor of a Type II (56,28,12)-code, since 12 is the best possible minimum distance for a self-dual code of length 56.
Corollary 9.If there exists a Type I (72,36,14)-code, then it has a doubly-even neighbor with greater minimum distance, i.e., it has a Type II (72,36,16)-code as a neighbor.

Neighborhood is not unique
The neighborhood of a self-dual code is not unique for permutation equivalent codes.Equivalent codes can be contained in neighborhoods whose members have different properties (minimum distance).For example, the wellknown (24,12,8) Golay code is unique up to permutation equivalence, but it is contained in (at least) two distinct neighborhoods.The first neighborhood is { } where C 1 and C 2 are two permutation equivalent (24,12,8) -codes and C 3 is a (24,12,2)-code.The second neighborhood is where C 4 is a (24,12,6)-code, C 5 is a (24,12,4)-code, and C 6 is a (24,12,8)-code.Generator matrices for C 1 , C 2 , C 3 , C 4 , C 5 , and C 6 are the following matrices G 1 , G 2 , G 3 , G 4 , G 5 , and G 6 respectively.
If C is a Type I code, then its maximal doubly-even subcode is unique, thus its neighborhood is unique as well.

Conclusion and further research
We are convinced that the investigation of binary self-dual codes through their neighborhoods opens new possibilities, both in finding codes whose existence is not known yet and in understanding the relations between selfdual codes better.
By Theorem 6, we know that the minimum distance of a singly-even self-dual code cannot be greater than the minimum distance of its best doubly-even neighbor, but it can be equal.
There are three non-equivalent (32,16,8) Type I codes [16].All of them have a doubly-even neighbor with minimum distance 8, which we computed by the program package TORCH [17].
Therefore, the following questions arise, whose solutions may help to solve open questions like the existence of a Type I (56,28,12)-code or a Type II (72,36,16)-code.
Problem 11.Is there a condition for the codelength n, such that the minimum distance of the best Type II code coincides with the minimum distance of its Type I neighbor?Problem 12.Given a neighborhood { }

Lemma 1 .
Let a,b and c be vectors of the same length.Then, Using the following equations in the given order, the proposition can be directly shown:

C 1 ,
C 2 , and C 3 are self-dual, we immediately get that 1

1 γ
Thus, C 1 is a singly-even code, and C 2 and C 3 are doubly-even codes.By assumption, we have ( ) 1 2, d C = and thus, we can choose such that ( ) 1 2. w γ = Let us denote the minimum distance of C 2 by d.Then, we have for all max

,
of self-dual codes of length divisible by 8. We assume that C 1 is Type I, and C 2 and C 3 are Type II.Then, For technical reasons and to keep the proof as simple as possible, we assume now that and since C max is doubly-even, we may assume that ( ) =  is it possible that all three minimum distances coincide, i.e., d(C 1 ) = d (C 2 ) = d (C 3 )?