New binary self-dual codes of lengths 56, 58, 64, 80 and 92 from a modification of the four circulant construction
Introduction
Self-dual codes are types of linear codes which possess many interesting properties and are closely related to many other mathematical structures. Much research in particular has been invested into developing techniques for constructing new extremal binary self-dual codes. One of the most well-known and extensively applied of the techniques is the four circulant construction, which was first introduced in [1] and uses a matrix G defined by and where A and B are circulant matrices. It follows that G is a generator matrix of a self-dual code of length 4n if and only if . In this work, we give a modification of the four circulant construction which we apply to construct extremal, optimal and best known binary self-dual codes that have previously not been known to exist. The new technique can be used to construct self-dual codes over any commutative Frobenius ring R. We introduce this technique and provide the conditions needed to produce a self-dual code.
For the proof of this technique, we utilise a specialised mapping Θ which was used in [2]. This mapping is inherently associated with the matrix product , where A and B are λ-circulant matrices over R such that . If A is the λ-circulant matrix generated by , then using Θ allows us to verify the equality by computing the values of quantities in terms of a. This eliminates the need to construct A from its generating vector as well as computing the matrix product , which improves computational efficiency. We give and prove our own results concerning Θ as done so in [2].
Using the new technique together with the building-up construction, we find many self-dual codes with weight enumerator parameters of previously unknown values (relative to referenced sources). In total, 93 new codes are found, including
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29 singly-even binary self-dual codes;
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1 binary self-dual code;
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1 singly-even binary self-dual code;
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50 singly-even binary self-dual codes;
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12 binary self-dual codes.
The rest of the work is organised as follows. In Section 2, we give preliminary definitions and results on self-dual codes, Gray maps, circulant matrices, the specialised mapping Θ and the alphabets which we use. In Section 3, we introduce the new technique and conditions needed for producing a self-dual code. In Section 4, we apply the new technique and the building-up construction to obtain the new self-dual codes of lengths 56, 58, 64, 80 and 92, whose weight enumerator parameter values and automorphism group orders we detail. We also tabulate the results in this section. We finish with concluding remarks and discussion of possible expansion on this work.
Section snippets
Self-dual codes
Let R be a commutative Frobenius ring (see [3] for a full description of Frobenius rings and codes over Frobenius rings). Throughout this work, we always assume R has unity. A code of length n over R is a subset of whose elements are called codewords. If is a submodule of , then we say that is linear. Let where and . The (Euclidean) dual of is given by where denotes the Euclidean inner product defined by
The construction
In this section, we present the new technique for constructing self-dual codes. We will hereafter always assume R is a commutative Frobenius ring. Theorem 3.1 Let and where , , and with and . Then G is a generator matrix of a self-dual code over R if and only if where .
Proof We know that G is a generator matrix of a self-dual
Results
In this section, we apply Theorem 3.1 to obtain the following types of new codes
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best known singly-even binary self-dual codes of lengths 56 and 80;
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an extremal singly-even binary self-dual code of length 64;
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extremal binary self-dual codes of length 92.
We also apply the following well-known technique for constructing self-dual codes referred to as the building-up construction. Theorem 4.1 ([11]) Let R be a commutative Frobenius ring. Let be a generator matrix of a self-dual code over R and let
Conclusion
In this work, we presented a technique for constructing self-dual codes which was derived as a modification of the four circulant construction utilising λ-circulant matrices. We proved the necessary conditions required for this technique to produce self-dual codes using a specialised mapping Θ related to λ-circulant matrices. We proved the ability of this technique by using it to construct many best known, optimal and extremal binary self-dual codes which were previously not known to exist. In
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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