New binary self-dual codes of lengths 56, 58, 64, 80 and 92 from a modification of the four circulant construction

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Abstract

In this work, we give a new technique for constructing self-dual codes over commutative Frobenius rings using λ-circulant matrices. The new construction was derived as a modification of the well-known four circulant construction of self-dual codes. Applying this technique together with the building-up construction, we construct singly-even binary self-dual codes of lengths 56, 58, 64, 80 and 92 that were not known in the literature before. Singly-even self-dual codes of length 80 with β{2,4,5,6,8} in their weight enumerators are constructed for the first time in the literature.

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 byG=(I2nX),where X=(ABBTAT), 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 AAT+BBT=In. 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 BAT, where A and B are λ-circulant matrices over R such that λ2=1. If A is the λ-circulant matrix generated by aRn, then using Θ allows us to verify the equality AAT=In by computing the values of n/2+1 quantities in terms of a. This eliminates the need to construct A from its generating vector as well as computing the matrix product AAT, 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

  • 29 singly-even binary self-dual [56,28,10] codes;

  • 1 binary self-dual [58,29,10] code;

  • 1 singly-even binary self-dual [64,32,12] code;

  • 50 singly-even binary self-dual [80,40,14] codes;

  • 12 binary self-dual [92,46,16] 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 C of length n over R is a subset of Rn whose elements are called codewords. If C is a submodule of Rn, then we say that C is linear. Let x,yRn where x=(x1,x2,,xn) and y=(y1,y2,,yn). The (Euclidean) dual C of C is given byC={xRn:x,y=0,yC}, where , denotes the Euclidean inner product defined byx,y=i=1nxiy

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

LetG=(I2nX),whereX=(ATCJBBTCJA) and where J=Jn, A=circλ(a), B=circλ(b) and C=circμ(c) with a,b,cRn and λ,μR:λ2=μ2=1. Then G is a generator matrix of a self-dual [4n,2n] code over R if and only if{xSΘ(x,x,j)[λ]={1,j=0,0,j[1..n/2],Θ(c,c,j)[μ]={1,j=0,0,j[1..n/2], where S={a,b}.

Proof

We know that G is a generator matrix of a self-dual [4n,2n]

Results

In this section, we apply Theorem 3.1 to obtain the following types of new codes

  • best known singly-even binary self-dual codes of lengths 56 and 80;

  • an extremal singly-even binary self-dual code of length 64;

  • 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 G be a generator matrix of a self-dual [2n,n] code C over R and let ri

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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