# On self-dual four circulant codes
^{†}^{†}thanks: This research is supported by National Natural Science Foundation of China (61672036), Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China (05015133) and the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (2015D11) and Key projects of support program for outstanding young talents in Colleges and Universities (gxyqZD2016008).

###### Abstract

Four circulant codes form a special class of -generator, index , quasi-cyclic codes. Under some conditions on their generator matrices they can be shown to be self-dual. Artin primitive root conjecture shows the existence of an infinite subclass of these codes satisfying a modified Gilbert-Varshamov bound.

Keywords: Quasi-cyclic codes; Self-dual codes; Circulant matrices; Artin primitive root conjecture.

MSC(2010): 94 B15, 94 B25, 05 E30

## 1 Introduction

A matrix over a finite field is said to be circulant if its rows are obtained by successive shifts from the first row. If two circulant matrices and of order satisfy , where the exponent denotes transposition, then the code generated by

(1) |

is a self-dual code. It is a code. This so-called four circulant construction was introduced in [5], revisited in [7], and many self-dual codes with good parameters have been constructed that way.

In the above generator matrix , and are circulant matrices and are determined by the polynomials whose -expansion are the first row of the matrices and , respectively. A code of length , dimension is a four circulant code if it is generated by .

A linear code of length over is a quasi-cyclic code of index for if for each codeword in , the vector , where the subscripts are taken modulo . Hence four circulant codes are quasi-cyclic codes of index . Quasi-cyclic codes form an important class of codes, which have been extensively studied [11, 13, 14]. By the Chinese Remainder Theorem(CRT), quasi-cyclic codes can be decomposed into a “CRT product” of shorter codes over larger alphabets [13]. Such a linear code affords a natural structure of module over the auxiliary ring A linear code of length over is an -submodule of . The ring is never a finite field except if . The CRT shows us that, if is coprime with the characteristic of , then the ring is a direct product of finite fields. If this -submodule is generated by generator sets, the code will be called -generator code. In view of the generator matrix structures of four circulant codes, we see that these codes are -generator.

In the present paper, we study index quasi-cyclic codes of parameters inspired by [1, 3, 8]. When has only two irreducible factors, we derive an enumeration formula of the self-dual four circulant codes over based on that decomposition, and derive an asymptotic lower bound on the minimum distance of these four circulant codes. The asymptotic lower bound is in the spirit of the Gilbert-Varshamov bound. Previous articles [6, 12] also explore this expurgated random coding technique for other families of structured codes.

The material is organized as follows. Section 2 collects the necessary notions and definitions. Section 3 discusses the algebraic structure of four circulant codes. Section 4 presents enumeration formula when only has two irreducible factors over and establishes the asymptotic of self-dual four circulant codes. Section 5 concludes this article.

## 2 Definitions and notations

Let denote a finite field of characteristic and . In the following, we consider codes over of length with coprime to . The code is generated by a matrix of the form (1).

From an algebraic perspective, we can view such a code as an -submodule in and its two generators are , where mod , mod .

If is a family of codes parameters , the rate and relative distance are defined as

and

Both limits are finite as limits of bounded quantities.

## 3 Algebraic structure of self-dual four circulant codes

Throughout this paper, we assume . According to [13], we can cast the factorization of into distinct irreducible polynomials over in the form

where is a self-reciprocal polynomial for , and is the reciprocal polynomial of for .

By the Chinese Remainder Theorem (CRT), we have

Let for simplicity. Note that all of these fields are extensions of . This decomposition naturally extends to as

In particular, each -linear code of length can be decomposed as the “CRT sum”

where for each , is a linear code over of length , and for each , is a linear code over of length and is a linear code over of length , which are called the constituents of .

Note that in terms of constituents of , we have is self-dual only if is self-dual relative to Hermitian product in for , and , for Similar to the proof of [8], we can get the following theorem.

###### Theorem 3.1.

If be a four circulant code over , then is self-dual if and only if .

###### Remark 3.2.

Under the condition of Theorem 3.1, then is a linear complementary-dual four circulant code if and only if .

## 4 Asymptotic of self-dual four circulant codes

### 4.1 Enumeration

In this subsection, we will give enumerative results for self-dual four circulant codes. Recall that the so-called quadratic character of is defined as if is square and if not.

For our purpose, we first give some lemmas without proofs as follows.

###### Lemma 4.1.

([2, Appendix]) If is odd, then the number of solutions in of the equation is .

The proof of the next Lemma while given in odd characteristic in [2] is easily seen to hold more generally, for all prime powers

###### Lemma 4.2.

([2, Appendix]) If is coprime with , then the number of solutions in of the equation is .

On the basis of Lemmas 4.1 and 4.2, we are now ready to state and prove the main result of this subsection.

###### Theorem 4.3.

Let be an odd prime, and be a primitive root modulo . Then, can be factored as a product of two irreducible polynomials over and the number of self-dual four circulant codes of length is is odd and is even. if if

###### Proof.

Since is a primitive root modulo , the cyclotomic cosets of modulo are only two in number, that is and . Due to , the number of monic irreducible factors of over is equal to the number of cyclotomic cosets of modulo . Hence can be factored as a product of two irreducible polynomials over . Let , where is a irreducible polynomial and .

By the CRT approach of [13], can be decomposed as , where and . In the case , we count self-dual four circulant codes of parameters over We can obtain the equation When is odd, it follows from Lemma 4.1 that the number of the solutions of that equation is . When is even, then the equation becomes Hence the number of the solutions of that equation is .

In the case , the factor is a self-reciprocal polynomial of degree , and the number of self-dual four circulant codes of parameters over is equal to the number of solutions of the equation is is odd and is even. ∎ if if Hence the number of self-dual four circulant codes of length . It follows from Lemma 4.2 that the number of the solutions of that equation is

### 4.2 Distance bounds

In number theory, Artin’s conjecture on primitive roots states that a given integer which is neither a perfect square nor is a primitive root modulo infinitely many primes [9]. It was proved conditionally under by Hooley [4]. In this subsection, we study the case when factors as a product of two irreducible polynomials over , i.e. , where is an irreducible polynomial over . We call constant vectors if the codewords of the cyclic code of length generated by . For example, when , , then . Let and denote a polynomial of coprime with , and let be the self-dual four circulant code with generator .

###### Lemma 4.4.

If , where is not a constant vector, then there are at most generators such that

###### Proof.

The condition is equivalent to the system of equations

If is invertible mod , then

(2) |

In that case and are uniquely determined by this system of equations. If is zero or a zero divisor mod , we assume there are solutions to the system. So . Since , then there are only at most choices for Given , we will have choices for by the CRT since is not a constant vector. In total, there are at most choices for ∎

Recall the -ary entropy function defined for by

This quantity is instrumental in the estimation of the volume of high-dimensional Hamming balls when the base field is . The result we are using is that the volume of the Hamming ball of radius is, up to subexponential terms, , when , and goes to infinity [10, Lemma 2.10.3]. The main result obtained in this paper is as follows.

###### Theorem 4.5.

Let be odd prime, and be a primitive root modulo , then there are infinite families of self-dual four circulant codes of relative distance satisfying .

###### Proof.

The four circulant codes containing a vector of weight or less are by standard entropic estimates of [10, Lemma 2.10.3] and Lemma 4.4 bounded above by a quantity of order at most , up to subexponential terms. This number will be less than the total number of self-dual four circulant codes, which is, by Theorem 4.3 of the order as long as This ensures the existence of such codes of distance Letting the result follows. ∎

## 5 Conclusion and Open problems

In this paper, we have considered self-dual four circulant codes. Inspired by [8], we have studied four circulant codes from the algebraic perspective and given an enumeration formula of the self-dual subclass in a special case of the factorization of . This paper can be considered as a companion paper of [8]. The main difference between the two papers is the nontrivial nature of the existence of a factorization of into two irreducible polynomials, which requires Artin’s conjecture on primitive roots, while the same problem for can be solved by elementary means. We have only considered enumeration formulae in the case that the factorization of consists of two irreducible polynomials. It would be a worthwhile task to relax this condition by looking at lengths where the factorization of into irreducible polynomials contains more than two elements. In fact extending our enumerative results to a general factorization of seems to be a difficult task, leading to solving complex diagonal equations over finite fields.

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