# On cyclic codes of length 2^e over finite fields

Professor Cunsheng Ding gave cyclotomic constructions of cyclic codes with length being the product of two primes. In this paper, we study the cyclic codes of length n=2^e and dimension k=2^e-1. Clearly, Ding's construction is not hold in this place. We describe two new types of generalized cyclotomy of order two, which are different from Ding's. Furthermore, we study two classes of cyclic codes of length n and dimension k. We get the enumeration of these cyclic codes. What's more, all of the codes from our construction are among the best cyclic codes. Furthermore, we study the hull of cyclic codes of length n over F_q. We obtain the range of ℓ=( Hull(C)). We construct and enumerate cyclic codes of length n having hull of given dimension.

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04/07/2019

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## 1 Introduction

Throughout this paper, let be a finite field, where

is an odd prime power. Hamming weight of codeword

is the number of nonzero , and denoted by wt. If is a -dimensional subspace of , then is referred as an linear code over . If , the code is called MDS code. The (Euclidean) dual code of , denoted by , is defined by

 C⊥={u∈Fnq | u⋅c=0 ∀ c∈C},

where denotes the standard inner product. The hull of linear code is defined by

 Hull(C):=C∩C⊥.

If a linear code is invariant under cyclic shift on , then is called a cyclic code. By identifying a codeword with its polynomial representation in , a linear code over of length is cyclic code if and only if the corresponding set in is just an ideal in . Let be a cyclic code over . Then there is a monic polynomial of minimal degree in such that , where . Furthermore, dimdeg.

Since cyclic codes have efficient encoding and decoding algorithms, they have wide application in storage and communication systems. The cyclic codes were investigated in many literatures [1, 11, 12, 18, 25]. But the enumeration of cyclic codes is in general unknown.

Classical cyclotomy was considered by Gauss in his Disquisitiones Arithmeticae [9] and cyclotomy has a wide application in literatures [7, 8, 2, 19, 14]. The cyclotomic cyclic code is a simple construction of the best cyclic code [3, 5, 6, 26]. Ding and Pless presented a cyclotomic approach to the construction of all binary duadic codes of prime lengths [3]. Ding and Helleseth gave the new generalized cyclotomy [4] and then they studied generalized cyclotomic codes of length and the bound on the minimum odd-like weight of these codes with length [5]. After then, Ding gave three types of generalized cyclotomy and studied three classes of cyclic codes of length being the product of two primes and dimension and proved bound on their minimum odd-like weight[6]

. Recently, Xiong explained some of the numerical data by developing a general method on cyclic codes of composite length and on estimating the minimum distance

[26].

The hull has been studied in many literatures [16, 17, 20, 21, 22, 23, 15, 10, 24]. It is well known that the hull is important in many fields, especially when the hull is small [16, 17, 20, 21]. Sangwisut et. al. obtained the enumerations of cyclic and negacyclic codes of length over having hulls of a given dimension [24]. Recently, Li and Zeng presented sufficient and necessary condition that linear codes and cyclic codes have one-dimensional hull, and they constructed some linear codes and cyclic codes with one-dimension hull [15]. Inspired by these latter works, we consider the cyclic codes of length over finite field .

In this paper, we give some background and recall some basic results in Section 2. In Sections 3 and 4, we get the enumeration of cyclic codes over . We also give a new cyclotomy of order two and construct some best cyclic codes with and , respectively. In Section 5, we construct and enumerate cyclic codes of length having given dimensional hull. We conclude the paper in Section 6.

We will compare some of the codes presented in this paper with the tables of best known linear codes (referred to as the Databse later) maintained by Markus Grassl at http://www.codetab les.de. The example in this paper is computed by Magma.

## 2 Preliminaries

In this paper, we always assume that gcd. A cyclic code of length over is referred as a simple-root cyclic code if and are relatively coprime, otherwise a repeated-root cyclic code. Let be a primitive -th root of unity. Then we have

 xn−1=∏i∈Zn(x−ξi).

Let denote the ring with integer addition and integer multiplication modulo . For each , denote by the -cyclotomic coset of modulo containing . mod , where is the smallest positive integer such that (mod ), and it is the size of the -cyclotomic coset. The smallest integer in is called the coset leader of . Let be the set of all the coset leaders. Then we have

 Zn=⋃a∈Γ(q,n)Ca.

It is well know that is a monic irreducible polynomial over , which is called minimum polynomial of over . Then we have

 xn−1=∏a∈Γ(q,n)ma(x),

which is the factorization of into irreducible factors over . This canonical factorization of over is crucial for the study of cyclic codes. Then we only need to choose the cyclotomic cosets to obtain the generator polynomial of cyclic codes . The set is referred to as the defining set of .

It is well known that is the non-negative minimal complete set of residues modulo . Let denote all the invertible elements of . Denoted by a reduced set of residues modulo . Then we have

 Z¯n={z∈Zn| gcd(z,n)=1}.

We recall the definition of the Euler function . For , is defined to be the number of integers between and relatively prime to . Then we give following lemmas, which can be found in literature [13].

###### Lemma 2.1.

Let symbols be the same as before. Then

 Z∗n=Z¯n.

What’s more

 ∣∣Z∗n∣∣=φ(n),

where denotes the cardinality of .

###### Lemma 2.2.

Let , then has primitive root if and only if is one of following from

 2, 4, pl, 2pl,

where and is an odd prime.

For any sets and , define

 A∖B={x∈A | x∉B}.

## 3 Cyclic codes C(n,n2,q) over Fq with q=2sf+1

Throughout this section let , and , where , and is an odd integer. In this section, let denote the code with length and dimension over .

### 3.1 All cyclic codes C(n,n2,q) over Fq

In this subsection, we study all cyclic codes by the generator polynomial. Assume that the multiplicative order of modulo is , i.e., ord. In the previous section, we give the definitions of and . Next we give definition of new sets as follows

 Ti={ 2ik| 1≤k≤2e−i  and k is odd }, for 0≤i≤e−1,

let and we have (). Then we have the following lemma immediately.

###### Lemma 3.1.

Let symbols be the same as before. Then

 Zn=e⋃i=0Ti.
###### Proof.

For any , we have . Then we have . Note that for , . Furthermore, . Then

From the definition of , we give following lemma, which will be used in the sequel.

###### Lemma 3.2.

For any , then we have

(1) If , ;

(2) If , .

###### Proof.

(1) For any , where and is odd. Then we obtain , Thus . Clearly, we have

(2) The previous subsection is similar to (1). Note that

###### Theorem 3.3.

Let symbols be the same as before. Then the total number of cyclic codes with length over is equal to

 2t2s−1+2s.
###### Proof.

From Lemmas 3.1 and 3.2, we obtain the total number of cyclotomic cosets is

 t2s−1+2s.

Thus we can easily get the desired conclusion.∎

We always assume that . Then we obtain the following theorem.

###### Theorem 3.4.

Let symbols be the same as before. Then the total number of cyclic codes with length and dimension over is equal to

 ∑∑ti=0ji2t−i=2e−1(2s−1j0)(2s−1)j1⋯(2s−1jt−1)(2sjt), (1)

where , and .

###### Proof.

From Lemmas 3.1 and 3.2, we have the number of cyclotomic cosets with is , for , and the number of cyclotomic cosets with is . Next we choose cyclotomic cosets with such that , for . Thus we can easily get the desired conclusion.∎

In order to show that all of the codes from our construction are the best cyclic codes, we provide information about all cyclic codes of length 8 and dimension 4 over , and all codes of length 16 and dimension 8 over in the sequel.

A. All cyclic codes

From formula (1), we know that the total number of is 14. We obtain the factorization of over as follows

 x8−1=f11(x)f12(x)f13(x)f14(x)f21(x)f22(x),

where polynomial as follows

 f11(x)=x+1 f12(x)=x+2 f13(x)=x+3 f14(x)=x+4 f21(x)=x2+2 f22(x)=x2+3

Let and denote the generator polynomial and minimum distance of cyclic codes, respectively. Then we list all cyclic codes in Table 1.

B. All cyclic codes

From formula (1), we know that the total number of is 30. We obtain the factorization of over as follows

 x16−1=f11(x)f12(x)f13(x)f14(x)f21(x)f22(x)f41(x)f42(x),

where polynomial as follows

 f11(x)=x+1 f12(x)=x+2 f13(x)=x+3 f14(x)=x+4 f21(x)=x2+2 f22(x)=x2+3 f41(x)=x4+2 f42(x)=x4+3

Let and denote the generator polynomial and minimum distance of cyclic codes, respectively. Then we list all cyclic codes in Table 2.

### 3.2 A cyclotomy of order two and its codes

From Lemma 2.2, we know that doesn’t have primitive root, where . Then the Ding’s construction in [6] can not be used in this place. In this subsection, we give a new generalized cyclotomic classes of order by the number theory. First of all, we give a important lemma from the number theory [13].

###### Lemma 3.5.

Let , , then the following sentences are hold

1) the degree of modulo is ;

2) the reduced set of residues modulo as follows

 Z¯n={±5b (mod n) | 0≤b<2e−2}.

Note that . Then we the definition of a generalized cyclotomic classes of order 2 as follows

 Wi={(−1)i5b | 0≤b<2e−2}, i=0,1.

Notice that is a subgroup of and . Thus the sets and form a cyclotomy of order 2. From Lemma 3.1 and the proof of the Theorem 3.3. Clearly, If , every set is a union of -cyclotomic set , where If , every set is a union of -cyclotomic set , where In other words, there are -cyclotomic sets , where

Next we can use this cyclotomy () to construct some cyclic codes. We give a definition as follows

 wj(x)=∏i∈Wj(x−ξi), j∈{0,1}.
###### Lemma 3.6.

Let symbols be the same as before. Then we have , where .

###### Proof.

Since , , we obtain . Then we have , where . It then follows that

 wj(x)q=∏i∈Wj(xq−ξqi)=∏i∈qWj(xq−ξi)=wj(xq),

then we have .∎

Let is a union of any -cyclotomic sets , where , and , for . Let is a union of any -cyclotomic sets , where and , and for . Let and . Next we define for each ,

 d(i)0(x)=∏j∈D(i)0(x−ξj), d(i)1(x)=∏j∈D(i)1(x−ξj).

In fact, from Lemma 3.1, we have

 xn−1=w0(x)w1(x)e−1∏i=1d(i)0(x)d(i)1(x).

Next we construct cyclic codes from the cyclotomy . We can obtain cyclic codes with generator polynomials as follows

 g(x)=wj(x)e−1∏i=1d(i)li(x), (2)

where

Obviously, the cyclic codes with generator polynomial have length and dimension . The total number of this cyclic codes with generator polynomial as is equal to

 4(2s−12s−2)t−1e−2∏i=t(2e−i−12e−i−2).

From the construction and the definitions of and , the desired conclusion is obtained immediately. In the sequel, we give some example show that the cyclic codes from this construction are the best cyclic codes.

###### Example 3.7.

Let . Then from the construction we have 8 cyclic codes in Table 3.

By comparing Tables 1 and 3, we know that all of cyclic codes from our construction are the best cyclic codes.

###### Example 3.8.

Let . Then from the construction we have 16 cyclic codes in Table 4.

By comparing Tables 2 and 4, we know that all of cyclic codes from our construction are the best cyclic codes.

## 4 Cyclic codes C(n,n2,q) over Fq with q=2sf+3

Throughout this section let , , where , and is an odd integer. In this section, let denote the code with length and dimension over .

### 4.1 All cyclic codes C(n,n2,q) over Fq

In this subsection, we study all cyclic codes by the generator polynomial. In the previous section, we give the definitions of , and . Then we have the following lemma, which will be used in the sequel.

###### Lemma 4.1.

For any , then we have , what’s more

(1) If , ;

(2) and .

###### Proof.

For any , where and is odd. Then we obtain , Thus . Clearly, we have Note that and . ∎

###### Theorem 4.2.

Let symbols be the same as before. Then the total number of cyclic codes with length over is equal to

 22e−1.
###### Proof.

From Lemmas 3.1 and 4.1, we obtain the total number of cyclotomic cosets is

 2e−1.

Thus we can easily get the desired conclusion.∎

Then we obtain the following theorem.

###### Theorem 4.3.

Let symbols be the same as before. Then the total number of cyclic codes with length and dimension over is equal to

 ∑∑e−2i=0ji2e−i−2=2e−1(2j0)(2j1)⋯(3je−3)(2je−2), (3)

where when or and .

###### Proof.

From Lemmas 3.1 and 4.1, we have the number of cyclotomic cosets with is , for , the number of cyclotomic cosets with is and the number of cyclotomic cosets with is . Next we choose cyclotomic cosets with such that . Thus we can easily get the desired conclusion.∎

Next we give some examples as follows.

A. All cyclic codes

From formula (3), we know that the total number of is 14. We obtain the factorization of over as follows

 x16−1=f11(x)f12(x)f21(x)f22(x)f23(x)f41(x)f41(x),

where polynomial as follows

 f11(x)=x+1 f12(x)=x+10 f21(x)=x2+1 f22(x)=x2+3x+10 f23(x)=x2+8x+10 f41(x)=x4+3x2+10 f42(x)=x4+8x2+10

Let and denote the generator polynomial and minimum distance of cyclic codes, respectively. Then we list all cyclic codes in Table 5.

B. All cyclic codes

From formula (3), we know that the total number of is 30. We obtain the factorization of over as follows

 x32−1=f11(x)f12(x)f13(x)f14(x)f21(x)f22(x)f41(x)f42(x),

where polynomial as follows

 f11(x)=x+1 f12(x)=x+18 f21(x)=x2+1 f22(x)=x2+6x+18 f23(x)=x2+13x+18 f41(x)=x4+6x2+18 f42(x)=x4+13x2+18 f81(x)=x8+6x4+18 f82(x)=x8+13x4+18

Let and denote the generator polynomial and minimum distance of cyclic codes, respectively. Then we list all cyclic codes in Table 6.