Four classes of linear codes from cyclotomic cosets

Abstract

This paper presents four classes of linear codes from coset representatives of subgroups and cyclotomic coset families of certain finite field, and determines their weight enumerators. These linear codes may have applications in consumer electronics, communications and secret sharing schemes.

1 Introduction

Let q be a power of a prime p and ${\mathbb F}_q$ be a finite field of size q. An [n, k, d] linear code ${\mathcal C}$ over ${\mathbb F}_q$ is a k-dimensional subspace of ${\mathbb F}_q^{n}$ with minimum distance d. Let $A_{i}$ denote the number of codewords with Hamming weight i in ${\mathcal C}$. The weight enumerator of ${\mathcal C}$ is defined by $1+ A_1 x+ A_{2}x^{2} + \cdots + A_{n}x^{n}$. The weight distribution of a code not only gives the error correcting ability of the code, but also allows the computation of the probability of error detection and correction [12]. So the study of the weight distributions of linear codes is important in both theory and applications.

Let ${\mathbb F}_{{q}^{m}}$ be a finite field with ${q}^{m}$ elements. Note that ${\mathbb F}_{{q}^{m}}$ is the unique degree m extension of ${\mathbb F}_q$. Let $\mathrm{Tr}_q^{{q}^{m}}$ be the trace function from ${\mathbb F}_{{q}^{m}}$ onto ${\mathbb F}_q$. From a subset $D=\{d_1, d_2, \ldots , d_n\}\subset {\mathbb F}_{{q}^{m}}$, we define a class of generic linear code of length $n=|D|$ over ${\mathbb F}_q$ as follows:

$$\begin{aligned} {\mathcal C}_D = \left\{ \mathbf{c}_y = \left( \mathrm{Tr}_q^{{q}^{m}}( yd_1), \mathrm{Tr}_q^{{q}^{m}}( yd_2), \ldots , \mathrm{Tr}_q^{{q}^{m}}(yd_n) \right) : y\in {\mathbb F}_{{q}^{m}} \right\} . \end{aligned}$$

(1)

The dimension of ${\mathcal C}_D$ equals the dimension of the vector space $V_D$ generated by $d_1, d_2, \ldots , d_n \in {\mathbb F}_{{q}^{m}}$ over ${\mathbb F}_q$ [5]. If $e=dim_{ {\mathbb F}_q} V_D<m$, then ${\mathcal C}_D$ has repeated codewords and each codeword repeats $q^{m-e}$ times. If $e=m$ then each codeword occurs once. One may get different codes ${\mathcal C}_D$ from different orderings of the elements of D, but these codes are permutation equivalent and have the same lengths, dimensions and weight distributions. Hence, the orderings of the elements of D will not affect the results in this correspondence.

Let $\zeta _p$ be a primitive p-th root of an unity, and let

$$\begin{aligned} \chi _m(y) = \zeta _p^{\mathrm{Tr}_p^{{q}^{m}}(y)} \,\,\, \mathrm{and} \,\,\, \chi (y) = \zeta _p^{\mathrm{Tr}_p^{q}(y)} \end{aligned}$$

be canonical additive characters over ${\mathbb F}_{{q}^{m}}$ and ${\mathbb F}_q$, respectively. For a codeword $\mathbf{c}_y$ in ${\mathcal C}_D$, its Hamming weight is equal to

$$\begin{aligned} wt( \mathbf{c}_y)= & {} n - | \{ 1\le i\le n \,: \, \mathrm{Tr}_q^{{q}^{m}}(yd_i) = 0\} | \nonumber \\= & {} n-\frac{1}{q}\sum _{d\in D} \sum _{z\in {\mathbb F}_q} \chi \left( z \mathrm{Tr}_q^{{q}^{m}}(d y ) \right) \nonumber \\= & {} \frac{(q-1)n}{q} -\frac{1}{q} \sum _{d\in D} \sum _{z\in {\mathbb F}_{q}^{*}} \chi _m \left( zdy \right) . \end{aligned}$$

(2)

From (2) the weight distribution of ${\mathcal C}_D$ is directly derived from the value distribution of the exponential sum over the subsets ${\mathbb F}_{q}^{*}Dy$ of ${\mathbb F}_{{q}^{m}}^{*}$ for $y\in {\mathbb F}_{{q}^{m}}$. If the set D is well chosen, ${\mathcal C}_D$ may have good parameters. This construction technique was employed in [5, 7] for obtaining linear codes with only a few weights. Following this generic construction, in the past three years, many authors present works [2,3,4, 9,10,11, 14, 15, 20, 21, 24] on constructions of linear codes with few weights. Along this line we will give four classes of linear codes with a defining set from representatives of coset decomposition of subgroups or cyclotomic cosets families of some finite fields. By using some combinatorial techniques and Gauss sums with index 2 case, we determine the weight distributions of these linear codes.

The rest of the paper is organized as follows. Section 2 recalls the theory of Gauss sums over finite fields. Section 3 presents a class of linear codes from coset representatives of some subgroup of ${\mathbb F}_{q}^{*}$, and determines their weight distributions. In Sect. 4 we give three classes of cyclic codes from cyclotomic cosets and determine their weight distributions. Section 5 concludes this paper.

2 Gauss sums

Let ${\mathbb F}_{{q}^{m}}$ be a finite field with ${q}^{m}$ elements, where q is a power of a prime p. The canonical additive character $\chi _m$ over ${\mathbb F}_{{q}^{m}}$ is defined by

$$\begin{aligned} \chi _m(y) =\zeta _p^{\mathrm{Tr}_p^{{q}^{m}}(y)}, \end{aligned}$$

where $\zeta _p$ is a primitive p-th root of an unity and $\mathrm{Tr}_p^{{q}^{m}}(\cdot )$ is the trace function from ${\mathbb F}_{{q}^{m}}$ to ${\mathbb F}_p$. As $m=1$ the $\chi _1$ is the canonical additive character over ${\mathbb F}_q$, and we denote it by $\chi$ for short. The additive characters have the following orthogonal property,

$$\begin{aligned} \sum _{y\in {\mathbb F}_q} \chi (ay) = \left\{ \begin{array}{ll} q, &{}\quad \mathrm{if} \,\, a=0, \\ 0, &{}\quad \mathrm{otherwise }.\end{array} \right. \end{aligned}$$

Let $\psi$ be a multiplicative character of ${\mathbb F}_{q}^{*}$. The Gauss sum over ${\mathbb F}_q$ is defined by

$$\begin{aligned} g(\psi )=\sum \limits _{y\in \mathbb {F}_q^{*}}\psi (y)\chi (y). \end{aligned}$$

Gauss sums can be viewed as the Fourier coefficients in the Fourier expansion of $\psi |_{\mathbb {F}_q^{*}}$ in terms of the multiplicative characters of $\mathbb {F}_q^{*}$.

Lemma 1

(see [16]) Let q be a prime power, $\chi$ the canonical additive character of $\mathbb {F}_q$ is defined by $\chi (y) =\zeta _p^{\mathrm{Tr}_p^{q}(y)},$ where $y\in \mathbb {F}_q.$ And let $\psi$ be a multiplicative character of $\mathbb {F}_q$. Then, for every $y\in \mathbb {F}_q^{*}$,

$$\begin{aligned} \chi (y)=\frac{1}{q-1}\sum \limits _{\psi \in \widehat{\mathbb {F}}_q^{*}} g(\bar{\psi })\psi (y), \end{aligned}$$

where $\bar{\psi }=\psi ^{-1}$ and $\widehat{\mathbb {F}}_q^{*}$ denotes the character group of $\mathbb {F}_q^{*}$.

In general, the explicit evaluation of Gauss sums is very difficult. There are only a few cases where the Gauss sums have been evaluated [13, 17, 18]. We state here some results in the index 2 case which will be used in our constructions. Below we use $\phi (N)$ to denote the number of integers k with $0<k\le N$ such that gcd$(k,N)=1$ and $ord_N(p)$ to denote the order of p modulo N, which is the smallest positive integer f such that $p^f\equiv 1 \pmod N$.

Lemma 2

([22], Theorem 4.12) Let $N=2p_1^{m}$, where m is a positive integer, and $p_1>3$ is a prime such that $p_1\equiv 3 \pmod 4$. Assume that p is a prime such that $[\mathbb {Z}_N^{*}:\langle p \rangle ]=2$. Let $f=\phi (N)/2, q=p^f$, and let $\psi$ be a multiplicative character of order N of $\mathbb {F}_q$. Then for $0\le t<m$ we have

$$\begin{aligned} \begin{array}{l} g(\psi ^{p_1^t})= \left\{ \begin{array}{ll} (-1)^{\frac{p-1}{2}(m-1)}p^{\frac{f-1}{2}-hp_1^t}\sqrt{p^{*}}(\frac{b+c\sqrt{-p_1}}{2})^{2p_1^t}, &{} \mathrm{if}\,\, p_1\equiv 3 \pmod 8, \\ (-1)^{\frac{p-1}{2}m}p^{\frac{f-1}{2}}\sqrt{p^{*}}, &{} \mathrm{if}\,\, p_1\equiv 7 \pmod 8. \\ \end{array} \right. \\ g(\psi ^{2p_1^{t}})= p^{\frac{f-p_1^th}{2}}\big (\frac{b+c\sqrt{-p_1}}{2}\big )^{p_1^t}; \\ g(\psi ^{p_1^{m}})= (-1)^{\frac{p-1}{2}\frac{f-1}{2}}p^{\frac{f-1}{2}}\sqrt{p^{*}}, \end{array} \end{aligned}$$

where $p^{*}=(-1)^{\frac{p-1}{2}}p$, and h is the class number of $\mathbb {Q}(\sqrt{-p_1})$, and b, c are integers determined by $4p^{h}=b^2+p_1c^2$ and $bp^{\frac{f-h}{2}}\equiv -2 \pmod {p_1}$.

Lemma 3

([22], Theorem 4.10) Let $N=p_1^{m_1}p_2^{m_2}$, where $m_1, m_2$ are positive integers, $p_1, p_2$ are primes such that $p_1\equiv 3 \pmod 4$ and $p_2 \equiv 1 \pmod 4$. Assume that p is a prime such that $[\mathbb {Z}_N^{*}:\langle p \rangle ]=2$, ord$_{p_1^{m_1}}(p)=\phi (p_1^{m_1})$ and ord$_{p_2^{m_2}}(p)=\phi (p_2^{m_2})$. Let $f=\phi (N)/2, q=p^f$, and $\psi$ be a multiplicative character of order N of $\mathbb {F}_q$. Then for $0\le s<m_1$ and $0\le t<m_2,$ we have

$$\begin{aligned} g(\psi ^{p_{1}^{s}p_2^t})= & {} p^{\frac{1}{2}(f-h_{12}p_{1}^{s}p_2^t)}\left( \frac{b+c \sqrt{-p_1p_2}}{2}\right) ^{p_{1}^{s}p_2^t}, \\ g(\psi ^{p_1^{m_1}p_2^t})= & {} p^{\frac{f}{2}}; \\ g(\psi ^{p_{1}^{s}p_2^{m_2}})= & {} -p^{\frac{f}{2}}, \end{aligned}$$

where $h_{12}$ is the class number of $\mathbb {Q}(\sqrt{-p_1p_2})$, and b, c are integers determined by $b,c\not \equiv 0 \pmod {p}$, $4p^{h_{12}}=b^2+p_1p_2c^2$ and $b\equiv 2p^{\frac{1}{2}h_{12}} \pmod {p_1}$.

Lemma 4

([22], Theorem 4.14) Let $N=4p_1^m$, where m is a positive integer, and $p_1$ is a prime such that $p_1\equiv 1 \pmod 4$. Assume that p is a prime such that $p\equiv 3\pmod 4$, $[\mathbb {Z}_N^{*}: \langle p \rangle ]=2$ and ord$_{p_1^m}(p)=\phi (p_1^m)$. Let $f=\phi (N)/2, q=p^f$, and let $\psi$ be a multiplicative character of order N of $\mathbb {F}_q$. Then for $0\le s<m$ we have

$$\begin{aligned} g(\psi ^{p_{1}^{s}})= & {} p^{\frac{f}{4}}(b+c\sqrt{p_1})^{p_{1}^{s}},\\ g(\psi ^{p_1^m})= & {} g(\psi ^{2p_1^{s}})=g(\psi ^{2p_1^m})=-p^{\frac{f}{2}}; \\ g(\psi ^{4p_{1}^{s}})= & {} p^{\frac{f}{2}}, \end{aligned}$$

where b, c are integers determined by $p^{\frac{f}{2}}=b^2+p_1c^2$ and $b\equiv -p^{\frac{f}{4}} \pmod {p_1}$.

3 A class of linear codes from coset representatives of some subgroup of ${\mathbb F}_{q}^{*}$

Let G be a subgroup of ${\mathbb F}_{q}^{*}$ with order $\lambda$. Then $\lambda | (q-1)$ and $\lambda | ({q}^{m}-1)$. Let $\alpha$ be a primitive element of ${\mathbb F}_{{q}^{m}}$, then $\beta =\alpha ^{\frac{{q}^{m}-1}{q-1}}$ is a primitive element of ${\mathbb F}_{q}$. It is clear that ${\mathbb F}_{{q}^{m}}^{*}$ has the following coset decomposition,

$$\begin{aligned} {\mathbb F}_{{q}^{m}}^{*} = \mathop \cup \limits _{i=0}^{\frac{{q}^{m}-q}{q-1}} \alpha ^i {\mathbb F}_{q}^{*} = \mathop \cup \limits _{i=0}^{\frac{{q}^{m}-q}{q-1}} \mathop \cup \limits _{j=0}^{\frac{q-\lambda -1}{\lambda }}\alpha ^i \beta ^j G. \end{aligned}$$

(3)

In this section we will study a class of linear codes over ${\mathbb F}_q$ as follows,

$$\begin{aligned} {\mathcal C}_D = \left\{ c_y=( \mathrm{Tr}_q^{{q}^{m}}( yd_1), \mathrm{Tr}_q^{{q}^{m}}( yd_2), \ldots , \mathrm{Tr}_q^{{q}^{m}}(yd_n)) : y\in {\mathbb F}_{{q}^{m}} \right\} , \end{aligned}$$

(4)

where $D=\{d_1, d_2, \ldots , d_n\}\subset {\mathbb F}_{{q}^{m}}$ is a defining set. By choosing proper coset representatives in (3) as the defining set D, we can obtain linear codes ${\mathcal C}_D$ with only few weights. To this end, we introduce some preliminary lemmas.

Lemma 5

Let k be a positive integer with $k\le m$, and let $\alpha _1, \alpha _2, \ldots , \alpha _k\in {\mathbb F}_{{q}^{m}}$ be linearly independent over ${\mathbb F}_{q}$. Denote by

$$\begin{aligned} V_i = \{ y\in {\mathbb F}_{{q}^{m}} \,\, |\,\, \mathrm{Tr}_q^{{q}^{m}} (\alpha _i y) = 0\}, \, i=1, 2, \ldots , k . \end{aligned}$$

Let I and J be subsets of $\{1,2,\ldots , k\}$ such that $I \cap J =\emptyset$ and $I \cup J = \{1,2,\ldots , k\}$. Let $S_I= \mathop \cap \limits _{i\in I} V_i$ and $S_\emptyset = {\mathbb F}_{{q}^{m}}$. Then

$$\begin{aligned} \mid S_I{\setminus } (\mathop \cup _{j\in J} V_j) \mid = q^{m-k} (q-1)^{k-|I|} . \end{aligned}$$

Proof

It is clear that $|I|+|J|=k$, and $|S_I| = q^{m-|I|}$. Let $A_j = S_I\cap V_j$ for $j\in J$.

$$\begin{aligned}&\mid S_I{\setminus } (\mathop \cup _{j\in J} V_j) \mid =\mid S_I\mid - \mid \mathop \cup \limits _{j \in J} A_j \mid \\&\quad = q^{m-|I|} -\left( \sum \limits _{j\in J} |A_j| - \sum \limits _{j_1<j_2} |A_{j_1}\cap A_{j_2}| +\cdots + (-1)^{|J|-1}|\mathop \cap \limits _{j\in J} A_j|\right) \\&\quad = q^{m-|I|}- {|J|\atopwithdelims ()1} q^{m-|I|-1} + {|J| \atopwithdelims ()2} q^{m-|I|-2} +\cdots + (-1)^{|J|} {|J| \atopwithdelims ()|J|} q^{m-|I|-|J|} \\&\quad =q^{m-k} (q-1)^{|J|}=q^{m-k} (q-1)^{k-|I|}. \end{aligned}$$

$\square$

Lemma 6

Let $D_k$ be a set with k elements in ${\mathbb F}_{{q}^{m}}$ which are linearly independent over ${\mathbb F}_{q}$. Let $\chi _m$ be the canonical additive character over ${\mathbb F}_{{q}^{m}}$. For y running through ${\mathbb F}_{{q}^{m}}$, the value distribution of the exponential sum

$$\begin{aligned} S_{D_k} (y) = \sum _{d\in D_k} \sum _{z\in {\mathbb F}_{q}^{*}}\chi _m(z d y) \end{aligned}$$

is as follows,

Values	Frequency
$iq-k$	$q^{m-k} {k \atopwithdelims ()i}(q-1)^{k-i}$,    $i=0,1,\ldots , k$

Proof

It is clear that

$$\begin{aligned} S_{D_k}(y)= & {} \sum _{d\in D_k}\sum _{z\in {\mathbb F}_{q}^{*}}\chi _m( dz y)\\= & {} \sum _{d\in D_k} \sum _{z\in {\mathbb F}_{q}^{*}}\chi \left( \mathrm{Tr}_q^{{q}^{m}}(d y) z\right) , \end{aligned}$$

where $\chi$ is the canonical additive character on ${\mathbb F}_q$. With notations introduced in Lemma 5 we set $\varDelta _{IJ}= S_I{\setminus } (\mathop \cup \nolimits _{j\in J} V_j)$. If $y \in \varDelta _{IJ}$ with $|I|=i$, then

$$\begin{aligned} S_{D_k}(y) = i(q-1)-(k-i) = iq-k . \end{aligned}$$

Moreover, the number of such y is ${ k \atopwithdelims ()i } |\varDelta _{IJ}|$ for $|I|=i$, and by Lemma 5 it equals ${ k \atopwithdelims ()i } q^{m-k} (q-1)^{k-i}$. $\square$

Let $\alpha$ and $\beta$ be as in (3), and $T=\{\beta ^j, \beta ^j\alpha , \ldots , \beta ^j\alpha ^{\frac{{q}^{m}-q}{q-1}}\}$ for some j with $0\le j\le \frac{q-1}{\lambda }-1$. Let $D_n$ with $1\le n\le m$ be a n-subset of T, and elements in $D_n$ are linearly independent over ${\mathbb F}_q$.

Theorem 1

With notations as above, we have

• (1) If $D=T$, then ${\mathcal C}_D$ is a one-weight code with parameters $[\frac{{q}^{m}-1}{q-1}, m , q^{m-1}]$.

• (2) If $D=D_n$, then ${\mathcal C}_D$ is an [n, n, 1] code with the weight enumerator as follows,

  $$\begin{aligned} \sum _{i=0}^n {n \atopwithdelims ()i}(q-1)^{n-i} x^{n-i} . \end{aligned}$$

• (3) If $D= T {\setminus } D_n$ and ${q}^{m}>4$, then ${\mathcal C}_D$ is a $[\frac{{q}^{m}-1}{q-1}-n, m, q^{m-1}-n]$ code with the weight enumerator as follows,

  $$\begin{aligned} 1+ \sum _{i=0}^{n-1} q^{m-n}{n \atopwithdelims ()i} (q-1)^{n-i} x^{q^{m-1}+i-n} + (q^{m-n}-1) x^{q^{m-1}}. \end{aligned}$$

Proof

• (1) When $D= \{\beta ^j, \beta ^j\alpha , \ldots , \beta ^j\alpha ^{\frac{{q}^{m}-q}{q-1}}\}$, we have ${\mathbb F}_{{q}^{m}}^{*}= D {\mathbb F}_{q}^{*}$. For $y\in {\mathbb F}_{{q}^{m}}^{*}$,

  $$\begin{aligned} S_{D}(y) = \sum _{d\in D} \sum _{z\in {\mathbb F}_{q^{*}}} \chi (dz y)=\sum _{z'\in {\mathbb F}_{{q}^{m}}^{*}} \chi (z'y)=-1 . \end{aligned}$$

  It is clear that the dimension of ${\mathcal C}_D$ is m and it has a constant weight $q^{m-1}$ from equation (2).

• (2) When $D= D_n$, from equation (2) and Lemma 6 we can easily get the weight enumerator. Since n elements in D are linearly independent over ${\mathbb F}_q$ we know that the dimension of ${\mathcal C}_D$ is n.

• (3) When $D= T {\setminus } D_n$, for $y\in {\mathbb F}_{{q}^{m}}^{*}$ we have

$$\begin{aligned} S_{D}(y)= & {} \sum _{d\in D} \sum _{z\in {\mathbb F}_{q^{*}}} \chi _m (dz y)\\= & {} \sum _{z'\in {\mathbb F}_{{q}^{m}}^{*}} \chi _m(z'y) -\sum _{d\in D_n} \sum _{z\in {\mathbb F}_{q^{*}}} \chi \left( \mathrm{Tr}_{q}^{{q}^{m}}(dz y)\right) \\= & {} -1 - S_{D_n}(y) . \end{aligned}$$

By (2) and Lemma 6 we get the weight enumerator of ${\mathcal C}_D$.

Remark 1

Let E be a subset of ${\mathbb F}_{q}^{*}$, and $ED=\{ed\,\, |\,\, e\in E, d\in D\}$ where D is introduced in Theorem 1. For each case in Theorem 1, if choose ED as a new defining set we get new linear codes with longer codewords. This result also can be derived from Theorem 1 in [21].

Remark 2

When $|D_n|$ is small, and $D=T{\setminus } D_n$, from (3) of Theorem 1 we get linear codes ${\mathcal C}_D$ with only few weights. Moreover, the code constructed in (3) of Theorem 1 has minimum nonzero weight $w_{min} = q^{m-1}-n$ and maximum weight $w_{max}= q^{m-1}$, and so it satisfies that $w_{min}/w_{max}>\frac{q-1}{q}$ whenever $n<q^{m-2}$. Thus the code can be employed to obtain secret sharing schemes with interesting access structures using the framework in [23].

Example 1

Using the primitive polynomial $f(x)=x^3+x^2-x+1$ $\in \mathbb {F}_{3}[x]$, we construct the $\mathbb {F}_{3^3}$ as $\mathbb {F}_{3}[\alpha ]$ where $f(\alpha )=0$. Let $D=\{1,\alpha \}$ be a defining set. Then each codeword in $\mathcal{C}_D$ repeats once and its weight enumerator of the irreducible cyclic code ${\mathcal C}_D$ defined by (4) is

$$\begin{aligned} 1+4x^{1}+4x^{2}. \end{aligned}$$

This is confirmed by Magma.

4 Three classes of linear codes from cyclotomic cosets

In this section, we investigate the weight distribution of linear codes with a defining set from some cyclotomic cosets, and their Hamming weights of the codewords can be expressed by Gauss sums with index 2 case.

Let N be a divisor of $q-1$ and $\beta$ a fixed primitive element of ${\mathbb F}_{q}$. Define $C_i^{(N, q)}=\beta ^i \langle \beta ^N\rangle$ for $i=0,1,\ldots , N-1$, where $\langle \beta ^N\rangle$ is a subgroup of ${\mathbb F}_{q}^{*}$ generated by $\beta ^N$. Let ${\mathcal C}_D$ be a linear code as follows,

$$\begin{aligned} {\mathcal C}_D = \left\{ \mathbf{c}_x = ( \mathrm{Tr}^q_p( x d_1), \mathrm{Tr}^q_p( x d_2), \ldots , \mathrm{Tr}^q_p( x d_n)), \,\, x\in {\mathbb F}_q \right\} \end{aligned}$$

(5)

with a defining set

$$\begin{aligned} D=\left\{ d_1,d_2,\ldots ,d_n \right\} =C_0^{(N,q)}=\{\beta ^{Nj}:~ 0\le j <n\}, \end{aligned}$$

(6)

where $n=\frac{q-1}{N}$. It is known that ${\mathcal C}_D$ is an irreducible cyclic code. When $\gcd (\frac{q-1}{p-1}, N)$ is small or N is small, Ding and Yang in [6] studied the weight distribution of ${\mathcal C}_D$ by some known Gauss sums and Gauss periods. In this paper, we will use Gauss sum of index 2 case to determine Hamming weight of the codeword in ${\mathcal C}_D$. To this end, we firstly introduce some preliminary lemmas.

Lemma 7

Let q be a prime power and let s be an integer such that $s \,|\,(q-1)$. Let $\tau$ be a multiplicative character of ${\mathbb F}_{q}^{*}$ with order s and $x\in {\mathbb F}_q$, then

$$\begin{aligned} \begin{array}{l} \sum \limits _{0< i < s} \tau ^{i}(x) =\left\{ \begin{array}{ll} s-1, &{}\quad \mathrm{if}\ x\in C_0^{(s,q)}, \\ -1, &{}\quad \mathrm{if}\ x\notin C_0^{(s,q)}\cup \{0\}. \end{array} \right. \end{array} \end{aligned}$$

Lemma 8

Let q be a prime power and let $p'$ be an odd prime such that $p'|(q-1)$. Let $\tau$ be a multiplicative character of $\mathbb {F}_q^{*}$ defined by $\tau (\beta )=\zeta _{p'}$, where $\beta$ is a primitive element of $\mathbb {F}_q$ and $\zeta _{p^\prime }$ is a primitive $p^\prime$-th root of an unity. Then

$$\begin{aligned} \begin{array}{l} \sum \limits _{0< s< p'}\left( \frac{s}{p'} \right) \tau ^{s}(x) =\left\{ \begin{array}{ll} 0, &{}\quad \mathrm{if}\ x\in C_0^{(p',q)}, \\ \\ \left( \frac{i}{p'}\right) \sqrt{(-1)^{\frac{p'-1}{2}}p'}, &{}\quad \mathrm{if}\ x\in C_i^{(p',q)}\ \mathrm{and}\ i\not =0 , \end{array} \right. \end{array} \end{aligned}$$

where $\left( \frac{ \cdot }{p'}\right)$ is the Legendre symbol.

Proof

If $\tau (x)=1$, we have $\sum \limits _{0< s< p'}\left( \frac{s}{p'}\right) \tau ^{s}(x)=\sum \limits _{0< s< p'}\left( \frac{s}{p'}\right) =0.$ Since $\tau (\beta )=\zeta _{p'}$, $x\in C_i^{(p',q)}$ and $i\not =0$, we have $\tau (x)=\tau ^i(\beta )=\zeta _{p'}^i$. It follows that

$$\begin{aligned} \sum \limits _{0< s< p'}\left( \frac{s}{p'}\right) \tau ^{s}(x)= & {} \sum \limits _{0< s< p'}\left( \frac{s}{p'}\right) \tau ^{is}(\beta )\\= & {} \left( \frac{i}{p'}\right) \sum \limits _{0< s< p'}\left( \frac{s}{p'}\right) \zeta _{p'}^s =\left( \frac{i}{p'}\right) \sqrt{(-1)^{\frac{p'-1}{2}}p'}. \end{aligned}$$

This completes the proof. $\square$

In order to determine Hamming weight of the codeword $\mathbf{c}_x$ in (5) we need to calculate the following number.

$$\begin{aligned} S_D(x)= |\left\{ d\in D \,\, |\,\, \mathrm{Tr}_p^q(xd) = 0\right\} |=\frac{n}{p}+\frac{1}{p}\sum \limits _{z\in \mathbb {F}_p^{*}}\sum \limits _{d\in D}\chi (zxd). \end{aligned}$$

(7)

Let $\varphi$ be a multiplicative character of ${\mathbb F}_{q}^{*}$ of order $q-1$ and $\widehat{\mathbb {F}}_q^{*}=\{\varphi ^j:~ 0\le j<q-1\}$. By Fourier expansion formula, Lemma 1 and Lemma 7,

$$\begin{aligned} S_D(x)= & {} \frac{n}{p}+\frac{1}{p(q-1)}\sum \limits _{z\in \mathbb {F}_p^{*}}\sum \limits _{d\in D}\sum \limits _{\psi \in \widehat{\mathbb {F}}_q^{*}} g(\bar{\psi })\psi (zxd)\nonumber \\= & {} \frac{n}{p}+\frac{1}{p(q-1)} \sum \limits _{0\le j<q-1}g(\bar{\varphi }^{j})\varphi ^{j}(x)\sum \limits _{z\in \mathbb {F}_p^{*}}\varphi ^{j}(z)\sum \limits _{d\in D}\varphi ^{j}(d)\nonumber \\= & {} \frac{n}{p}+\frac{1}{pN} \sum \limits _{0\le j<N}g(\bar{\varphi }^{nj})\varphi ^{nj}(x) \sum \limits _{z\in \mathbb {F}_p^{*}}\varphi ^{nj}(z)\nonumber \\= & {} \frac{n}{p}+\frac{p-1}{pN} \mathop {\mathop {\sum }\limits _{0\le j<N,}}\limits _{(p-1)|nj}g(\bar{\varphi }^{nj})\varphi ^{nj}(x)\nonumber \\= & {} \frac{n}{p}+\frac{p-1}{pN} \mathop {\mathop {\sum }\limits _{0\le j<N,}}\limits _{l|j}g(\bar{\varphi }^{nj})\varphi ^{nj}(x)\nonumber \\= & {} \frac{n}{p}+\frac{p-1}{pN} \sum \limits _{0\le j<\frac{N}{l}}g(\bar{\varphi }^{nlj})\varphi ^{nlj}(x), \end{aligned}$$

(8)

where $l=\frac{p-1}{\gcd (n, p-1)}$. From $q-1=n\cdot N$ we know that $l \,\,|\,\, N$.

Next we will construct three classes of irreducible cyclic codes as in (5) and (6) by choosing a proper defining set D, and determine their weight distributions.

Theorem 2

Let $N=2p_1$, where $p_1>3$ is a prime such that $p_1\equiv 3 \pmod 4$. Assume that p is a prime such that $[\mathbb {Z}_N^{*}:\langle p\rangle ]=2$. Let $f=\phi (N)/2$ and $q=p^f$. Let h be the class number of $\mathbb {Q}(\sqrt{-p_1})$, and b, c be integers determined by $4p^{h}=b^2+p_1c^2$ and $bp^{\frac{f-h}{2}}\equiv -2 \pmod {p_1}$. Let ${\mathcal C}_D$ be a linear code defined in (5) and (6), then the weight enumerator of $\mathcal{C}_D$ is as follows:

$$\begin{aligned}&1+ \frac{q-1}{p_1} x^{\frac{p-1}{pN}\left( q- \frac{b\left( p_1-1\right) }{2} p^{\frac{f-h}{2}}\right) }+ \frac{\left( q-1\right) \left( p_1-1\right) }{2p_1}x^{\frac{p-1}{pN}\left( q+\frac{b+cp_1}{2} p^{\frac{f-h}{2}} \right) }\nonumber \\&\quad +\, \frac{\left( q-1\right) \left( p_1-1\right) }{2p_1}x^{\frac{p-1}{pN}\left( q+\frac{b-cp_1}{2} p^{\frac{f-h}{2}}\right) } . \end{aligned}$$

(9)

Proof

Since $p_1$ is a prime with $p_1\equiv 3 \pmod 4$, we have that $f=\frac{p_1-1}{2}$ is odd, and $\frac{q-1}{p-1}=p^{f-1}+\cdots +1\equiv 1 \pmod 2$. Since $p_1>3$ and $[\mathbb {Z}_N^{*}:\langle p\rangle ]=2$, we know that $p\not \equiv 1 \pmod {p_1}$. Since $N=2p_1\, |\, (q-1)$, we have $l=\frac{p-1}{\gcd (n, p-1)}=\frac{(p-1)2p_1}{\gcd (q-1, (p-1)2p_1)}=\frac{2p_1}{\gcd (\frac{q-1}{p-1}, 2p_1)}=2$.

Let $\psi$ be a multiplicative character of $\mathbb {F}_q^{*}$ defined by $\psi (\beta )=\zeta _{2p_1}$. By Lemma 2, we have

$$\begin{aligned} g(\bar{\psi }^{2})= p^{\frac{f-h}{2}}\frac{b+c\sqrt{-p_1}}{2}, \end{aligned}$$

where h is the class number of $\mathbb {Q}(\sqrt{-p_1})$, and b, c are integers determined by $4p^{h}=b^2+p_1c^2$ and $bp^{\frac{f-h}{2}}\equiv -2 \pmod {p_1}$. By the same argument as in Theorem 5.1 of [8] we have

$$\begin{aligned} {\begin{matrix} g(\bar{\psi }^{2s})= \frac{1}{2}p^{\frac{f-h}{2}}\left( b+\left( \frac{s}{p_1}\right) c\sqrt{-p_1} \right) , \end{matrix}} \end{aligned}$$

(10)

where $0< s<p_1$ and $(\frac{\cdot }{p_1})$ is the Legendre symbol.

Let $\tau =\varphi ^{2n}$ in (8). Since $l=2$ and $N=2p_1$, by (8) the Hamming weight of codeword $\mathbf{c}_x$ equals

$$\begin{aligned} wt(\mathbf{c}_x )= & {} n-S_D(x) =\frac{(p-1)n}{p}-\frac{p-1}{pN} \sum \limits _{0\le j<\frac{N}{l}}g(\bar{\tau }^{j})\tau ^{j}(x)\nonumber \\= & {} \frac{(p-1)q}{pN}-\frac{p-1}{pN} \sum \limits _{0< j<\frac{N}{l}}g(\bar{\tau }^{j})\tau ^{j}(x)\nonumber \\= & {} \frac{(p-1)q}{pN}-\frac{p-1}{pN} \sum \limits _{0< j<p_1}g(\bar{\tau }^{j})\tau ^{j}(x). \end{aligned}$$

(11)

By (10) it is followed that

$$\begin{aligned} \sum \limits _{0< j<p_1}g(\bar{\tau }^{j})\tau ^{j}(x)= & {} \sum \limits _{0<s< p_1 }\frac{1}{2}p^{\frac{f-h}{2}} \left( b+\left( \frac{s}{p_1}\right) c\sqrt{-p_1}\right) \tau ^{s}(x) \nonumber \\= & {} \frac{b}{2} p^{\frac{f-h}{2}}\sum \limits _{0<s< p_1 }\tau ^{s}(x) + \frac{c}{2}\sqrt{-p_1} p^{\frac{f-h}{2}} \sum \limits _{0<s< p_1 } \left( \frac{s}{p_1}\right) \tau ^{s}(x).\qquad \quad \end{aligned}$$

(12)

By Lemma 7 and Lemma 8, we have

$$\begin{aligned} \sum \limits _{0< j<p_1}g(\bar{\tau }^{j})\tau ^{j}(x)= \left\{ \begin{array}{ll} \frac{b(p_1-1)}{2} p^{\frac{f-h}{2}}, &{} \ x\in C_0^{(p_1, q)}, \\ - \left( \frac{b}{2} + \frac{cp_1}{2}\left( \frac{i}{p_1}\right) \right) p^{\frac{f-h}{2}}, &{} \ x \ \in C_{i}^{(p_1,q)}\ \mathrm{and}\ i\not =0. \end{array}\right. \end{aligned}$$

(13)

Hence, we obtain

$$\begin{aligned} wt(\mathbf{c}_x ) = \left\{ \begin{array}{ll} \frac{p-1}{pN}\left( q- \frac{b(p_1-1)}{2} p^{\frac{f-h}{2}} \right) , &{} \mathrm{if}\ x\in C_0^{(p_1, q)}, \\ \frac{p-1}{pN}\left( q+ \frac{b}{2} p^{\frac{f-h}{2}}+\frac{cp_1}{2}\left( \frac{i}{p_1}\right) p^{\frac{f-h}{2}}\right) , &{} \mathrm{if}\ x\in C_i^{(p_1, q)}\ \mathrm{and}\ i\not =0. \end{array}\right. \end{aligned}$$

(14)

Frequency of each Hamming weight is derived from (14) directly. So we get the weight enumerator of ${\mathcal C}_D$ as in (9). $\square$

Remark 3

When $b,c\in \{-1,1\}$, the obtained codes are two-weighted.

Example 2

Let $p=11$, $p_1=7$ and $N=14$. Then $f=3$ and $q=11^{3}$. The class number h of $\mathbb {Q}(\sqrt{-7})$ is equal to 1 (see [19]). Since $4p^{h}=b^2+p_1c^2$ and $bp^{\frac{f-h}{2}}\equiv -2 \pmod {p_1}$, we have $c^2=4$ and $b=-4$. By Theorem 2, the weight enumerator of the irreducible cyclic code ${\mathcal C}_D$ defined by (5) is

$$\begin{aligned} 1+190x^{95}+570x^{90}+570x^{80}. \end{aligned}$$

This is confirmed by Magma.

Example 3

Let $p=3$, $p_1=11$ and $N=22$. Then $f=5$ and $q=3^{5}$. The class number h of $\mathbb {Q}(\sqrt{-11})$ is equal to 1 (see [19]). Since $4p^{h}=b^2+p_1c^2$ and $bp^{\frac{f-h}{2}}\equiv -2 \pmod {p_1}$, we have $c^2=1$ and $b=1$. By Theorem 2, the weight enumerator of the irreducible cyclic code ${\mathcal C}_D$ defined by (5) is

$$\begin{aligned} 1+132x^{6}+110x^{9}. \end{aligned}$$

This is confirmed by Magma.

Theorem 3

Let $N=p_1p_2$, where $p_1, p_2$ are primes such that $p_1\equiv 3 \pmod 4$ and $p_2 \equiv 1 \pmod 4$. Assume that p is a prime such that $[\mathbb {Z}_N^{*}:\langle p\rangle ]=2$, ord$_{p_1}(p)=\phi (p_1)$ and ord$_{p_2}(p)=\phi (p_2)$. Let $f=\phi (N)/2, q=p^f$. Let $h_{12}$ be the class number of $\mathbb {Q}(\sqrt{-p_1p_2})$, and let b, c be integers determined by $b,c\not \equiv 0 \pmod {p}$, $4p^{h_{12}}=b^2+p_1p_2c^2$ and $b\equiv 2p^{\frac{1}{2}h_{12}} \pmod {p_1}$. Let $h=\frac{1}{2}(f-h_{12})$ and ${\mathcal C}_D$ be a linear code defined in (5) and (6), then the weight enumerator of the linear code ${\mathcal C}_D$ is as follows:

$$\begin{aligned}&1+ \frac{q-1}{p_1p_2} x^{\frac{p-1}{pN}( q- \frac{b}{2}p^{h}(p_1-1)(p_2-1)-p^{\frac{f}{2}}(p_2-p_1))}\nonumber \\&\quad +\frac{(q-1)(p_2-1)}{p_1p_2} x^{\frac{p-1}{pN}( q+\frac{b}{2}p^{h}(p_1-1)+p^{\frac{f}{2}}p_1)} + \frac{(q-1)(p_1-1)}{p_1p_2} x^{\frac{p-1}{pN}( q+\frac{b}{2}p^{h}(p_2-1)-p^{\frac{f}{2}}p_2)}\nonumber \\&\quad + \frac{(q-1)(p_1-1)(p_2-1)}{2p_1p_2} \left( x^{\frac{p-1}{pN}( q-\frac{b}{2}p^{h}+\frac{cp_1p_2}{2}p^{h})} + x^{\frac{p-1}{pN}( q-\frac{b}{2}p^{h}-\frac{cp_1p_2}{2}p^{h})}\right) . \end{aligned}$$

(15)

Proof

Since $p_1,p_2$ are primes, ord$_{p_1}(p)=\phi (p_1)$ and ord$_{p_2}(p)=\phi (p_2)$, we have $p\not \equiv 1 \pmod {p_1}$ and $p\not \equiv 1 \pmod {p_2}$. Since $N=p_1p_2\,|\,(q-1)$, we have $p_1p_2|\frac{q-1}{p-1}$ and $l=\frac{p-1}{\gcd (n, p-1)}=\frac{(p-1)p_1p_2}{\gcd (q-1, (p-1)p_1p_2)}=\frac{p_1p_2}{\gcd (\frac{q-1}{p-1}, p_1p_2)}=1$.

Let $\tau _1$ be a multiplicative character of $\mathbb {F}_q^{*}$ defined by $\tau _1(\beta )=\zeta _{p_1}$ and let $\tau _2$ be a multiplicative character of $\mathbb {F}_q^{*}$ defined by $\tau _2(\beta )=\zeta _{p_2}$. By Lemma 3, we have

$$\begin{aligned} g(\bar{\tau }_1\bar{\tau }_2)= p^{h}\frac{b+c \sqrt{-p_1p_2}}{2}, \,\, g(\bar{\tau }_2)= p^{\frac{f}{2}}, \,\, g(\bar{\tau }_1)= -p^{\frac{f}{2}}, \end{aligned}$$

where $h_{12}$ is the class number of $\mathbb {Q}(\sqrt{-p_1p_2})$, $h=\frac{1}{2}(f-h_{12})$ and b, c are integers determined by $b,c\not \equiv 0 \pmod {p}$, $4p^{h_{12}}=b^2+p_1p_2c^2$ and $b\equiv 2p^{\frac{1}{2}h_{12}} \pmod {p_1}$. Every multiplicative character of ${\mathbb F}_{q}^{*}$ with order $p_1p_2$ is of the form $\tau _{1}^{s} \tau _2^t$, where $s\in \mathbb {Z}_{p_1}^{*}$ and $t\in \mathbb {Z}_{p_2}^{*}$. By the same argument as in Theorem 5.1 in [8], we have

$$\begin{aligned} g(\bar{\tau }_1^{s}\bar{\tau }_2^t)=\frac{p^{h}}{2}\left( b+c\left( \frac{s}{p_1}\right) \left( \frac{t}{p_2}\right) \sqrt{-p_1p_2}\right) , \,\, g(\bar{\tau }_2^{t})= p^{\frac{f}{2}}, \,\, g(\bar{\tau }_1^{s})= -p^{\frac{f}{2}}, \end{aligned}$$

(16)

where $(\frac{.}{p_1})$ is the Legendre symbol.

Let $\tau =\varphi ^{n}$ in (8). Since $l=1$ and $N=p_1p_2$, from (8) we have

$$\begin{aligned} wt(\mathbf{c}_x )= & {} n-S_D(x) =\frac{(p-1)n}{p}-\frac{p-1}{pN} \sum \limits _{0\le j<\frac{N}{l}}g(\bar{\tau }^{j})\tau ^{j}(x)\nonumber \\= & {} \frac{(p-1)q}{pN}-\frac{p-1}{pN} \sum \limits _{0< j<\frac{N}{l}}g(\bar{\tau }^{j})\tau ^{j}(x)\nonumber \\= & {} \frac{(p-1)q}{pN}-\frac{p-1}{pN} \sum \limits _{0< j<N}g(\bar{\tau }^{j})\tau ^{j}(x). \end{aligned}$$

(17)

Since $\tau$ is a character with order N, we get

$$\begin{aligned} \sum \limits _{0< j<N}g(\bar{\tau }^{j})\tau ^{j}(x)= & {} \mathop {\mathop {\sum }\limits _{0< j< N,}}\limits _{\mathrm{gcd}(j,N)=1} g(\bar{\tau }^{j})\tau ^{j}(x) +\mathop {\mathop {\sum }\limits _{0< j< N,}}\limits _ {\mathrm{gcd}(j,N)=p_2} g(\bar{\tau }^{j})\tau ^{j}(x)\nonumber \\&+\mathop {\mathop {\sum }\limits _{0< j< N,}}\limits _{\mathrm{gcd}(j,N)=p_1} g(\bar{\tau }^{j})\tau ^{j}(x). \end{aligned}$$

(18)

Below we will compute each sum individually. By (16) and Lemma 7 and Lemma 8,

$$\begin{aligned}&\mathop {\mathop {\sum }\limits _{0< j< N,}}\limits _{\mathrm{gcd}(j,N)=1 } g(\bar{\tau }^{j})\tau ^{j}(x) =\mathop {\mathop {\sum }\limits _{0< s< p_1,}}\limits _{0<t<p_2 } g(\bar{\tau }_1^{s}\bar{\tau }_2^t)\tau _1^{s}\tau _2^t(x)\nonumber \\&\quad =\mathop {\mathop {\sum }\limits _{0< s< p_1,}}\limits _{0<t<p_2 } \frac{p^{h}}{2}\left( b+c\left( \frac{s}{p_1}\right) \left( \frac{t}{p_2}\right) \sqrt{-p_1p_2}\right) \tau _1^{s}\tau _2^t(x) \nonumber \\&\quad = \frac{bp^h}{2}\mathop {\mathop {\sum }\limits _{0< s< p_1,}}\limits _{0<t<p_2 }\tau _1^{s}\tau _2^t(x)+\frac{cp^h \sqrt{-p_1p_2}}{2}\sum \limits _{0< s< p_1} \left( \frac{s}{p_1}\right) \tau _1^{s}(x)\sum \limits _{0<t<p_2 }\left( \frac{t}{p_2}\right) \tau _2^t(x) \nonumber \\&\quad =\left\{ \begin{array}{ll} \frac{b}{2}p^{h}(p_1-1)(p_2-1), &{} \mathrm{if}\,\, x\in C_0^{(p_1p_2, q)}, \\ -\frac{b}{2}p^{h}(p_1-1), &{} \mathrm{if}\,\, x\in C_{0}^{(p_1,q)}\,\, \mathrm{and}\,\, x\notin C_{0}^{(p_2, q)}, \\ -\frac{b}{2}p^{h}(p_2-1), &{} \mathrm{if}\,\, x\notin C_{0}^{(p_1, q)}\,\, \mathrm{and}\,\, x\in C_{0}^{(p_2, q)}, \\ \frac{b}{2}p^{h}-\frac{cp_1p_2p^h}{2} \left( \frac{i}{p_1}\right) \left( \frac{j}{p_2}\right) , &{} \mathrm{if}\ x\in C_{i}^{(p_1,q)},\ x\in C_{j}^{(p_2,q)}\ \mathrm{and}\ ij\not =0. \end{array} \right. \end{aligned}$$

(19)

From (16) and Lemma 7 we have

$$\begin{aligned} \mathop {\mathop {\sum }\limits _{0< j< N,}}\limits _{\mathrm{gcd}(j,N)=p_2 } g(\bar{\tau }^{j})\tau ^{j}(x)= & {} \sum \limits _{0<i< p_1 } g(\bar{\tau }_1^{i})\tau _1^{i}(x)=-p^{\frac{f}{2}} \sum \limits _{0<i< p_1 }\tau _1^{i}(x) \nonumber \\= & {} \left\{ \begin{array}{ll} -p^{\frac{f}{2}}(p_1-1), &{} \mathrm{if}\,\, x\in C_0^{(p_1, q)}, \\ p^{\frac{f}{2}}, &{} \mathrm{if}\,\, x\notin C_{0}^{(p_1, q)}\cup \{0\}, \end{array} \right. \end{aligned}$$

(20)

and

$$\begin{aligned} \mathop {\mathop {\sum }\limits _{0< j< N,}}\limits _{\mathrm{gcd}(j,N)=p_1 } g(\bar{\tau }^{j})\tau ^{j}(x)= & {} \sum \limits _{0<i< p_2 } g(\bar{\tau }_2^{i})\tau _2^{i}(x) =p^{\frac{f}{2}} \sum \limits _{0<i< p_2 }\tau _2^{i}(x) \nonumber \\= & {} \left\{ \begin{array}{ll} p^{\frac{f}{2}}(p_2-1), &{} \mathrm{if}\,\, x\in C_0^{(p_2, q)}, \\ -p^{\frac{f}{2}}, &{} \mathrm{if}\,\, x\notin C_{0}^{(p_2, q)}\cup \{0\}. \\ \end{array} \right. \end{aligned}$$

(21)

Combing (17), (18), (19), (20) and (21) we obtain

$$\begin{aligned} wt(\mathbf{c}_x ) = \left\{ \begin{array}{ll} \frac{p-1}{pN}\left( q- \frac{b}{2}p^{h}(p_1-1)(p_2-1)-p^{\frac{f}{2}} (p_2-p_1)\right) , &{} \mathrm{if}\ x\in C_0^{(p_1p_2, q)}, \\ \frac{p-1}{pN}\left( q+\frac{b}{2}p^{h}(p_1-1)+p^{\frac{f}{2}}p_1\right) , &{} \mathrm{if}\ x\in C_{0}^{(p_1,q)}\ \mathrm{and}\ x\notin C_{0}^{(p_2,q)}, \\ \frac{p-1}{pN}\left( q+\frac{b}{2}p^{h}(p_2-1)-p^{\frac{f}{2}}p_2\right) , &{} \mathrm{if}\ x\notin C_{0}^{(p_1,q)}\ \mathrm{and}\ x\in C_{0}^{(p_2,q)}, \\ \frac{p-1}{pN}\left( q-\frac{b}{2}p^{h} + \frac{cp_1p_2}{2}p^{h}\left( \frac{i}{p_1}\right) \left( \frac{j}{p_2}\right) \right) , &{} \mathrm{if}\ x\in C_{i}^{p_1},\ x\in C_{j}^{p_2}\ \mathrm{and}\ ij\not =0. \end{array}\right. \end{aligned}$$

(22)

Frequency of each Hamming weight is derived from equation (22) directly. So we get the weight enumerator of ${\mathcal C}_D$ as in (15). $\square$

Remark 4

When $c^2=1,b=\frac{p_2-p_1}{2}$ and $p^{\frac{h_{12}}{2}}=\frac{p_1+p_2}{4}$, the obtained codes are three-weighted.

Example 4

Let $p=2$, $p_1=3$, $p_2=13$ and $N=39$. Then $f=12$ and $q=2^{12}$. The class number $h_{12}$ of $\mathbb {Q}(\sqrt{-39})$ is equal to 4 (see [1]). Since $b^2+p_1p_2c^2=4p^{h_{12}}$ and $b\equiv 2p^{\frac{h_{12}}{2}}\pmod {p_1}$, we have $c^2=1$ and $b=5$. By Theorem 3, the weight enumerator of the irreducible cyclic code ${\mathcal C}_D$ defined by (5) is

$$\begin{aligned} 1+ \frac{2^{12}-1}{39} x^{32}+ \frac{8(2^{12}-1)}{13} x^{56} + \frac{14(2^{12}-1)}{39} x^{48}. \end{aligned}$$

This weight enumerator coincides with that computed by Magma.

Example 5

Let $p=2$, $p_1=11$, $p_2=5$ and $N=55$. Then $f=20$ and $q=2^{20}$. The class number $h_{12}$ of $\mathbb {Q}(\sqrt{-55})$ is equal to 4 ([1]). Since $b^2+p_1p_2c^2=4p^h$ and $b\equiv 2p^{\frac{h_{12}}{2}} \pmod {p_1}$, we have $c^2=1$ and $b=-3$. By Theorem 3, the complete weight enumerator of the irreducible cyclic code ${\mathcal C}_D$ defined by (5) is

$$\begin{aligned} 1+ \frac{2^{20}-1}{55} x^{9728}+ \frac{24(2^{20}-1)}{55} x^{9600} + \frac{30(2^{20}-1)}{55} x^{9472}, \end{aligned}$$

This weight enumerator coincides with that computed by Magma.

Theorem 4

Let $N=4p_1$, where $p_1$ is prime such that $p_1\equiv 1 \pmod 4$. Assume that p is a prime such that $p\equiv 3\pmod 4$, $[\mathbb {Z}_N^{*}:\langle p \rangle ]=2$ and ord$_{p_1}(p)=\phi (p_1)$. Let $f=\phi (N)/2,\ q=p^f$, and b, c be integers determined by $p^{\frac{f}{2}}=b^2+p_1c^2$ and $b\equiv -p^{\frac{f}{4}} \pmod {p_1}$. Let ${\mathcal C}_D$ be a linear code defined in (5) and (6), then the weight enumerator of $\mathcal{C}_D$ is as follows:

$$\begin{aligned}&1+ \frac{q-1}{4p_1} x^{\frac{\left( p-1\right) }{pN}\left( q +3p^{\frac{f}{2}}-2b\left( p_1-1\right) p^{\frac{f}{4}}\right) }+ \frac{q-1}{2p_1}x^{\frac{\left( p-1\right) }{pN}\left( q- \left( 2p_1-1\right) p^{\frac{f}{2}}\right) } \nonumber \\&\quad +\frac{q-1}{4p_1} x^{\frac{\left( p-1\right) }{pN}\left( q +2 b\left( p_1-1\right) p^{\frac{f}{4}} -p^{\frac{f}{2}}\right) } + \frac{\left( q-1\right) \left( p_1-1\right) }{4p_1} x^{\frac{\left( p-1\right) }{pN}\left( q+ 3p^ {\frac{f}{2}} +2 b p^{\frac{f}{4}}\right) } \nonumber \\&\quad + \frac{\left( q-1\right) \left( p_1-1\right) }{4p_1}x^{\frac{\left( p-1\right) }{pN}\left( q + p^{\frac{f}{2}} + 2 cp_1 p^{\frac{f}{4}}\right) } + \frac{\left( q-1\right) \left( p_1-1\right) }{4p_1}x^{\frac{\left( p-1\right) }{pN}\left( q-p^{\frac{f}{2}} - 2b p^{\frac{f}{4}}\right) } \nonumber \\&\quad + \frac{\left( q-1\right) \left( p_1-1\right) }{4p_1}x^{\frac{\left( p-1\right) }{pN}\left( q+ p^{\frac{f}{2}} -2 cp_1 p^{\frac{f}{4}}\right) }. \end{aligned}$$

(23)

Proof

Since $p_1$ is a prime, and ord$_{p_1}(p)=\phi (p_1)$, we have $p\not \equiv 1 \pmod {p_1}$. Since $4\,|\,f=p_1-1$ and $p\equiv 3\pmod 4$, it follows that $\frac{q-1}{p-1}=p^{f-1}+p^{f-2}+\cdots +1\equiv 0\pmod 4$. Since $N=4p_1\,| \,(q-1)$, we imply that $4p_1|\frac{q-1}{p-1}$ and $l=\frac{p-1}{\gcd (n, p-1)}=\frac{(p-1)4p_1}{\gcd (q-1, (p-1)4p_1)}=\frac{4p_1}{\gcd (\frac{q-1}{p-1}, 4p_1)}=1$.

Let $\tau _1$ be the multiplicative character of $\mathbb {F}_q^{*}$ defined by $\tau _1(\beta )=\zeta _{p_1}$ and let $\tau _2$ be the character of $\mathbb {F}_q^{*}$ defined by $\tau _2(\beta )=\zeta _{4}$. By Lemma 4, we know that

$$\begin{aligned} g(\bar{\tau }_1\bar{\tau }_2)=p^{\frac{f}{4}}(b+c\sqrt{-p_1}),\, g(\bar{\tau }_1)= p^{\frac{f}{2}},\, g(\bar{\tau }_2)=g(\bar{\tau }_{2}^{2})=g(\bar{\tau }_1\bar{\tau }_{2}^{2})= -p^{\frac{f}{2}}, \end{aligned}$$

where b, c are integers determined by $p^{\frac{f}{2}}=b^2+p_1c^2$ and $b\equiv -p^{\frac{f}{4}} \pmod {p_1}$. Every multiplicative character of $\mathbb {F}_q^{*}$ with order $4p_1$ is of the form $\tau _{1}^{s}\tau _2^t$, where $s\in \mathbb {Z}_{p_1}^{*}$ and $t\in \mathbb {Z}_{4}^{*}$. It follows that for any $s\in \mathbb {Z}_{p_1}^{*}$ and $t\in \mathbb {Z}_{4}^{*}$,

$$\begin{aligned}&g(\bar{\tau }_1^{s}\bar{\tau }_2) =p^{\frac{f}{4}}( b+c\left( \frac{s}{p_1}\right) \sqrt{-p_1}),\,\, g(\bar{\tau }_1^{s}\bar{\tau }_{2}^{3}) =p^{\frac{f}{4}}( b-c\left( \frac{s}{p_1}\right) \sqrt{-p_1}),\nonumber \\&g(\bar{\tau }_2^t)=g(\bar{\tau }_2^{2t})=g(\bar{\tau }_{1}^{s}\bar{\tau }_2^{2t})= -p^{\frac{f}{2}}, \,\, g(\bar{\tau }_1^{s})= p^{\frac{f}{2}}, \end{aligned}$$

(24)

where $\left( \frac{.}{p_1}\right)$ is the Legendre symbol.

Let $\tau =\varphi ^{n}$ in (8). Since $l=1$ and $N=4p_1$, by (8) the Hamming weight of codeword $\mathbf{c}_x$ equals

$$\begin{aligned} wt(\mathbf{c}_x )= & {} n-S_D(x) =\frac{(p-1)n}{p}-\frac{p-1}{pN} \sum \limits _{0\le j<\frac{N}{l}}g(\bar{\tau }^{j})\tau ^{j}(x)\nonumber \\= & {} \frac{(p-1)q}{pN}-\frac{p-1}{pN} \sum \limits _{0< j<\frac{N}{l}}g(\bar{\tau }^{j})\tau ^{j}(x)\nonumber \\= & {} \frac{(p-1)q}{pN}-\frac{p-1}{pN} \sum \limits _{0< j<N}g(\bar{\tau }^{j})\tau ^{j}(x). \end{aligned}$$

(25)

Since $\tau$ is a character with order N, it follows that every multiplicative character of $\mathbb {F}_q^{*}$ with order $4p_1$ is of the form $\tau _{1}^{s}\tau _2^{t}$, where $s\in \mathbb {Z}_{p_1}^{*}$ and $t\in \mathbb {Z}_4^{*}$. By (24) we have

$$\begin{aligned}&\sum \limits _{0< j<N}g(\bar{\tau }^{j})\tau ^{j}(x) = \sum \limits _{\tau , o(\tau ) =2} g(\bar{\tau })\tau (x) + \sum \limits _{\tau , o(\tau )=4} g(\bar{\tau })\tau (x)+ \sum \limits _{\tau , o(\tau )=p_1 }g(\bar{\tau })\tau (x)\nonumber \\&\quad +\sum \limits _{\tau , o(\tau )=2p_1 }g(\bar{\tau })\tau (x)+\sum \limits _{\tau , o(\tau )=4p_1 }g(\bar{\tau })\tau (x)\nonumber \\&\quad = g(\bar{\tau _2}^2)\tau _{2}^{2}(x)+ \left( g(\bar{\tau _2})\tau _2(x) + g(\bar{\tau _2}^3)\tau _{2}^{3}(x) \right) + \sum \limits _{0<s<p_1} g(\bar{\tau _1}^s) \tau _{1}^{s}(x) \nonumber \\&\qquad +\sum \limits _{0<s<p_1} g(\bar{\tau _1}^s \bar{\tau _2}^2) \tau _{1}^{s}\tau _{2}^{2}(x) + \sum \limits _{0<s<p_1} \left( g(\bar{\tau _1}^s \bar{\tau _2}) \tau _{1}^{s}\tau _2(x)+g(\bar{\tau _1}^s \bar{\tau _2}^3) \tau _{1}^{s}\tau _{2}^{3}(x)\right) \nonumber \\&\quad = -p^{\frac{f}{2}}\left( \tau _2(x) + \tau _{2}^{2}(x) + \tau _{2}^{3}(x)\right) + p^{\frac{f}{2}} \sum \limits _{0<s<p_1} \tau _{1}^{s}(x) - p^{\frac{f}{2}} \tau _{2}^{2}(x)\sum \limits _{0<s<p_1} \tau _{1}^{s}(x) \nonumber \\&\qquad + \sum \limits _{0<s<p_1} p^{\frac{f}{4}} \left( b+c\left( \frac{s}{p_1}\right) \sqrt{-p_1}\right) \tau _{1}^{s}\tau _2(x) + \sum \limits _{0<s<p_1} p^{\frac{f}{4}} \left( b-c\left( \frac{s}{p_1}\right) \sqrt{-p_1}\right) \tau _{1}^{s}\tau _{2}^{3}(x) \nonumber \\&\quad = -p^{\frac{f}{2}}\left( \tau _2(x) + \tau _{2}^{2}(x) + \tau _{2}^{3}(x)\right) + \left( p^{\frac{f}{2}}(1-\tau _{2}^{2}(x))+bp^{\frac{f}{4}}(\tau _2(x)+\tau _{2}^{3}(x))\right) \sum \limits _{0<s<p_1} \tau _{1}^{s}(x) \nonumber \\&\qquad + cp^{\frac{f}{4}}(\tau _2(x)-\tau _{2}^{3}(x))\sqrt{-p_1} \sum \limits _{0<s<p_1}\left( \frac{s}{p_1}\right) \tau _{1}^{s}(x), \end{aligned}$$

(26)

where $o(\tau )$ denotes the order of the character $\tau$. Next we evaluate above sum according to x.

If $x\in C_0^{(p_1, q)}$ and $x\in C_0^{(4, q)}$ from (26) we have

$$\begin{aligned} \sum \limits _{0< j<N}g(\bar{\tau }^{j})\tau ^{j}(x) = -3p^{\frac{f}{2}} + 2b (p_1-1) p^{\frac{f}{4}}. \end{aligned}$$

If $x\in C_0^{(p_1, q)}$ and $x\in C_1^{(4, q)}$ from (26) we get

$$\begin{aligned} \sum \limits _{0< j<N}g(\bar{\tau }^{j})\tau ^{j}(x) =(2p_1-1)p^{\frac{f}{2}}. \end{aligned}$$

If $x\in C_0^{(p_1, q)}$ and $x\in C_2^{(4, q)}$ from (26) we obtain

$$\begin{aligned} \sum \limits _{0< j<N}g(\bar{\tau }^{j})\tau ^{j}(x) = p^{\frac{f}{2}} - 2b (p_1-1) p^{\frac{f}{4}} . \end{aligned}$$

If $x\in C_0^{(p_1, q)}$ and $x\in C_3^{(4, q)}$ from (26) we imply

$$\begin{aligned} \sum \limits _{0< j<N}g(\bar{\tau }^{j})\tau ^{j}(x) = (2p_1-1)p^{\frac{f}{2}} . \end{aligned}$$

If $x\in C_i^{(p_1, q)}, i\ne 0$ and $x\in C_0^{(4, q)}$ from (26) we have

$$\begin{aligned} \sum \limits _{0< j<N}g(\bar{\tau }^{j})\tau ^{j}(x) = -3p^{\frac{f}{2}}-2b p^{\frac{f}{4}}. \end{aligned}$$

If $x\in C_i^{(p_1, q)}, i\ne 0$ and $x\in C_1^{(4, q)}$ from (26) we get

$$\begin{aligned} \sum \limits _{0< j<N}g(\bar{\tau }^{j})\tau ^{j}(x) = -p^{\frac{f}{2}}+2cp_1 p^{\frac{f}{4}}\left( \frac{i}{p_1}\right) . \end{aligned}$$

If $x\in C_i^{(p_1, q)}, i\ne 0$ and $x\in C_2^{(4, q)}$ from (26) we obtain

$$\begin{aligned} \sum \limits _{0< j<N}g(\bar{\tau }^{j})\tau ^{j}(x) = p^{\frac{f}{2}}+2b p^{\frac{f}{4}} .\quad \ \end{aligned}$$

If $x\in C_i^{(p_1, q)}, i\ne 0$ and $x\in C_3^{(4, q)}$ from (26) we imply

$$\begin{aligned} \ \ \sum \limits _{0< j<N}g(\bar{\tau }^{j})\tau ^{j}(x) = -p^{\frac{f}{2}}-2cp_1 p^{\frac{f}{4}}\left( \frac{i}{p_1}\right) . \end{aligned}$$

Combing (25), (26) and above discussions, we obtain

$$\begin{aligned} wt(\mathbf{c}_x )=\left\{ \begin{array}{ll} \frac{(p-1)}{pN}( q + 3p^{\frac{f}{2}}-2b(p_1-1)p^{\frac{f}{4}} ) , &{} \mathrm{if}\ x\in C_0^{(p_1, q)} \cap C_0^{(4, q)}, \\ \frac{(p-1)}{pN}( q -(2p_1-1)p^{\frac{f}{2}}), &{} \mathrm{if}\ x\in C_0^{(p_1, q)}\cap ( C_1^{(4, q)} \cup C_3^{(4, q)}), \\ \frac{(p-1)}{pN}( q + 2 b(p_1-1)p^{\frac{f}{4}}-p^{\frac{f}{2}}), &{} \mathrm{if}\ x\in C_{0}^{(p_1,q)}\cap C_{2}^{(4,q)}, \\ \frac{(p-1)}{pN}( q +3p^{\frac{f}{2}} +2b p^{\frac{f}{4}}), &{} \mathrm{if}\, x\in C_{i}^{(p_1, q)}\cap C_0^{(4,q)}, i\not =0, \\ \frac{(p-1)}{pN}\left( q+p^{\frac{f}{2}}- 2 c p_1 p^{\frac{f}{4}}\left( \frac{i}{p_1}\right) \right) , &{} \mathrm{if}\ x\in C_{i}^{(p_1,q)}\cap C_1^{(4,q)}, i\not =0, \\ \frac{(p-1)}{pN}( q - p^{\frac{f}{2}} -2b p^{\frac{f}{4}}), &{} \mathrm{if}\ x\in C_{i}^{(p_1,q)} \cap C_{2}^{(4,q)}, i\not =0, \\ \frac{(p-1)}{pN}\left( q+p^{\frac{f}{2}}+2 c p_1 p^{\frac{f}{4}}\left( \frac{i}{p_1}\right) \right) , &{} \mathrm{if}\ x\in C_{i}^{(p_1,q)}\cap C_3^{(4,q)}, i\not =0. \\ \end{array} \right. \end{aligned}$$

(27)

Frequency of each Hamming weight is derived from (27) directly. So we get the weight enumerator of ${\mathcal C}_D$ as in (23).

Example 6

Let $p=3$, $p_1=5$ and $N=20$. Then $f=4$ and $q=3^{4}$. Since $b^2+p_1c^2=p^{\frac{f}{2}}$ and $b\equiv -p^{\frac{f}{4}} \pmod {p_1}$, we have $c^2=1$ and $b=2$. By Theorem 4, the weight enumerator of ${\mathcal C}_D$ defined by (5) is

$$\begin{aligned} 1+ 4 x^{2} + 4 x^{4}. \end{aligned}$$

This result is confirmed by Magma, and shows that ${\mathcal C}_D$ has dimension 2. In fact, the dimension of the vector space generated by $\{1, \alpha ^{20}, \alpha ^{40}, \alpha ^{60}\}$ is 2 for some primitive element $\alpha \in {\mathbb F}_{3^4}$, which is equal to the dimension of the code ${\mathcal C}_D$ by Theorem 6 in [5].

Example 7

Let $p=3$, $p_1=17$ and $N=68$. Then $f=16$ and $q=3^{16}$. Since $b^2+p_1c^2=p^{\frac{f}{2}}$ and $b\equiv -p^{\frac{f}{4}} \pmod {p_1}$, we have $c^2=81$ and $b=72$. By Theorem 4, the weight enumerator of the irreducible cyclic code ${\mathcal C}_D$ defined by (5) is

$$\begin{aligned}&1+ \frac{3^{16}-1}{34} x^{419904}+\frac{3^{16}-1}{68} x^{420390} + \frac{8(3^{16}-1)}{17} x^{421848}+ \\&\frac{8(3^{16}-1)}{17} x^{422334}+\frac{3^{16}-1}{68}x^{423792}. \end{aligned}$$

This weight enumerator coincides with that computed by Magma.

5 Concluding remark

In this paper we present four classes of linear codes from coset decomposition of subgroups and cyclotomic coset families of certain finite field, and determine their weight distributions by Gauss sums with index 2 cases. Many other similar linear codes maybe constructed and their corresponding weight distributions maybe determined by Gauss sums with other index 2 cases.