The b-symbol weight distributions of all semiprimitive irreducible cyclic codes

Abstract

Up to a new invariant $\mu (b)$, the complete b-symbol weight distribution of a particular kind of two-weight irreducible cyclic codes, was recently obtained by Zhu et al. (Des Codes Cryptogr 90(5):1113–1125, 2022). The purpose of this paper is to simplify and generalize the results of Zhu et al., and obtain the b-symbol weight distributions of all one-weight and two-weight semiprimitive irreducible cyclic codes.

1 Introduction

Let $\mathrm{I\!F}_q$ be the finite field with q elements. An [n, l] linear code, $\mathscr {C}$, over $\mathrm{I\!F}_q$ is an l-dimensional subspace of $\mathrm{I\!F}_q^n$ (see for example [4, Section 1.4]). In this context, the vectors of $\mathscr {C}$ are called codewords. Let $A_i$ be the number of codewords with Hamming weight i in $\mathscr {C}$ (recall that the Hamming weight of a codeword c is the number of nonzero coordinates in c). Then, the sequence $A_0$, $A_1$, $\ldots$, $A_n$ is called the Hamming weight distribution of $\mathscr {C},$ and the polynomial $A_0+A_1T+\ldots +A_nT^n$ is called the Hamming weight enumerator of $\mathscr {C}$. An N-weight code is a code such that the cardinality of the set of nonzero weights is N. That is, $N=|\{i:A_i\ne 0,i=1,2,3,\ldots ,n\}|$.

A linear code $\mathscr {C}$ of length n is cyclic if $(c_0,c_1,$ $\ldots ,$ $c_{n-1})\in \mathscr {C}$ implies $(c_{n-1},c_0,c_1,$ $\ldots ,c_{n-2})\in \mathscr {C}$. Cyclic codes have wide applications in storage and communication systems because, unlike encoding and decoding algorithms for linear codes, encoding/decoding algorithms for cyclic codes can be implemented easily and efficiently by employing shift registers with feedback connections (see for example [6, p. 209]). As usual in cyclic codes, we always assume that the length n of any cyclic code is relatively prime to q.

By identifying any vector $(c_0,c_1,\ldots ,c_{n-1})\in \mathrm{I\!F}_q^n$ with the polynomial $c_0+c_1x+\cdots +c_{n-1}x^{n-1}\in \mathrm{I\!F}_q[x],$ it follows that any linear code $\mathscr {C}$ of length n over $\mathrm{I\!F}_q$ corresponds to a subset of the residue class ring $\mathrm{I\!F}_q[x]/\langle x^n-1\rangle$. Moreover, it is well known that the linear code $\mathscr {C}$ is cyclic if and only if the corresponding subset is an ideal of $\mathrm{I\!F}_q[x]/\langle x^n-1\rangle$ (see for example [5, Theorem 9.36]).

Note that every ideal of $\mathrm{I\!F}_q[x]/\langle x^n-1\rangle$ is principal. Thus, if $\mathscr {C}$ is a cyclic code of length n over $\mathrm{I\!F}_q$, then $\mathscr {C}=\langle g(x)\rangle ,$ where g(x) is a monic polynomial, such that $g(x)\mid (x^n-1)$. This polynomial is unique, and it is called the generator polynomial of $\mathscr {C}$ ( [6, Theorem 1, p. 190]). On the other hand, the polynomial $h(x)=(x^n-1)/g(x)$ is referred to as the parity check polynomial of $\mathscr {C}$. A cyclic code over $\mathrm{I\!F}_q$ is called irreducible (reducible) if its parity check polynomial is irreducible (reducible) over $\mathrm{I\!F}_q$.

Denote by $w_H(\cdot )$ the usual Hamming weight function. For $1\le b<n$, let the Boolean function $\bar{\mathcal{Z}}:\mathrm{I\!F}_{q}^b \rightarrow \{0,1\}$ be defined by $\bar{\mathcal{Z}}(v)=0$ iff v is the zero vector in $\mathrm{I\!F}_{q}^b$. The b-symbol Hamming weight, $w_b(\textbf{x})$, of $\textbf{x}=(x_0,\cdots ,x_{n-1}) \in \mathrm{I\!F}_{q}^n$ is defined as

$$\begin{aligned} \begin{aligned} w_b({\textbf {x}})\!:=\!w_H(\bar{\mathcal {Z}}(x_0,\ldots ,x_{b-1}),\bar{\mathcal {Z}}(x_1,\ldots ,x_{b}),\cdots ,\bar{\mathcal {Z}}(x_{n-1},\ldots ,x_{b+n-2 \!\pmod {n}}))\,. \end{aligned} \end{aligned}$$

When $b=1$, $w_1(\textbf{x})$ is exactly the Hamming weight of $\textbf{x}$, that is $w_1(\textbf{x})=w_H(\textbf{x})$. For any $\textbf{x}, \textbf{y} \in \mathrm{I\!F}_{q}^n$, we define the b-symbol distance (b-distance for short) between $\textbf{x}$ and $\textbf{y}$, $d_b(\textbf{x},\textbf{y})$, as $d_b(\textbf{x},\textbf{y}):=w_b(\textbf{x}-\textbf{y})$, and for a code $\mathscr {C}$ (linear or not) over $\mathrm{I\!F}_{q}$ of length n, the b-symbol minimum Hamming distance, $d_b(\mathscr {C})$, of $\mathscr {C}$ is defined as $d_b(\mathscr {C}):=\hbox {min } d_b(\textbf{x},\textbf{y})$, with $\textbf{x},\textbf{y} \in \mathscr {C}$ and $\textbf{x} \ne \textbf{y}$. In this context we will say that $\mathscr {C}$ is a b-symbol code with parameters $(n,M,d_b(\mathscr {C}))_q$, where $M=|\mathscr {C}|$. Let $A_i^{(b)}$ denote the number of codewords with b-symbol Hamming weight i in $\mathscr {C}$. The b-symbol Hamming weight enumerator of $\mathscr {C}$ is defined by

$$\begin{aligned} 1+A_1^{(b)}T+A_2^{(b)}T^2+\cdots +A_n^{(b)}T^n. \end{aligned}$$

Note that if $b=1$, then the b-symbol Hamming weight enumerator of $\mathscr {C}$ is the ordinary Hamming weight enumerator of $\mathscr {C}$. Some contributions to the b-symbol Hamming weight enumerator of a code can be found in [3, 9, 11, 12] and the references therein.

Up to a new invariant $\mu (b)$, the complete b-symbol weight distribution of some irreducible cyclic codes was recently obtained in [12]. The irreducible cyclic codes considered therein, belong to a particular kind of one-weight and two-weight irreducible cyclic codes that were recently characterized in terms of their lengths ([10]). Thus, the purpose of this paper is to present a generalization for the invariant $\mu (b)$, which will allow us to obtain the b-symbol Hamming weight distributions of all one-weight and two-weight irreducible cyclic codes, excluding only the exceptional two-weight irreducible cyclic codes studied in [8].

This work is organized as follows: In Sect. 2, we fix some notation and recall some definitions and some known results to be used in subsequent sections. Section 3 is devoted to presenting preliminary results. Particularly, in this section, we give an alternative proof of an already known result which determines the weight distributions of all one-weight and two-weight semiprimitive irreducible cyclic codes. In Sect. 4, we use such alternative proof, in order to determine the b-symbol weight distributions of all one-weight and two-weight semiprimitive irreducible cyclic codes.

2 Notation, definitions and known results

Unless otherwise specified, throughout this work we will use the following:

Notation. For integers v and w, with $\gcd (v,w)=1$, ${\text {Ord}}_v(w)$ will denote the multiplicative order of w modulo v. By using p, t, q, r, and $\varDelta$, we will denote positive integers such that p is a prime number, $q=p^t$ and $\varDelta =\frac{q^r-1}{q-1}$. From now on, $\gamma$ will denote a fixed primitive element of $\mathrm{I\!F}_{q^r}$. Let u be an integer such that $u|(q^r-1)$. For $i=0,1,\ldots ,u-1$, we define $\mathcal{C}_i^{(u,q^r)}:=\gamma ^i \langle \gamma ^u \rangle$, where $\langle \gamma ^u \rangle$ denotes the subgroup of $\mathrm{I\!F}_{q^r}^*$ generated by $\gamma ^u$. The cosets $\mathcal{C}_i^{(u,q^r)}$ are called the cyclotomic classes of order u in $\mathrm{I\!F}_{q^r}$. For an integer u, such that $\gcd (p,u)=1$, p is said to be semiprimitive modulo u if there exists a positive integer d such that $u|(p^d+1)$. Additionally, we will denote by “${\text {Tr}}_{\mathrm{I\!F}_{q^r}/\mathrm{I\!F}_q}$" the trace mapping from $\mathrm{I\!F}_{q^r}$ to $\mathrm{I\!F}_q$.

Main assumption. From now on, we are going to use n and N as integers in such a way that $nN=q^r-1$, with the important assumption that $r={\text {Ord}}_n(q)$. Under these circumstances, observe that if $h_N(x) \in \mathrm{I\!F}_{q}[x]$ is the minimal polynomial of $\gamma ^{-N}$ (see for example [6, p. 99]), then, due to Delsarte’s Theorem [1], $h_N(x)$ is parity-check polynomial of an irreducible cyclic code of length n and dimension r over $\mathrm{I\!F}_q$.

The following gives an explicit description of an irreducible cyclic code of length n and dimension r over $\mathrm{I\!F}_q$.

Definition 1

Let q, r, n, and N be as before. Then the set

is called an irreducible cyclic code of length n and dimension r over $\mathrm{I\!F}_q$.

An important kind of irreducible cyclic codes are the so-called semiprimitive irreducible cyclic codes:

Definition 2

[10, Definition 4] With our current notation and main assumption, fix $u=\gcd (\varDelta ,N)$. Then, any [n, r] irreducible cyclic code over $\mathrm{I\!F}_{q}$ is semiprimitive if $u \ge 2$ and the prime p is semiprimitive modulo u.

Apart from a few exceptional codes, it is well known that all two-weight irreducible cyclic codes are semiprimitive. In fact, it is conjectured in [8] that the number of these exceptional codes is eleven.

The canonical additive character of $\mathrm{I\!F}_q$ is defined as follows:

$$\begin{aligned} \chi (x):=e^{2\pi \sqrt{-1}\text {Tr}(x)/p} \ \ \ \ \hbox { for all } x \in \mathrm{I\!F}_{q} \end{aligned}$$

where “Tr" denotes the trace mapping from $\mathrm{I\!F}_{q}$ to the prime field $\mathrm{I\!F}_p$. Let $a\in \mathrm{I\!F}_q$. The orthogonality relation for the canonical additive character $\chi$ of $\mathrm{I\!F}_q$ is given by (see for example [5, Chapter 5]):

$$\begin{aligned} \sum _{x \in \mathrm{I\!F}_{q}} \chi (ax)=\left\{ \begin{array}{cl} \;q\; &{} \hbox { if a=0,} \\ \\ \;0\; &{} \hbox { otherwise.} \end{array} \right. \end{aligned}$$

This property plays an important role in numerous applications of finite fields. Among them, this property is useful for determining the Hamming weight of a given vector over a finite field; for example if $V=(a_0,a_1,\ldots ,a_{n-1}) \in \mathrm{I\!F}_{q}^n$, then

$$\begin{aligned} w_H(V)=n-\frac{1}{q}\sum _{i=0}^{n-1}\sum _{y \in \mathrm{I\!F}_{q}}\chi (ya_i). \end{aligned}$$

(1)

Let $\chi '$ be the canonical additive character of $\mathrm{I\!F}_{q^r}$ and let $u \ge 1$ be an integer such that $u|(q^r-1)$. For $i=0,1,\ldots ,u-1$, the i-th Gaussian period, $\eta _i^{(u,q^r)}$, of order u for $\mathrm{I\!F}_{q^r}$ is defined to be

$$\begin{aligned} \eta _i^{(u,q^r)}:=\sum _{x \in \mathcal{C}_i^{(u,q^r)}} \chi '(x). \end{aligned}$$

Suppose that $a\in \mathcal{C}_i^{(u,q^r)}$. Since $\sum _{x \in \mathrm{I\!F}_{q^r}}\chi '(ax^u)=u\eta _i^{(u,q^r)}+1$ and $\eta _0^{(1,q^r)}+1=0$, the following result is a direct consequence of Theorem 1 in [7]:

Theorem 1

With our notation suppose that $rt=2sd$ and $u | (p^d+1)$, for positive integers s, d and u. Then

$$\begin{aligned} \frac{u\eta _i^{(u,q^r)} + 1}{q^{r/2}}= \left\{ \begin{array}{cl} (-1)^{s-1}(u-1) &{} \hbox { if } i \equiv \delta \pmod {u} \,, \\ \\ (-1)^{s} &{} \hbox { if } i \not \equiv \delta \pmod {u} \,, \end{array} \right. \end{aligned}$$

where the integer $\delta$ is defined in terms of the following two cases:

$$\begin{aligned} \begin{aligned} \delta := \left\{ \begin{array}{cl} 0\; &{}{} \text{ if } u=1; \text{ or } p=2; \text{ or } p>2 \text{ and } 2|s; \text{ or } p>2, 2 \not \mid s, \text{ and } 2|\frac{p^d+1}{u} \,, \\ {} &{}{} \\ \frac{u}{2}\; &{}{} \text{ if } p>2, 2 \not \mid s \text{ and } 2 \not \mid \frac{p^d+1}{u} \,. \end{array} \right. \end{aligned} \end{aligned}$$

Remark 1

As shown below, by means of the previous theorem, it is possible to determine, in a single result, the Hamming weight enumerator of all one-weight and semiprimitive two-weight irreducible cyclic codes.

Under certain circumstances, and for a fixed coset $\mathcal{C}_i^{(N,q^r)}$, it is necessary to consider the set of products of the form xy, where $x \in \mathcal{C}_i^{(N,q^r)}$ and $y \in \mathrm{I\!F}_{q}^*$. The following result goes in this direction:

Lemma 1

[2, Lemma 5] Let N be a positive divisor of $q^r-1$ and let i be any integer with $0 \le i < N$. Fix $u=\gcd (\varDelta ,N)$. We have the following multiset equality:

$$\begin{aligned} \left\{ xy: x \in \mathcal{C}_i^{(N,q^r)}, \;y \in \mathrm{I\!F}_{q}^* \right\} =\frac{(q-1)u}{N} * \mathcal{C}_i^{(u,q^r)}, \end{aligned}$$

where $\frac{(q-1)u}{N} * \mathcal{C}_i^{(u,q^r)}$ denotes the multiset in which each element in the set $\mathcal{C}_i^{(u,q^r)}$ appears in the multiset with multiplicity $\frac{(q-1)u}{N}$.

The following definitions are inspired by and similar to those of [12].

Definition 3

Let b be an integer, with $1\le b\le r$. Let $\mathcal{P}(b)$ be the subset of cardinality $(q^b-1)/(q-1)$ in $\mathrm{I\!F}_{q^r}^*$ defined as

$$\begin{aligned} \mathcal{P}(b):=\bigcup _{j=1}^{b-1} \{\gamma ^{(j-1)N}+x_1\gamma ^{jN}+\cdots +x_{b-j}\gamma ^{(b-1)N}: x_1,\ldots ,x_j \in \mathrm{I\!F}_q \} \cup \{\gamma ^{(b-1)N}\}. \end{aligned}$$

Remark 2

Note that $\mathcal{P}(1)=\{1\}$.

Definition 4

Let b be as in Definition 3 and fix $u=\gcd (\varDelta ,N)$. For $0\le i < u$, we define $\mu _{(i)}(b)$ as

$$\begin{aligned} \mu _{(i)}(b):=|\{ x \in \mathcal{P}(b): x \in \mathcal{C}_i^{(u,q^r)} \}|. \end{aligned}$$

Remark 3

Since $\mathcal{C}_0^{(2,q^r)}=\{ x\in \mathrm{I\!F}_{q^r}^*: x \hbox { is a square in } \mathrm{I\!F}_{q^r}^*\}$, note that $\mu _{(i)}(b)$ is indeed a generalization of the invariant $\mu (b)$ in [12]. Furthermore, note that $\mu _{(0)}(1)=1$ and $\mu _{(i)}(1)=0$, for $1 \le i < u$.

The following important result from [12], is key in order to achieve our goals.

Lemma 2

[12, Lemma 4.3] Let $\mathscr {C}$ be as in Definition 1 and let $c(a) \in \mathscr {C}$ be a codeword. Then, for any integer $1\le b\le r$,

$$\begin{aligned} w_b(c(a))=\frac{1}{q^{b-1}}\sum _{\theta \in \mathcal{P}(b)}w_H(c(\theta a)). \end{aligned}$$

Remark 4

The previous lemma is key for us because, although the condition $\gcd (\frac{q^r-1}{q-1},N)=2$ is one of the main assumptions in [12], Lemma 4.3 is beyond such condition. However it is important to observe that there is a small misprint in the proof of Lemma 4.3; more specifically the equality

$$\begin{aligned} n-w_1(c(a))=\sum _{x \in I} \frac{1}{q} \sum _{y\in \mathrm{I\!F}_{q}} \chi (yax), \end{aligned}$$

should be

$$\begin{aligned} n-w_1(c(a))=\sum _{x \in I} \frac{1}{q} \sum _{y\in \mathrm{I\!F}_{q}} \chi (yax^N). \end{aligned}$$

3 Preliminary results

In the light of Remark 3, the following is a generalization of [12, Lemma 2.1].

Lemma 3

Let b and $\mu _{(i)}(b)$ be as in Definition 4. If $b=r$ then, for any $0\le i < u$, we have

$$\begin{aligned} \mu _{(i)}(r)=\frac{1}{u}|\mathcal{P}(b)|=\frac{\varDelta }{u}. \end{aligned}$$

Proof

Clearly

$$\begin{aligned} \mathrm{I\!F}_{q^r}^*=\bigsqcup _{x \in \mathcal{P}(b)} x\mathrm{I\!F}_{q}^*, \end{aligned}$$

where $\sqcup$ is a disjoint union. Now, since $u| \varDelta$ and $\langle \gamma ^{\varDelta } \rangle =\mathrm{I\!F}_{q}^*$, $x \in \mathcal{C}_i^{(u,q^r)}$ if and only if each element of $x\mathrm{I\!F}_{q}^*$ is also in $\mathcal{C}_i^{(u,q^r)}$. This implies that

$$\begin{aligned} \mu _{(i)}(r)(q-1)=\frac{q^r-1}{u}, \end{aligned}$$

which is the number of elements in $\mathcal{C}_i^{(u,q^r)}$. This completes the proof. $\square$

It is already known the Hamming weight enumerator of all one-weight and semiprimitive two-weight irreducible cyclic codes over any finite field (see for example [8, 10]). By means of the following theorem we recall such a result and give an alternative proof of it. As will be clear later, this alternative proof will be important for fulfilling our goals.

Theorem 2

Let $\mathscr {C}$ be as in Definition 1. Fix $u=\gcd (\varDelta ,N)$. Assume that $u=1$ or p is semiprimitive modulo u. Let d be the smallest positive integer such that $u | (p^d+1)$ and let $s=1$ if $u=1$ and $s=(rt)/(2d)$ if $u>1$. Fix

$$\begin{aligned} W_A=\frac{n q^{r/2-1}}{\varDelta }(q^{r/2}-(-1)^{s-1}(u-1))\;\;\; \hbox { and } \;\;\;W_B=\frac{n q^{r/2-1}}{\varDelta }(q^{r/2}-(-1)^s). \end{aligned}$$

Then, $\mathscr {C}$ is an [n, r] irreducible cyclic code whose Hamming weight enumerator is

$$\begin{aligned} 1+\frac{q^r-1}{u}T^{W_A}+\frac{(q^r-1)(u-1)}{u}T^{W_B}. \end{aligned}$$

(2)

Remark 5

Note that Theorem 2 gives, in a single result, an explicit description of the Hamming weight enumerators of all one-weight ($u=1$) and two-weight ($2\le u < \varDelta$) irreducible cyclic codes, excluding only the exceptional two-weight irreducible cyclic codes studied in [8]. Therefore observe that the two-weight irreducible cyclic codes considered in [12] ($u=\gcd (\varDelta ,N)=2$) belong also to Theorem 2.

Proof

First note that if $u>1$, then there must exist an integer s such that $rt=2sd$.

For $a \in \mathrm{I\!F}_{q^r}^*$, let $c(a) = ({\text {Tr}}_{\mathrm{I\!F}_{q^r}/\mathrm{I\!F}_q}(a \gamma ^{Ni}))_{i=0}^{n-1} \in \mathscr {C}$. Let $\chi$ and $\chi '$ be the canonical additive characters of $\mathrm{I\!F}_{q}$ and $\mathrm{I\!F}_{q^r}$, respectively. Thus, by the orthogonality relation for the character $\chi$ (see (1)) the Hamming weight of the codeword c(a), $w_H(c(a))$, is

where the last equality holds by Lemma 1. Now, suppose that $a \in \mathcal{C}_i^{(u,q^r)}$ for some $0\le i < u$. Thus

$$\begin{aligned} w_H(c(a))= & {} n-\frac{n}{q}-\frac{(q-1)}{qN}u \eta _i^{(u,q^r)}\\= & {} \frac{n}{\varDelta q}(q^r-1)-\frac{n}{\varDelta q}u \eta _i^{(u,q^r)} \\= & {} \frac{nq^{r-1}}{\varDelta }-\frac{n}{\varDelta q}(u \eta _i^{(u,q^r)}+1)\\= & {} \frac{nq^{r-1}}{\varDelta }-\frac{n q^{r/2-1}}{\varDelta }\frac{(u\eta _i^{(u,q^r)} + 1)}{q^{r/2}}\\= & {} \frac{n q^{r/2-1}}{\varDelta }(q^{r/2}-\frac{u\eta _i^{(u,q^r)} + 1}{q^{r/2}}). \end{aligned}$$

Let $\delta$ be as in Theorem 1 and observe that $i \equiv \delta \pmod {u}$ iff $a \in \mathcal{C}_{\delta }^{(u,q^r)}$. Therefore, owing to Theorem 1, we have

$$\begin{aligned} \begin{aligned} w_H(c(a))= \left\{ \begin{array}{cl} W_A\; &{}{} \text{ if } a \in \mathcal {C}_{\delta }^{(u,q^r)}\,, \\ {} &{}{} \\ W_B\; &{}{} \text{ if } a \in \mathrm {I\!F}_{q^r}^* \setminus \mathcal {C}_{\delta }^{(u,q^r)}\,. \end{array} \right. \end{aligned} \end{aligned}$$

(3)

The result now follows from the fact that $|\mathcal{C}_{\delta }^{(u,q^r)}|=\frac{q^r-1}{u}$ and $|\mathrm{I\!F}_{q^r}^* {\setminus }\mathcal{C}_{\delta }^{(u,q^r)}|=\frac{(q^r-1)(u-1)}{u}$. $\square$

4 The b-symbol weight distribution of all one-weight and two-weight semiprimitive irreducible cyclic codes

We are now in conditions to present our main results.

Theorem 3

Assume the same notation and assumptions as in Theorem 2. Let $\mathcal{P}(b)$, $\mu _{(i)}(b)$, and $\delta$ be as before. For $0 \le i < u$ and $1 \le b \le r$, let

$$\begin{aligned} \begin{aligned} W_i^{(b)} = \frac{(q-1)q^{r/2-b}}{N}\left[ |\mathcal {P}(b)|\left( q^{r/2}-(-1)^s\right) +(-1)^s u\mu _{((\delta -i)\!\pmod {u})}(b)\right] \end{aligned} \end{aligned}$$

(4)

Then, the b-symbol Hamming weight enumerator of $\mathscr {C}$ is

$$\begin{aligned} A(T)=1+\frac{q^r-1}{u}\sum _{i=0}^{u-1}T^{W_i^{(b)}}. \end{aligned}$$

(5)

Proof

Let $a\in \mathrm{I\!F}_{q^r}^*$ and let $c(a) \in \mathscr {C}$. Let $W_A$ and $W_B$ be as in Theorem 2 and suppose that $a\in \mathcal{C}_i^{(u,q^r)}$, for some $0 \le i < u$. Thus, from (3), $w_H(c(\theta a))=W_A$ iff $\theta a \in \mathcal{C}_\delta ^{(u,q^r)}$ iff $\theta \in \mathcal{C}_{(\delta -i)\!\!\pmod {u}}^{(u,q^r)}$. But there are exactly $\mu _{((\delta -i)\!\!\pmod {u})}(b)$ elements $\theta$ in $\mathcal{P}(b)$ that satisfy the condition $\theta \in \mathcal{C}_{(\delta -i)\!\!\pmod {u}}^{(u,q^r)}$. Therefore, owing to Lemma 2, $w_b(c(a))=W_i^{(b)}$ where

$$\begin{aligned} \begin{aligned} W_i^{(b)}=\frac{1}{q^{b-1}}\left[ \mu _{((\delta -i)\!\pmod {u})}(b)W_A+\left( |\mathcal {P}(b)|-\mu _{((\delta -i)\!\pmod {u})}(b)\right) W_B \right] . \end{aligned} \end{aligned}$$

Hence, (4) follows by considering the explicit values of $W_A$ and $W_B$ in Theorem 2. Finally, the b-symbol Hamming weight enumerator of $\mathscr {C}$ follows from (3) and from the fact that $|\mathcal{C}_i^{(u,q^r)}|=\frac{q^r-1}{u}$, for any $0\le i < u$. $\square$

Note that the previous theorem is also valid for $b=1$. In fact, in this case, the ordinary Hamming weight enumerator in (2) is exactly the same as the 1-symbol Hamming weight enumerator of (5) (take into consideration Remarks 2 and 3). Therefore we see that Theorem 3 not only simplifies and generalizes [12, Corollary 3.1] but also generalizes Theorem 2.

Example 1

The following are some examples of Theorem 3.

• (a) Let $(q,r,N,b)=(3,4,2,3)$. Thus $u=\gcd (\varDelta ,N)=2$, $s=2$, $\delta =0$, and $|\mathcal{P}(b)|=q^2+q+1=13$. Since $\mu _{(0)}(b)=8$ (see [12, Example 2.3]), $\mu _{(1)}(b)=|\mathcal{P}(b)|-\mu _{(0)}(b)=5$. Therefore, owing to Theorems 2 and 3, $W_A=30$, $W_B=24$, $W_0^{(3)}=40$, $W_1^{(3)}=38$, and $\mathscr {C}$ is a $[40,4]_3$ irreducible cyclic code whose ordinary and 3-symbol Hamming weight enumerators are $1+40T^{24}+40T^{30}$ and $1+40T^{38}+40T^{40}$, respectively.

• (b) Let $(q,r,N,b)=(2,4,3,2)$. Thus $u=\gcd (\varDelta ,N)=3$, $s=2$, $\delta =0$, and $|\mathcal{P}(b)|=q+1=3$. We take $\mathrm{I\!F}_{16}=\mathrm{I\!F}_2(\gamma )$ with $\gamma ^4+\gamma +1=0$. Hence $\mathcal{P}(b)=\{1=\gamma ^0,\gamma ^3,1+\gamma ^3=\gamma ^{14}\}$. This means that $\mu _{(0)}(b)=2$, $\mu _{(1)}(b)=0$, and $\mu _{(2)}(b)=1$. Therefore, owing to Theorems 2 and 3, $W_A=4$, $W_B=2$, $W_0^{(2)}=5$, $W_1^{(2)}=4$, $W_2^{(2)}=3$, and $\mathscr {C}$ is a $[5,4]_2$ irreducible cyclic code whose ordinary and 2-symbol Hamming weight enumerators are $1+10T^{2}+5T^{4}$ and $1+5(T^{3}+T^{4}+T^{5})$, respectively.

• (c) Let $(q,r,N,b)=(4,3,9,2)$. Thus $u=\gcd (\varDelta ,N)=3$, $s=3$, $\delta =0$, and $|\mathcal{P}(b)|=q+1=5$. Let $\mathrm{I\!F}_{4}=\mathrm{I\!F}_2(\alpha )$ with $\alpha ^2+\alpha +1=0$. We take $\mathrm{I\!F}_{64}=\mathrm{I\!F}_4(\gamma )$ with $\gamma ^3+\gamma ^2+\gamma +\alpha =0$. Hence $\mathcal{P}(b)=\{1=\gamma ^0,\gamma ^9,1+\gamma ^9=\gamma ^{27},1+\alpha \gamma ^9=\gamma ^{5},1+\alpha ^2\gamma ^9=\gamma ^{40}\}$. This means that $\mu _{(0)}(b)=3$, $\mu _{(1)}(b)=1$, and $\mu _{(2)}(b)=1$. Therefore, owing to Theorems 2 and 3, $W_A=4$, $W_B=6$, $W_0^{(2)}=6$, $W_1^{(2)}=W_2^{(2)}=7$, and $\mathscr {C}$ is a $[7,3]_4$ irreducible cyclic code whose ordinary and 2-symbol Hamming weight enumerators are $1+21T^{4}+42T^{6}$ and $1+21T^{6}+42T^{7}$, respectively.

• (d) Let $(q,r,N,b)=(5,5,4,3)$. Thus $u=\gcd (\varDelta ,N)=1$ and $|\mathcal{P}(b)|=\mu _{(0)}(b)=q^2+q+1=31$. Therefore, owing to Theorems 2 and 3, $W_A=625$, $W_0^{(3)}=775$, and $\mathscr {C}$ is a $[781,5]_5$ one-weight irreducible cyclic code whose ordinary and 3-symbol Hamming weight enumerators are $1+3124T^{625}$ and $1+3124T^{775}$, respectively.

Remark 6

With the help of a C program, the previous numerical examples were corroborated. Such C program is available via email upon request.

As Example 1-(d) has shown, it is quite easy to obtain the b-symbol Hamming weight enumerator of a one-weight irreducible cyclic code (that is, when $u=1$). The following result shows it in the general case.

Theorem 4

Assume the same notation as in Theorem 3. If $u=\gcd (\varDelta ,N)=1$, then, for any $1\le b\le r$, the b-symbol Hamming weight enumerator of $\mathscr {C}$ is

$$\begin{aligned} A(T)=1+(q^r-1)T^{\frac{q^r-q^{r-b}}{N}}. \end{aligned}$$

Proof

If $u=1$, then $\mu _{(0)}(b)=|\mathcal{P}(b)|=\frac{q^b-1}{q-1}$. Thus the result now follows from (4). $\square$

Remark 7

If $\mathscr {C}$ is an $(n,M,d_b(\mathscr {C}))_q$ b-symbol code, with $b\le d_b(\mathscr {C})\le n$, then Ding et al. [3] established the Singleton-type bound $M\le q^{n-d_b(\mathscr {C})+b}$. Therefore, an $(n,M,d_b(\mathscr {C}))_q$ b-symbol code $\mathscr {C}$ with $M=q^{n-d_b(\mathscr {C})+b}$ is called a maximum distance separable (MDS for short) b-symbol code.

Similar to Theorem 3.3 in [12] we also have:

Theorem 5

Let $\mathscr {C}$ be as in Definition 1. Let $a\in \mathrm{I\!F}_{q^r}^*$ and consider the codeword $c(a)=({\text {Tr}}_{\mathrm{I\!F}_{q^r}/\mathrm{I\!F}_q}(a \gamma ^{Ni}))_{i=0}^{n-1}$ in $\mathscr {C}$. Then

$$\begin{aligned} w_r(c(a))=n, \end{aligned}$$

(6)

and $\mathscr {C}$ is an MDS b-symbol code.

Proof

Suppose that $a\in \mathcal{C}_i^{(u,q^r)}$ for some $0 \le i < u$. Thus, by the proof of Theorem 3, $w_r(c(a))=W_i^{(r)}$ where

$$\begin{aligned} \begin{aligned} W_i^{(r)} = \frac{(q-1)q^{r/2-r}}{N}\left[ |\mathcal {P}(r)|\left( q^{r/2}-(-1)^s\right) +(-1)^s u\mu _{((\delta -i)\!\pmod {u})}(r)\right] \end{aligned} \end{aligned}$$

But, owing to Lemma 3, $\mu _{((\delta -i)\!\pmod {u})}(r)=\frac{\varDelta }{u}$. On the other hand, $|\mathcal{P}(r)|=\varDelta =\frac{q^r-1}{q-1}$ and $n=\frac{q^r-1}{N}$. Thus, (6) now follows. Finally, since $d_b(\mathscr {C})=n$ and $|\mathscr {C}|=q^r$, $\mathscr {C}$ is an MDS b-symbol code by Remark 7. $\square$