Strongly regular graphs from reducible cyclic codes

Abstract

Let p be a prime number. Reducible cyclic codes of rank 2 over $\mathbb {Z}_{p^m}$ are shown to have exactly two Hamming weights in some cases. Their weight distribution is computed explicitly. When these codes are projective, the coset graphs of their dual codes are strongly regular. The spectra of these graphs are determined.

1 Introduction

Since the seminal paper of Delsarte [9] two-weight codes have been studied in symbiosis with combinatorial objects like strongly regular graphs (SRGs) [5, 6, 9] or geometrical structures like caps in projective spaces [8]. An important class of codes for construction of two-weight codes is that of irreducible cyclic codes [1, 18]. In particular, it is conjectured that all two-weight projective irreducible cyclic codes are known [12, 18]. More recently, reducible cyclic codes have been used as a source of construction of two-weight codes. The connection between SRG’s and two-weight codes over finite fields was extended recently to two-weight codes over rings for the homogeneous weight [6, 7].

In a companion paper [15], the authors have studied irreducible cyclic codes of rank 2 over $\mathbb {Z}_{p^m}$ that have two nonzero weights for the Hamming metric. In the present paper, we study reducible cyclic codes of rank 2 over $\mathbb {Z}_{p^m}$ that have two nonzero weights for the Hamming metric. Like in [13] the main enumeration technique is the connection with second-order recurrences. An extra difficulty in comparison with the finite field case is the size of the code which can be, in the case $m=2,$ either $p^4$ or $p^3.$ This difficulty is overcome by using the notion of residue and torsion codes which are familiar in the domain of codes over rings [14]. Generalizing this approach to $m>2,$ is possible but very technical. In the section, on codes over $\mathbb {Z}_{p^m}$ we will not make any claims on the size of the code. Under a mild condition (see Proposition 3), our codes are projective, and thus SRGs can be constructed from them. Their size, degrees, their two restricted eigenvalues with their multiplicities are determined explicitly. They provide alternative coding constructions to some of the SRGs with parameters in Brouwer’s table of SRGs [17].

The material is organized as follows. The next section collects notations, definitions and basic facts. Section 3 studies codes over $\mathbb {Z}_{p^2}.$ Section 4 deals with their attached SRGs. Section 5 indicates how the material can be partially generalized to $\mathbb {Z}_{p^m}.$ Section 5 recapitulates our results and presents some challenging open problems.

2 Definitions and notation

2.1 Linear codes over rings

Throughout the paper, let p be an odd prime. Let $\mathbb {Z}_{p^m}$ denote the ring of integers modulo $p^m,$ and let $\mathbb {Z}_{p^m}^\times$ denote its multiplicative group. The function $\phi ()$ is Euler totient function. If u and v are two coprime integers, the notation $o_u(v)$ means the multiplicative order of v modulo u,  or in other words, its order as an element of the multiplicative group of the residue class ring $\mathbb {Z}_u.$ A linear code C of length n over $\mathbb {Z}_{p^m}$ is a submodule of $\mathbb {Z}_{p^m}^n$. The residue and torsion codes of $C\le \mathbb {Z}_{p^2}^n,$ denoted R(C) and Tor(C) are two codes of length n over $\mathbb {F}_p$ defined as

$$\begin{aligned}&R(C)=\{r \in \mathbb {F}_p^n \mid \exists y \in C, \, y \equiv r \pmod {p}\},\\&Tor(C)= \{r \in \mathbb {F}_p^n \mid p r \in C\}. \end{aligned}$$

Some elementary facts about these codes are as follows. For a proof of more general statements in the context of chain rings see [14, Chap. 5].

Proposition 1

If C is a linear code over $\mathbb {Z}_{p^2},$ the following properties hold.

• 1. R(C) are linear codes;

• 2. $R(C)\subseteq Tor(C)$;

• 3. $|C|=|R(C)||Tor(C)|$;

• 4. $\dim (Tor(C))$ is at most the rank of C as a $\mathbb {Z}_{p^2}$-module.

Proof

If $r \in R(C),$ then there is $s \in \mathbb {F}_p^n$ such that $r+ps \in C.$ Hence $p(r+ps)=pr \in C,$ and $r\in Tor(C).$ That proves 2. To prove point 3, we apply the first isomorphism theorem to the map $\alpha : C \rightarrow \mathbb {F}_p^n,$ defined by $x \mapsto x \pmod {p}.$ Note that $\alpha (C)=R(C)$ and that $Ker(\alpha )=pTor(C).$ This remark also shows point 1. Point 4 is immediate from $pTor(C)\subseteq C.$ $\square$

2.2 Weights

The Hamming weight of $\mathbf{x}\in \mathbb {Z}_{p^2}^n$ is denoted by $w_H(\mathbf{x}).$ The weight distribution of a code C of length n over $\mathbb {Z}_{p^2}$ is defined as the list

$$\begin{aligned} {[}\langle 0,1\rangle , \dots ,\langle w_i,A_i \rangle ,\dots , \langle w_n,A_n \rangle ], \end{aligned}$$

where $A_i$ is the numbers of $c \in C$ with $w_H(c)=w_i.$ The number $A_i$ is called the frequency of the weight $w_i.$ The dual $C^\bot$ of C is understood with respect to the standard inner product. The minimum distance of a linear code is its minimum nonzero Hamming weight. A linear code is projective if any pair of columns of its generator matrix are linearly independent.

2.3 Cyclic codes

A code is cyclic if it is linear and invariant under the cyclic shift. We consider cyclic codes of the form $\langle g(x)\rangle$ with g(x) a divisor of $x^n-1.$ All the cyclic codes in this paper are reducible, in the sense that their check polynomial $h(x)=\frac{x^n-1}{g(x)}$ is not irreducible. The parameters of a two-weight code C over an alphabet A of size q are listed as $[n,k,\{w_1,w_2\}]_q$ if A is a finite field, and C is of dimension k,  and $(n, |C|, \{w_1,w_2\})_q$ if A is a finite ring, but not a finite field.

2.4 Graphs

A simple graph on v vertices is called a strongly regular graph with parameters $(v, \eta , \lambda , \mu )$ if

• 1. each vertex is adjacent to $\eta$ vertices;

• 2. for each pair of adjacent vertices, there are $\lambda$ vertices adjacent to both;

• 3. for each pair of non-adjacent vertices, there are $\mu$ vertices adjacent to both.

An eigenvalue of a graph $\Gamma$ (i.e., an eigenvalue of its adjacency matrix) is called a restricted eigenvalue if there is a corresponding eigenvector which is not a multiple of the all-one vector 1. Note that for an $\eta$-regular connected graph, the restricted eigenvalues are simply the eigenvalues different from $\eta$. The two restricted eigenvalues of an SRG are usually denoted by $r,\,s,$ with respective multiplicities $f,\,g.$ The spectrum of an SRG is then compactly denoted by $\{\eta ^1,r^f,\,s^g\}.$ Given the spectrum, the parameters $\lambda ,\, \mu$ are uniquely determined by the formulas of [2, §1.1.1], or [5, Th. 1.3.1]. The coset graph of a projective code $C\subseteq \mathbb {Z}_{p^m}^n$ has for vertices the cosets of C, two vertices being connected iff they differ by a coset of minimum Hamming weight one.

3 Codes

Let p an odd prime, and let $N>1$ be a divisor of $\phi (p^m)=p^{m-1}(p-1).$ Let a, b be two distinct elements of $\mathbb {Z}_{p^m}^\times ,$ both of order a divisor of N. Let C(a, b) denote the cyclic code of length N over $\mathbb {Z}_{p^m}$ of check polynomial $h(x)=\frac{(1-ax)(1-bx)}{ab}.$ Its generator matrix G can then be described as

$$\begin{aligned} G=\begin{pmatrix} a &{} a^2&{} \dots &{} a^N\\ b &{} b^2&{} \dots &{} b^N\\ \end{pmatrix}. \end{aligned}$$

Different pairs (a, b) can construct equivalent codes C(a, b) as the next result shows.

Proposition 2

If $(a,b)=\lambda (a',b')$ for some $\lambda \in \mathbb {Z}_{p^m}^\times ,$ then C(a, b) and $C(a',b')$ are monomially equivalent. If $(a,b)= (a'^\theta ,b'^\theta )$ for some $\theta \in \mathbb {Z}_{p^m}^\times ,$ coprime with N,  then C(a, b) and $C(a',b')$ are permutation equivalent.

Proof

If $(a,b)=\lambda (a',b')$ then the column i of G is scaled by $\lambda ^i.$ Hence, coordinate i of each codeword is scaled by $\lambda ^i.$ This is monomial equivalence with an identity permutation part [11, §1.7]. The second assertion is a special case of multiplier equivalence of cyclic codes [11, §4.3]. $\square$

For the next section, we need the following result.

Proposition 3

Assume $m=2.$ If $a \equiv b \pmod {p}$, the code C(a, b) is not projective. If $a \ne b \pmod {p},$ the code C(a, b) is a projective code iff $ord_{p^2}(\frac{b}{a})\ge N$ .

Proof

Suppose there is a nontrivial linear combination between the columns of indices i and j of G of the form

$$\begin{aligned} \lambda \begin{pmatrix} a^i \\ b^i \end{pmatrix} +\mu \begin{pmatrix} a^j \\ b^j \end{pmatrix}=0. \end{aligned}$$

Four cases can occur depending on the invertibility of $\lambda$ and $\mu .$

• 1. Since both a and b are invertible, the two cases $\lambda \notin \mathbb {Z}_{p^2}^\times ,\,\mu \in \mathbb {Z}_{p^2}^\times ,$ or $\lambda \in \mathbb {Z}_{p^2}^\times ,\,\mu \notin \mathbb {Z}_{p^2}^\times ,$ cannot happen.

• 2. If both $\lambda , \mu \in p\mathbb {Z}_{p^2}^\times ,$ then letting $\lambda =p\lambda ',$ and $\mu =p\mu '$ yields

  $$\begin{aligned} \lambda ' \begin{pmatrix} a^i \\ b^i \end{pmatrix} +\mu ' \begin{pmatrix} a^j \\ b^j \end{pmatrix} \equiv 0 \pmod {p}. \end{aligned}$$

  • (a) If $a \equiv b \pmod {p}$, the system reduces to

    $$\begin{aligned} \lambda ' a^i +\mu ' a^j \equiv 0 \pmod {p}. \end{aligned}$$

    Given $i<j,$ it is easy to find $\lambda ',\,\mu '$ that satisfy that equation. So the code is not projective in that case.

  • (b) If, on the other hand, $a \ne b \pmod {p},$ the system above forces $\lambda '=\mu '=0,$ entailing in turn $\lambda =\mu =0.$

• 3. If both $\lambda , \mu \in \mathbb {Z}_{p^2}^\times ,$ then getting rid of $\lambda ,\,\mu$ yields $(\frac{b}{a})^{j-i}=1,$ which is impossible if $ord_{p^2}(\frac{b}{a})\ge N$ and $j-i<N.$

Case 2(a) proves the first assertion. From cases 2(b) and 3, the second assertion follows. $\square$

3.1 $a \not \equiv b \pmod {p}$

In this subsection, assume $m=2.$ We give next a case where C(a, b) is a two-weight code, and, in that case, compute its weight distribution.

Theorem 1

Let $e'=ord_p(\frac{b}{a}),$ and $e=ord_{p^2}(\frac{b}{a}).$ Suppose that $e'=e,$ and that $a \ne b \pmod {p}.$ Then the code C(a, b) has nonzero weights $\{N-\frac{N}{e},N\}.$ Their respective frequencies are

$$\begin{aligned} A_1=e(p^2-1), A_2=(p^2-1)(p^2-e+1). \end{aligned}$$

Proof

Write $c=(c_n)$ for a codeword of $C=C(a,b).$ By linearity $c_n=\lambda a^n+\mu b^n,$ for some $\lambda ,\mu \in \mathbb {Z}_{p^2}.$.

• 1. $\underline{\mu \not \equiv 0 \pmod {p}}$. Hence $\mu \in \mathbb {Z}_{p^2}^\times ,$ and we can rewrite the equation for $c_n$ as

  $$\begin{aligned} c_n=\mu a^n \left( \frac{\lambda }{\mu }+ (\frac{b}{a})^n\right) . \end{aligned}$$

  If $\frac{\lambda }{\mu } \in \langle \frac{b}{a}\rangle ,$ this equation in n admits $\frac{N}{e}$ solutions, and $w_H(c)=N-\frac{N}{e}.$

  If, on the other hand, $\frac{\lambda }{\mu } \notin \langle \frac{b}{a}\rangle ,$ it has no solution and $w_H(c)=N.$

• 2. $\underline{\mu \equiv 0 \pmod {p},\, \lambda \not \equiv 0 \pmod {p}}.$ By reduction $\pmod {p}$ we see that $c_n\ne 0,$ and $w_H(c)=N.$

• 3. $\underline{\mu \equiv 0 \pmod {p},\, \lambda \equiv 0 \pmod {p}}.$ That means that $c=pc'$ for some $c'\in Tor(C).$ But Tor(C) is a replication of a reducible cyclic code of dimension 2 over $\mathbb {F}_p.$ Thus, its possible nonzero weights are, by [15], $\{N',\, N'-\frac{N'}{e'}\},$ for some divisor $N'$ of N. Since $e'=e,$ we see that

  $$\begin{aligned} \frac{N}{N'}\{N',\, N'-\frac{N'}{e'}\}=\{N,\, N-\frac{N}{e}\}. \end{aligned}$$

  We claim that $|C|=p^4.$ This will be proved upon using Proposition 1 as follows. Since $R(C)\subseteq Tor(C)$ and $\dim R(C)=2,$ we see that $Tor(C)=R(C)$ (note that the dimension of Tor(C) is at most 2 by point 4). Then $|C|=|R(C)| |Tor(C)|=p^2\times p^2=p^4,$ by the above considerations.

  We can now compute the weight distribution of C as a function of N and p. Since $a,b \in \mathbb {Z}_{p^2}^\times ,$ we see that the dual distance of C(a, b) is at least two. The frequencies of the weights can then be computed by the first two Pless power moments [10, §7.3]. They are thus solutions of the system

  $$\begin{aligned} A_1+A_2=p^4-1,(N-\frac{N}{e})A_1+NA_2=p^2(p^2-1)N.\qquad \qquad \qquad \qquad \qquad \ \ \square \end{aligned}$$

Example For $p=3,$ $N=6$ and $a=1\,,b=8,$ we have $e=e'=2$ and we obtain the weight distribution

The case $e=N$ of Theorem 1 gives the best parameters.

Corollary 1

Let $e'=ord_p(\frac{a}{b}),$ and $e=ord_{p^2}(\frac{a}{b}).$ Suppose that $e'=e=N,$ and that $a \ne b \pmod {p}.$ Then the code C(a, b) is optimal.

Proof

In that case, the minimum distance is $N-1.$ The code is MDR in the sense of [14, Chap. 12]. $\square$

Example For $p=5,$ $N=4$ and $a=24\,,b=7,$ we have $e=e'=4$ and we obtain the weight distribution

$$\begin{aligned} {[}\langle 0, 1\rangle , \langle 3, 96\rangle , \langle 4, 528\rangle ] . \end{aligned}$$

3.2 $a \equiv b \pmod {p}$

In this subsection, assume $m=2.$ We give a construction with hypotheses exclusive from that of Theorem 1.

Theorem 2

Let $e=ord_{p^2}(\frac{a}{b}).$ Suppose that $a \equiv b \pmod {p}.$ Then the code C(a, b) has nonzero weights $\{N-\frac{N}{e},N\}.$ Their respective frequencies are

$$\begin{aligned} A_1=e(p-1), A_2=(p-1)(p^2+p+1-e). \end{aligned}$$

Proof

Write $c=(c_n)$ for a codeword of $C=C(a,b).$ By linearity $c_n=\lambda a^n+\mu b^n,$ for some $\lambda ,\mu \in \mathbb {Z}_{p^2}.$ We have the same three cases as in the proof of Theorem 1, that is

• 1. $\underline{\mu \ne 0 \pmod {p}}$;

• 2. $\underline{\mu \equiv 0 \pmod {p},\, \lambda \ne 0 \pmod {p}}$;

• 3. $\underline{\mu \equiv 0 \pmod {p},\, \lambda \equiv 0 \pmod {p}}.$

Points 1 and 2 can be treated as in the proof of Theorem 1. To deal with case three, we write $c=pc'$ with $c'_n=\lambda ' a^n+\mu ' b^n.$ Now $c_n=0$ iff $c'_n\equiv 0 \pmod {p},$ which happens iff $\lambda ' +\mu ' \equiv 0 \pmod {p},$ since $a^n \equiv b^n \pmod {p},$ and $a^n \not \equiv 0 \pmod {p}.$ Thus either $c=0$ or $c'_n \not \equiv 0,$ for all $n=1,2,\dots ,N.$ In that case $w_H(c)=N.$

We claim that $|C|=p^3.$ Indeed $|R(C)|=p,$ by the condition $a \equiv b \pmod {p}.$ That $Tor(C)=R(C)$ is impossible since then the code would be generated by

$$\begin{aligned} \begin{pmatrix} a &{} a^2&{} \dots &{} a^N\\ pa &{} pa^2&{} \dots &{} pa^N\\ \end{pmatrix}, \end{aligned}$$

which generates a one-weight code, contradicting the previous paragraph. Thus, Tor(C) has dimension 2,  by 4. of Proposition 1, and by 3. of Proposition 1 we obtain

$$\begin{aligned} |C|=|R(C)| |Tor(C)|=p\times p^2=p^3. \end{aligned}$$

We can now compute the weight distribution of C. Since $a,b \in \mathbb {Z}_{p^2}^\times ,$ we see that the dual distance of C(a, b) is at least two. The frequencies of the weights can then be computed by the first two Pless power moments [11, §7.3]. They are thus solutions of the system

$$\begin{aligned} A_1+A_2=|C|-1, (N-\frac{N}{e})A_1+NA_2=N|C|\frac{(p^2-1)}{p^2}. \square \end{aligned}$$

Example For $p=3,$ $N=6$ and $a=1\,,b=4,$ or $a=1\,,b=7,$ we have $e=3$ and we obtain the weight distribution

4 Strongly regular graphs

Define the graph $\Gamma (a,b)$ as the coset graph of $C(a,b)^\bot .$ The codes of Theorem 1 give the following graphs.

Theorem 3

Keep the notation and hypotheses of Theorem 1. Assume, furthermore, that $ord_{p^2}(\frac{b}{a})\ge N$. Then the graph $\Gamma (a,b)$ is a SRG on $p^4$ vertices of degree $N(p^2-1).$ Its restricted eigenvalues are $\frac{N}{e}(q-e),-N$ of respective multiplicities $A_1,\,A_2$ of Theorem 1.

Proof

By Proposition 3 the code C(a, b) is projective, which shows that $\Gamma (a,b)$ has no multiple edges. By Theorem 11.1.11 of [2], the restricted eigenvalues are computed as $\lambda _i=n(p^2-1)-p^2w'_i$ for $i=1,2$ with the weights $w'_1=N-\frac{N}{e}$ and $w'_2=N$ from Theorem 1, and their multiplicities equal the frequencies of the corresponding weights. $\square$

Examples

• For $p=3,$ $N=2$ and $a=1\,,b=8,$ we have $e=e'=N=2$ and we obtain a SRG of parameters (81, 16, 7, 2),  and spectrum $\{16^1,7^{16},{-2}^{64}\}.$ This SRG is unique with these parameters from [17]. Alternative constructions include a $[8,4,\{6,9\}]_3.$ The fact that $C(1,8)^\bot =\{0\},$ shows that it is a Hamming graph H(2, 9),  also called grid graph in [2, p.262].

• For $p=5,$ $N=4$ and $a=24\,,b=7,$ we have $e=e'=N=4$ and we obtain a SRG of parameters (625, 96, 29, 12),  and spectrum $\{96^1,{21}^{96},{-4}^{528}\}.$ Alternative constructions from [17] are from a $[24,4,\{15,20\}]_5,$ and [3]. It would be interesting to find a direct link between the $[24,4,\{15,20\}]_5,$ and our $(4,25^2,\{3,4\})_{25}.$

5 Generalization

In this section, we indicate briefly how the previous constructions generalize from $p^2$ to $p^m.$ The length of the codes considered is $N \mid \phi (p^m)=p^{m-1}(p-1).$ The generalization of Theorem 1 is as follows. Note that, since no claim is made on |C(a, b)| we can do without the hypothesis $a \ne b \pmod {p^{m-1}}.$

Theorem 4

Let $m\ge 2$ be an integer, and put $e_i=ord_{p^{m-i}}(\frac{a}{b}).$ Suppose that for all $i=0,1,\dots ,m-1$ we have $e_i=e.$ Then the code C(a, b) has nonzero weights $\{N-\frac{N}{e},N\}.$ Their respective frequencies are

$$\begin{aligned} A_1=e\left( \frac{M}{p^m}-1\right) , A_2=M-1-e\left( \frac{M}{p^m}-1\right) , \end{aligned}$$

where we have let $M=|C(a,b)|.$

Proof

The proof goes by induction on m. The base point $m=2$ of the induction is Theorem 3 of [15]. Write $c=(c_n)$ for a codeword of $C=C(a,b).$ By linearity $c_n=\lambda a^n+\mu b^n,$ for some $\lambda ,\mu \in \mathbb {Z}_{p^m}.$ We have the same three cases as in the proof of Theorem 1, that is

• 1. $\underline{\mu \not \equiv 0 \pmod {p}}$;

• 2. $\underline{\mu \equiv 0 \pmod {p},\, \lambda \not \equiv 0 \pmod {p}}$;

• 3. $\underline{\mu \equiv 0 \pmod {p},\, \lambda \equiv 0 \pmod {p}}$.

Points 1 and 2 are exactly like before. To deal with case three, we write $c=pc'$ with $c'_n=\lambda ' a^n+\mu ' b^n.$ In fact $c'_n \pmod {p^m}$ is a replication of a codeword in a code $C(a',b'),$ of length $N' \mid N$ with $a',b'$ the remainders of a and b,  respectively, under division by $p^{m-1}.$ Thus, its possible nonzero weights are, by induction hypothesis, $\{N',\, N'-\frac{N'}{e_1}\},$ for some divisor $N'$ of N. Since $e_1=e_0=e,$ we see that

$$\begin{aligned}\frac{N}{N'}\left\{ N',\, N'-\frac{N'}{e_1}\right\} =\left\{ N,\, N-\frac{N}{e}\right\} .\end{aligned}$$

Since $a,b \in \mathbb {Z}_{p^m}^\times ,$ we see that the dual distance of C(a, b) is at least two. The frequencies of the weights can then be computed by the first two Pless power moments [11, §7.3]. They are thus solutions of the system

$$\begin{aligned} A_1+A_2=M-1, \left( N-\frac{N}{e}\right) A_1+NA_2=NM\frac{(p^m-1)}{p^m}. \end{aligned}$$

$\square$

The generalization of Theorem 2 is as follows.

Theorem 5

Let $e=ord_{p^m}(\frac{a}{b}).$ Suppose that $a \equiv b \pmod {p^{m-1}}.$ Then the code C(a, b) has nonzero weights $\{N-\frac{N}{e},N\}.$ Their respective frequencies are

$$\begin{aligned} A_1=e\left( \frac{M}{p^m}-1\right) , A_2=M-1-e\left( \frac{M}{p^m}-1\right) , \end{aligned}$$

where we have let $M=|C(a,b)|.$

Proof

Write $c=(c_n)$ for a codeword of $C=C(a,b).$ By linearity $c_n=\lambda a^n+\mu b^n,$ for some $\lambda ,\mu \in \mathbb {Z}_{p^m}.$ We have the same three cases as in the proof of Theorem 1, that is

• 1. $\underline{\mu \not \equiv 0 \pmod {p}}$;

• 2. $\underline{\mu \equiv 0 \pmod {p},\, \lambda \not \equiv 0 \pmod {p}}$;

• 3. $\underline{\mu \equiv 0 \pmod {p},\, \lambda \equiv 0 \pmod {p}}$.

Points 1 and 2 are exactly like before. To deal with case three, we write $c=pc'$ with $c'_n=\lambda ' a^n+\mu ' b^n.$ Now $c_n=0$ iff $c'_n\equiv 0 \pmod {p^{m-1}},$ which happens iff $\lambda ' +\mu ' \equiv 0 \pmod {p^{m-1}},$ since $a^n \equiv b^n \pmod {p^{m-1}},$ and $a^n \not \equiv 0 \pmod {p^{m}}.$ Thus either $c=0,$ or $c'_n \not \equiv 0,$ for all $n=1,2,\dots ,N.$ In that case $w_H(c)=N.$

The computation of the weight distribution is the same as in the proof of Theorem 4.

$\square$

Denote by $\Gamma (a,b)$ the coset graph for $C(a,b)^\bot .$

Theorem 6

Keep the notation and hypotheses of Theorem 4. Assume, furthermore, that C(a, b) is projective. Then the graph $\Gamma (a,b)$ is a SRG on |C(a, b)| vertices of degree $N(p^{m}-1).$ Its restricted eigenvalues are $\frac{N}{e}(q-e),-N$ of respective multiplicities $A_1,\,A_2$ of Theorem 4.

The proof is similar to that of Theorem 3 and is omitted.

Example For $p=m=3,$ $N=2$ and $a=1\,b=8,$ we have $e=2$ and we obtain the weight distribution

$$\begin{aligned} {[} \langle 0, 1\rangle , \langle 1, 52\rangle , \langle 2, 676\rangle ], \end{aligned}$$

yielding a SRG of parameters (729, 52, 25, 2),  with the spectrum $\{{52}^1,{25}^{52},{-2}^{676}\}.$ As per [17], the SRG is unique with these parameters and can be also constructed from a $[26,6, \{ 9, 18\}]_3.$ The fact that $C(1,8)=\mathbb {Z}_{27}^2$ shows that it is the Hamming graph H(2, 27) in the notation of [5].

6 Double-root cyclic codes

Let p an odd prime, and let $N>1$ be a divisor of $p\phi (p^m)=p^{m}(p-1).$ Let a be an element of $\mathbb {Z}_{p^m}^\times .$ Let C((a)) denote the cyclic code of length $p^m(p-1)$ over $\mathbb {Z}_{p^m}$ of check polynomial $h(x)=(\frac{(1-ax)}{a})^2.$ Note that, since $\frac{(1-ax)}{a}\mid x^{\phi (p^m)}-1$ its square h(x) divides $x^{p\phi (p^m)}-1=(x^{\phi (p^m)}-1)^p.$ Consider the punctured code $C_a$ of length p of C((a)) defined by its generator matrix $G_a$ as

$$\begin{aligned} G_a=\begin{pmatrix} a &{} a^2&{} \dots &{} a^p\\ a &{} 2a^2&{} \dots &{} pa^p\\ \end{pmatrix}. \end{aligned}$$

Theorem 7

The code $C_a$ is a projective two-weight code of parameters $(p, p^{2m},\{p-1,p\},$ with weight distribution

$$\begin{aligned}.\end{aligned}$$

Proof

The code is projective as can be seen by adapting the proof of Proposition 3. Computing the determinant of $\begin{pmatrix} a^i &{} a^j\\ ia^i &{} ja^j \end{pmatrix},$ which equals $(j-i)a^{i+j},$ modulo $p^{m-1}$ concludes the last case of the proof.

The determination of the weights goes by induction on m. The base point $m=1$ of the induction is Theorem 3 of [13]. Write $c=(c_n)$ for a codeword of $C=C_a.$ By linearity $c_n=(\lambda + \mu n) a^n,$ for some $\lambda ,\mu \in \mathbb {Z}_{p^m}.$ We have the same three cases as in the proof of Theorem 1, that is

• 1. $\underline{\mu \not \equiv 0 \pmod {p}}$;

• 2. $\underline{\mu \equiv 0 \pmod {p},\, \lambda \not \equiv 0 \pmod {p}}$;

• 3. $\underline{\mu \equiv 0 \pmod {p},\, \lambda \equiv 0 \pmod {p}}$.

In case 1, the equation $c_n=0$ has at most one solution in n. In that case, $w_H(c)=p-1$ or $w_H(c)=p.$ Case 2 by reduction modulo p implies that $c_n \ne 0.$ In case 3, we write $c_n=pc'_n,$ where $c'_n$ belongs to the code $C_a$ over $\mathbb {Z}_{p^{m-1}}$ and conclude by induction hypothesis.

The weight distribution is determined by solving the system

$$\begin{aligned} A_1+A_2=p^{2m}-1, (p-1)A_1+pA_2=p^m(p^m-1)p.\square \end{aligned}$$

Examples

• 1. For $p=3,\,m=2$ with $a=1,$ we obtain a code of weight distribution

  $$\begin{aligned}{}[ \langle 0, 1 \rangle , \langle 2, 16 \rangle , \langle 3, 64 \rangle ],\end{aligned}$$

  and a SRG of parameters (81, 24, 9, 6) with spectrum $\{24^1,6^{16},{-3}^{64}\}.$ An alternate construction from [4] is by considering a $[12,4,\{6,9)]_3.$

• 2. For $p=3,\,m=3$ with $a=1,$ we obtain a code of weight distribution

  $$\begin{aligned}{}[ \langle 0, 1 \rangle , \langle 2, 78 \rangle , \langle 3, 650 \rangle ],\end{aligned}$$

  and a SRG of parameters (729, 78, 27, 6) with spectrum $\{78^1,24^{78},-3^{650}\}.$ Alternate construction from [17]: a $[39,6,\{18,27\}]_3.$

• 3. For $p=5,\,m=2$ with $a=1,$ we obtain a code of weight distribution

  $$\begin{aligned}{}[ \langle 0, 1\rangle , \langle 4, 120\rangle , \langle 5, 504\rangle ],\end{aligned}$$

  a SRG on parameters (625, 120, 35, 20) with spectrum $\{120^1,{20}^{120},{-5}^{504}\}.$ An alternate construction from [17] is by considering a $[30,4,\{20,25)\}]_5.$

7 Conclusion

In this paper, we have constructed projective two-weight codes from reducible cyclic codes of rank 2 over the ring $\mathbb {Z}_{p^m}.$ The weight distribution, and, in some cases, the size of the code, have been determined explicitly. In particular, when the check polynomial is a square, we have used a punctured code to construct projective two-weight codes.

In all these constructions, the same weight distributions are reached several times for different values of a and b,  or of a in the double-root case. It would be interesting to know how many of the codes with the same weight distribution are permutation inequivalent. Proposition 2 is a first step in that direction. From the projective two-weight codes constructed SRGs have been built, giving rise to alternative realizations of the parameters in [17], using shorter codes over larger alphabets. In such a case, finding a direct link between the two codes involved in the spirit of [16] might be illuminating.

At a more fundamental level, generalizing this work to other rings is a promising direction of research.