Strongly regular graphs from reducible cyclic codes
``Since the seminal paper of \textit{Ph. Delsarte} [Discrete Math. 3, 47--64 (1972; Zbl 0245.94010)] two-weight codes have been studied in symbiosis with combinatorial objects like strongly regular graphs (SRGs) [\textit{A. E. Brouwer} et al., Distance-regular graphs. Berlin etc.: Springer-Verlag (1989; Zbl 0747.05073); \textit{E. Byrne} et al., Des. Codes Cryptography 48, No. 1, 1--16 (2008; Zbl 1184.94266); \textit{E. Byrne} et al., Des. Codes Cryptography 48, No. 1, 1--16 (2008; Zbl 1184.94266); \textit{Ph. Delsarte}, Discrete Math. 3, 47--64 (1972; Zbl 0245.94010)] or geometrical structures like caps in projective spaces [\textit{R. Calderbank} and \textit{W. M. Kantor}, Bull. Lond. Math. Soc. 18, 97--122 (1986; Zbl 0582.94019)]. An important class of codes for construction of two-weight codes is that of irreducible cyclic codes [\textit{L. D. Baumert} and \textit{R. J. McEliece}, Inf. Control 20, 158--175 (1972; Zbl 0239.94007); \textit{G. Vega}, Finite Fields Appl. 33, 1--13 (2015; Zbl 1368.11134)]. In particular, it is conjectured that all two-weight projective irreducible cyclic codes are known [\textit{B. Schmidt} and \textit{C. White}, Finite Fields Appl. 8, No. 1, 1--17 (2002; Zbl 1023.94016); \textit{G. Vega}, Finite Fields Appl. 33, 1--13 (2015; Zbl 1368.11134)]. 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 [\textit{E. Byrne} et al., Des. Codes Cryptography 48, No. 1, 1--16 (2008; Zbl 1184.94266); Finite Fields Appl. 18, No. 4, 711--727 (2012; Zbl 1245.05135)].'' Let \(p\) be an odd prime. A code \(C\subseteq \mathbb{Z}_{p^m}\) is called linear if \(C\) is a submodule of \(\mathbb{Z}_{p^m}.\) As usual, a code \(C\) is called cyclic if \(C\) is linear and invariant under the cyclic shift. Let \(C=\langle g(x) \rangle\) be a cyclic code over \(\mathbb{Z}_{p^m},\) where \(g(x)\) is a divisor of \(x^n-1.\) A cyclic code \(C\) is called \textit{reducible} if its check polynomial \(\displaystyle h(x)=\frac{x^n-1}{g(x)}\) is not irreducible. A linear code is called \textit{projective} if any pair of columns of its generator matrix are linearly independent. Let \(N>1\) be a divisor of \(\phi(p^m)=p^m(p-1)\) and let \(a,b\) be two distinct nonzero elements in \(\mathbb{Z}_{p^m}\) 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 \(\displaystyle h(x) = \frac{(1-ax)(1-bx)}{ab}.\) Its generator matrix \(G\) can then be described as \[ G = \begin{pmatrix} a & a^2 & \cdots & a^N\\ b & b^2 & \cdots & b^N \end{pmatrix}. \] In the paper under review, the authors consider \textit{reducible} cyclic codes of rank \(2\) over the ring \(\mathbb{Z}_{p^m},\) with \(p\) is an odd prime, having two nonzero Hamming weights. Among the main results are as follows. The paper provides two constructions of two-weight codes \(C(a,b)\) over the ring \(\mathbb{Z}_{p^2},\) for \(a \not \equiv b \pmod p\) (Theorem 1) and for \(a \equiv b \pmod p\) (Theorem 2). Then, the paper shows that the coset graph of dual of the code \(C(a,b)\) constitutes a strongly regular graph (Theorem 3). The paper also gives a generalization of the previous results (Theorem 1, Theorem 2, and Theorem 3) to the ring \(\mathbb{Z}_{p^m}\) (see Theorem 4, Theorem 5, and Theorem 6). Moreover, the authors also derive a theorem similar to Theorem 1 and 4, for certain repeated-root, namely double-root, cyclic code over \(\mathbb{Z}_{p^m}\) (Theorem 7).
scientific article; zbMATH DE number 7490055
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