The weight distribution of families of reducible cyclic codes through the weight distribution of some irreducible cyclic codes (Q2194517)

A reducible code is a cyclic code that has a parity check polynomial which has at least two irreducible polynomials in its reduction. The authors tackle the problem of finding the weight distribution of reducible cyclic codes. For a primitive element of \(\mathbb{F}_{q^k}\) denoted by \(\gamma\) and \(a,e\) -- integers such that \(e>1\), \(e\vert q^{k}-1\), let \(m_{(a,e)}(x)\in\mathbb{F}_q[x]\) be the least common multiple of the \(e\) polynomials \(h_a(x)\), \(h_{a+\frac{q^k-1}{e}}(x), \ldots, \) \(h_{a+(e-1)\frac{q^k-1}{e}}(x)\), where \(h_a(x)\) denotes the minimal the minimal polynomials of \(\gamma^{-a}\). Denote by \({\mathfrak M}(a,e)\) the cyclic code whose parity check polynomial is \(m_{(a,e)}(x)\). This paper proves that the weight distribution of some newly discovered families of reducible cyclic codes in the form of \({\mathfrak{M}}(a,e)\) can be expressed in terms of the weight distributions of already studied irreducible codes \({\mathfrak C}_{(ae)}\) having parity check polynomial \(h_{ae}(x)\).