Recurrent methods for constructing irreducible polynomials over \(\text{GF}(2^s)\)
The author provides a method to construct irreducible polynomials over finite fields. Starting with a polynomial \(f\in \text{GF}(q)[x]\) of degree \(n\) irreducible over \(\text{GF}(q)\) he shows how to obtain an irreducible polynomial of degree \(np\), where \(p\) is the characteristic of the field. He gives several such constructions depending on the initial polynomial \(f\). All the polynomials that can be obtained by this method have composite degree leading to composite finite fields. For binary fields a recursive application is possible for some polynomials. Therefore, the construction leads to irreducible polynomials of degree \(2^mn\) over \(\text{GF}(2^s)\). The conditions are satisfied for linear polynomials \(f(x)=x-a, a\in \text{GF}(2^s)^*\) leading to irreducible polynomials of degree \(2^m\). The method is then applied and extended to lead to self-dual irreducible polynomials. The third construction starts with primitive polynomials of degrees \(n_1,\ldots ,n_\sigma\) and an irreducible polynomial of degree \(n\) over \(\text{GF}(2)\) leading to an irreducible polynomial of degree \(n\prod_{i=1}^\sigma (2^{n_i}-1)\). Some results are extensions of [\textit{R. R. Varshamov}, Sov. Phys., Dokl. 9, 426--428 (1964); translation from Dokl. Akad. Nauk SSSR 156, 1308--1311 (1964; Zbl 0147.29602) and Sov. Math., Dokl. 29, 334--336 (1984); translation from Dokl. Akad. Nauk SSSR 275, 1041--1044 (1984; Zbl 0594.12014)].
scientific article; zbMATH DE number 1763661
scientific article