Construction of isodual codes from polycirculant matrices

Double polycirculant codes are introduced here as a generalization of double circulant codes. When the matrix of the polyshift is a companion matrix of a trinomial, we show that such a code is isodual, hence formally self-dual. Numerical examples show that the codes constructed have optimal or quasi-optimal parameters amongst formally self-dual codes. Self-duality, the trivial case of isoduality, can only occur over \F_2 in the double circulant case. Building on an explicit infinite sequence of irreducible trinomials over \F_2, we show that binary double polycirculant codes are asymptotically good

On some classes of irreducible polynomials over finite fields

In this thesis we describe the work in literature on various aspects of the theory of polynomials over nite elds. We focus on properties like irreducibility and divisibility. We also consider existence and enumeration problems for irreducible polynomials of special types. After the introductory Chapter 1, we collect the well-known results on irreducibility of binomials and trinomials in Chapter 2, where we also present the number of irreducible factors of a xed degree k of xt due to L. Redei. Chapter 3 is on self-reciprocal polynomials. An in nite family of irreducible, self-reciprocal polynomials over F2 is presented in Section 3.2. This family was obtained by J. L. Yucas and G. L. Mullen. Divisibility of self-reciprocal polynomials over F2 and F3 is studied in Sections 3.3 and 3.4 following the work of R. Kim and W. Koepf. The last chapter aims to give a survey of recent results concerning existence and enumeration of irreducible polynomials with prescribed coefficients