Construction of n variable codes (Q1068790)

In this paper we study the construction of codes in algebras of the following form: \[ A={\mathbb{F}}_ p[X_ 1,...,X_ n]/(t_ 1(X_ 1),...,t_ n(X_ N)), \] where every \(t_ i(X_ i)\) is a polynomial in \({\mathbb{F}}_ p[X_ i]\) (p prime). We construct these codes by using the decomposition of A in principal ideals \((g_ i)\) [\textit{A. Poli}, Codes dans certaines algèbres modulaires, Thèse d'état, Univ. P. Sabatier, Toulouse (1978)]. The construction of the polynomials \(g_ i\) requires two polynomial factorizations. We demonstrate some proofs for the construction of the polynomials \(g_ i\). A consequence of these proofs is that we deduce all the irreducible factors in \({\mathbb{F}}_ q[X]\) of an irreducible polynomial in \({\mathbb{F}}_ p[X]\), from one factor, where \(q=p^ r\). These proofs improve the computation time for a constructing algorithm of n variable codes. We present this algorithm and we give results obtained from a program written at the AAECC group on a Burroughs 6700 in Fortran IV.