Ambiguity resistant polynomial matrices (Q1301269)

An error control code defined over a complex field which maps each \(K\) input samples into \(N\) output samples is called \(N\times K\) code. An \(N \times K\) \((N \geq K)\) ambiguity resistant \((AR)\) matrix \(G(z)\) is an irreducible polynomial matrix of size \(N\times K\) over a field \(F\) such that the equation \(EG(z)= G(z)V(z)\) with \(E\) an unknown constant matrix and \(V(z)\) an unknown polynomial matrix has only the trivial solution \(E=\alpha I_N\), \(V(z)= \alpha I_K\), where \(\alpha\in F\). \(AR\) matrices have been introduced and applied in modern digital communications as error control codes defined over the complex field. In this paper the authors systematically study \(AR\) matrices over an infinite field \(F\). They discuss the classification of \(AR\) matrices, define their normal forms, find their simplest canonical forms, and characterize all \((K+1)\times K\) \(AR\) matrices in the applications. The paper is organized as follows. In Section 2 they present the canonical forms for \(N\times K\) \(AR\) matrices. In Section 3, they provide the necessary and sufficient conditions for a polynomial matrix of size \(N\times (N-1)\) to be \(AR\) in terms of its systematic or canonical form.