Algebraic lower bounds on the free distance convolutional codes

Abstract

A new module structure for convolutional codes is introduced and used to establish further links with quasicyclic and cyclic codes. We show that the set of finite weight codewords of an (n,k) convolutional code over Fq is isomorphic to an Fq [x]-submodule of Fqn [x], where Fqn [x] is the ring of polynomials in indeterminate x over Fqn, an extension field of Fq. Such a module can then be associated with a quasicyclic code of index n and block length nL viewed as an Fq [x]-submodule of Fqn [x]/ (xL $1) , for any positive integer L. Using this new module approach we derive algebraic lower bounds on the free distance of a convolutional code which can be read directly from the choice of polynomial generators. Links between convolutional codes and cyclic codes over the field extension Fqn are also developed and BCH-type results are easily established in this setting. An example to illustrate the optimal choice of the parameter L is included