Algebraic lower bounds on the free distance of convolutional codes
Kristine Lally · IEEE Transactions on Information Theory · 2006
A new module structure for convolutional codes is introduced and used to establish further links with quasi-cyclic and cyclic codes. The set of finite weight codewords of an (n,k) convolutional code over Fqis shown to be 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 quasi-cyclic code of index n and block length nL viewed as an Fq[x]-submodule of Fqn[x]/langxL-1rang, for any positive integer L. Using this new module approach algebraic lower bounds on the free distance of a convolutional code are derived which can be read directly from the choice of polynomial generators. Links between convolutional codes and cyclic codes over the field extension Fqnare also developed and Bose-Chaudhuri-Hocquenghem (BCH)-type results are easily established in this setting. Techniques to find the optimal choice of the parameter L are outlined