Algebraic lower bounds on the free distance of convolutional codes

K Lally

doi:10.1109/tit.2006.872980

Abstract

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 F q is shown to be isomorphic to an F q [x]-submodule of F q n [x], where F q n [x] is the ring of polynomials in indeterminate x over F q n , an extension field of F q . Such a module can then be associated with a quasi-cyclic code of index n and block length nL viewed as an F q [x]-submodule of F q n [x]/langx L -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 F q n are 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

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