Structural analysis of convolutional codes via dual codes

Abstract

A linear correspondence is developed between the states of a rate- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k/n</tex> convolutional encoder <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</tex> and the states of a corresponding syndrome former <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H^T</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> is an encoder of the code dual to the code generated by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</tex> . This correspondence is used to find an expression for the number of all-zero paths of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau</tex> in the code trellis; the answer depends only on the constraint lengths of the dual code. A partial answer to the resynchronization problem also falls out of this development.