Preface

Error correcting codes represent a widely applied technique for assuring reliable electronic communications and data recording. Although often some codes are introduced without the need for a polynomial approach, the attempt to provide a description where polynomial theory is however present could be of some interest. The intent of this book is to develop such theory, in a unitary way, trying to find several conceptual bridges between different classes of codes (block, convolutional, concatenated,...). This goal requires the introduction of some rather unusual mathematical tools (like interleaved polynomial multiplication or interleaved polynomial division), able to support it. Also some structural transformations, like modified code lengthening or modified H-extension, are needed in order to construct a coherent model, able to support noticeable new interpretations. The introduction of quasi-cyclic codes as a true generalization of the concept of cyclic codes is only an example of the fruitful use of such mathematical tools and geometrical transformations. They are also important for giving an intuitive justification for the encoder circuits to be adopted. State diagrams, constructed on these encoder circuits, contribute to give a better understanding of the properties characterizing the codes under study, besides the measure of the decoding computational complexity. The distinction between well-designed and not well-designed convolutional codes represents another innovative concept. The latter family of codes is equivalent to catastrophic convolutional codes, but since they are systematic, the catastrophic behavior is no longer a problem. On the other hand, a more than linear increase in the number of low-weight code frames with the number of frame periods remains a drawback for not well-designed convolutional codes, together with some difficulties in their parity check matrix determination and tail-biting arrangement organization. Direct product codes between a pair of block codes are demonstrated to be a subset of not well-designed convolutional codes. Some further conceptual (and rather surprising) bridges between block codes and convolutional codes are constructed. The treatment is organized maintaining distinct the approaches based on the generator matrix and on the parity check matrix. The reader is gradually guided to