# Generator Matrix Approach to Linear Block Codes

## Abstract

A linear transformation of a vector is firstly introduced. A linear code, described by means of its generator matrix, turns us to be a generalization of this concept. The right-inverse of such matrix is associated to the inverse vector transformation. The polynomial description of a linear block code is given adopting a proper generator polynomial, factor of a certain binomial. This unique polynomial is the element shifting in the rows of the generator matrix of a strict-sense time-invariant code described in this way. Code shortening, code punctuation, the concatenation of more than one code are discussed. The concept of subcode is introduced. In this framework code extension appears as a particular form of direct serial code concatenation. The class of cyclic codes is widely investigated, starting from the factorization of any type of binomials in a binary field. Many families of cyclic codes are presented. The structure of BCH codes and their design procedure are analysed. The encoder circuits able to construct any type of polynomial code exhibit evolutions which can be displayed in a suitable state diagram. It is useful to distinguish between systematic and nonsystematic encoders. The main properties of a polynomial code are evidenced in its state diagram. Direct product codes are described. Their generator matrix is calculated. Expressions for the minimum distance and the numbers of low-weight code words are derived. Lengthened cyclic codes and modified lengthened cyclic codes are treated, the latter representing a first step towards the concept of convolutional codes. Generalized parity check codes are discussed together with lengthened cyclic codes. Direct product codes are interpreted as a particular class of modified lengthened codes. The above topics are extended to the case of non-binary fields. We can have here cyclic or, more in general, pseudo-cyclic codes. Minimum-distance-separable codes, and in particular Reed-Solomon codes, are described. Finally a generator trellis, representing the temporal evolution of the state diagram, is constructed. The number of states in its stages gives a measure of the decoding computational complexity in a soft-decision decoding algorithm.

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