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# 3Linear Block Codes

## 3.1. Introduction

The aim of channel coding is to protect the data delivered by the source coder against the transmission errors.

We have shown that the random coding allows us to reach the limit of the channel coding theorem when the size of the codewords N tends to + ∞. However, this technique is not feasible due to the complexity of the associated coder and decoder. Indeed, let us consider a random binary code composed of 2k codewords where K is the number of bits of the information word and each information word is associated with a codeword of N bits. To construct a random encoder, it is necessary to first build a set 2k codewords drawn randomly. The encoding will correspond to the association of an information word with a unique codeword using a look up table. Since the code has no specific structure, the decoding will consist of an exhaustive search between the received word and all the 2k codewords of the set in order to determine the maximum likelihood (ML) codeword. The complexity of this decoder increases exponentially with K and is almost always unfeasible in practice.

As a consequence, we have to use codes with an algebraic structure such as the linearity property in order to simplify the coder and decoder. These codes will have to be adapted to the class of errors (random, isolated, bursty, etc.). In this book, we will focus on the three following ...

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