TABLE OF CONTENTS As we can see it from the exercises in the preceding section, it is very difficult to construct unstructured codes. A code is equivalent to the data of a family of spheres with radius ?, disjoint two by two. The number of spheres is M, and the code corrects ? errors per word (with maximum likelihood decoding). The best possible code is equivalent to the best packing of spheres, which is a very complex problem. In order to be able to build codes more easily, we agree to lose some freedom by imposing an algebraic structure on the code. We will thus consider the binary codes with a particular property: stability during addition.
These codes have a structure of vector subspaces of ( F 2) n. If the code C is a vector subspace of dimension k, it is said that the dimension of the linear code C is k. The number of words in C is then 2 k. From now on we will speak of a linear code ( n, k, d) instead of ( n, M = 2 k, d).
These codes have properties used for their decoding or construction.
The following proposition makes it possible to simplify obtaining the minimum distance when the code is linear.
The minimum weight of a...
