On products of linear error correcting codes

In this thesis we study products of linear error correcting codes. Error correcting codes are used to correct the errors introduced by some noisy communication channel and are essential in all communications that, due to economic or practical constraints, do not allow data retransmission: for instance deep space communications, broadcasting and mass storage. Their products, throughout the last forty years, have appeared in many different fields, such as cryptography, complexity theory, additive combinatorics and cryptanalysis. We study such products and discuss applications to cryptography. First, we prove that typically the product operation generates trivial codes; then, we investigate and characterize some class of codes whose products are non trivial and satisfy interesting properties. Our methods are algebraic-combinatorial in nature, though sometimes probabilistic techniques will be involved.