Title: Generalized subspace subcodes with application in cryptology,
Abstract: Most of the codes that have an algebraic decoding algorithm are derived from
the Reed Solomon codes. They are obtained by taking equivalent codes, for
example the generalized Reed Solomon codes, or by using the so-called subfield
subcode method, which leads to Alternant codes and Goppa codes over the
underlying prime field, or over some intermediate subfield. The main advantages
of these constructions is to preserve both the minimum distance and the
decoding algorithm of the underlying Reed Solomon code. In this paper, we
propose a generalization of the subfield subcode construction by introducing
the notion of subspace subcodes and a generalization of the equivalence of
codes which leads to the notion of generalized subspace subcodes. When the
dimension of the selected subspaces is equal to one, we show that our approach
gives exactly the family of the codes obtained by equivalence and subfield
subcode technique. However, our approach highlights the links between the
subfield subcode of a code defined over an extension field and the operation of
puncturing the $q$-ary image of this code. When the dimension of the subspaces
is greater than one, we obtain codes whose alphabet is no longer a finite
field, but a set of r-uples. We explain why these codes are practically as
efficient for applications as the codes defined on an extension of degree r. In
addition, they make it possible to obtain decodable codes over a large alphabet
having parameters previously inaccessible. As an application, we give some
examples that can be used in public key cryptosystems such as McEliece.