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In cryptography, the property of randomness in pseudo-random generators is very important to avoid any pattern in output sequences, to provide security against attacks, privacy and anonymity. In this article, the randomness of the family of sequences obtained from the generalized self-shrinking generator is analyzed. Moreover, the characteristics, generalities and relationship between the t-modified self-shrinking generator and the generalized self-shrinking generator are presented. We find that the t-modified self-shrunken sequences can be generated from a generalized self-shrinking generator. Then, an in-depth analysis of randomness focused on the generalized sequences by means of complete and powerful batteries of statistical tests and graphical tools is done, providing a useful vision of the behaviour of these sequences and proving that they are suitable to be used in cryptography.

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A constant dimension code consists of a set of k-dimensional subspaces of F n q , where F q is a finite field of q elements. Orbit codes are constant dimension codes which are defined as orbits under the action of a subgroup of the general linear group on the set of all k-dimensional subspaces of F n q. If the acting group is Abelian, we call the corresponding orbit code Abelian orbit code. In this paper we present a construction of an Abelian non-cyclic orbit code for which we compute its cardinality and its minimum subspace distance. Our code is a partial spread and consequently its minimum subspace distance is maximal.

In this paper a new construction of MDS array codes is introduced. In order to obtain a code with this property, the parity-check matrix is constructed just using a superregular matrix of blocks composed of powers of the companion matrix of a primitive polynomial. Also a decoding algorithm for these codes is introduced.