Maximum-distance separable (MDS) convolutional codes have the property that their free distance is maximal among all codes of the same rate and the same degree. In this paper, a class of MDS convolutional codes is introduced whose column distances reach the generalized Singleton bound at the earliest possible instant. Such codes are called strongly-MDS convolutional codes. They also have a maximum or near-maximum distance profile. The extended row distances of these codes will also be discussed briefly.

A constant dimension code consists of a set of k-dimensional subspaces of \BBF qⁿ. Orbit codes are constant dimension codes which are defined as orbits of a subgroup of the general linear group, acting on the set of all subspaces of \BBF qⁿ. If the acting group is cyclic, the corresponding orbit codes are called cyclic orbit codes. In this paper, we show how orbit codes can be seen as an analog of linear codes in the block coding case. We investigate how the structure of cyclic orbit codes can be utilized to compute the minimum distance and cardinality of a given code and propose different decoding procedures for a particular subclass of cyclic orbit codes.

A generalization of the original Diffie-Hellman key exchange in $(\mathbb Z$∕$p\mathbb Z)$^* found a new depth when Miller [27] and Koblitz [16] suggested that such a protocol could be used with the group over an elliptic curve. In this paper, we propose a further vast generalization where abelian semigroups act on finite sets. We define a Diffie-Hellman key exchange in this setting and we illustrate how to build interesting semigroup actions using finite (simple) semirings. The practicality of the proposed extensions rely on the orbit sizes of the semigroup actions and at this point it is an open question how to compute the sizes of these orbits in general and also if there exists a square root attack in general. In Section 5 a concrete practical semigroup action built from simple semirings is presented. It will require further research to analyse this system.

We examine-the implications of us- ing a Low Density Parity Check Code (LDPCC) in place of the usual Goppa code in McEliece's cryp- tosystem. Using a LDPCC allows for larger block lengths and the possibility of a combined error cor- rection/encryption protocol.