On the classification and enumeration of self-dual codes

Graph decomposition

For a growing number of applications, such as cellular, peer-to-peer, and sensor networks, efficient error-free transmission of data through a network is essential. Toward this end, Kotter and Kschischang propose the use of subspace codes to provide error correction in the network coding context. The primary construction for subspace codes is the lifting of rank-metric or matrix codes, a process that preserves the structural and distance properties of the underlying code. Thus, to characterize the structure and error-correcting capability of these subspace codes, it is valuable to perform such a characterization of the underlying rank-metric and matrix codes. This paper lays a foundation for this analysis through a framework for classifying rank-metric and matrix codes based on their structure and distance properties. To enable this classification, we extend work by Berger on equivalence for rank-metric codes to define a notion of equivalence for matrix codes, and we characterize the group structure of the collection of maps that preserve such equivalence. We then compare the notions of equivalence for these two related types of codes and show that matrix equivalence is strictly more general than rank-metric equivalence. Finally, we characterize the set of equivalence maps that fix the prominent class of rank-metric codes known as Gabidulin codes. In particular, we give a complete characterization of the rank-metric automorphism group of Gabidulin codes, correcting work by Berger, and give a partial characterization of the matrix-automorphism group of the expanded matrix codes that arise from Gabidulin codes.

Equivalence for Rank-Metric and Matrix Codes and Automorphism Groups of Gabidulin Codes

The main tools in the theory of hyperstructues are the fundamental relations. The fundamental relation on a hyperring was introduced by Vougiouklis at the fourth AHA congress. The fundamental relation on a hyperring (H v -ring) is defined as the smallest equivalence relation so that the quotient would be the (fundamental) ring. Note that the commutativity with respect to both sum and product in the (fundamental) ring are not assumed. Now, in this article we would like the (fundamental) ring to be commutative with respect to both sum and product, that is, the fundamental ring should be an ordinary commutative ring. Therefore we introduce a new strongly regular equivalence relation on hyperrings (H v -rings). If we consider this relation on a hperring (H v -ring), then the set of quotients is a commutative ring. Some properties of such rings are investigated.

Commutative Rings Obtained from Hyperrings (H v -rings) with α∗-Relations

The purpose of this article is to present certain results arising from a study of theory of hyperrings. By a hyperring we mean a Krasner hyperring, that is, a triple (R, +,·) is such that (R, +) is a canonical hypergroup, (R, ·) is a semigroup with a zero 0 where 0 is the scalar identity of (R, +) and · is distributive over + . In this article, we define the notions of normal, prime, maximal, and Jacobson radical of a hyperring and by considering these notions we obtain some results. We define hyperring of fractions and hyper-valuation on a hyperring. For this, as in the classical case, we need a mapping from R onto an ordered group G. Finally, we shall state and prove the Chinese Remainder Theorem for the case of hyperrings.

A Realization of Hyperrings

In this paper codes with two control symbols, built over some classes of groups, are studied.

Codes and groups

Balanced and strongly balanced Pk-designs

The aim of this paper is to characterize the strongly distributive multiplicative hyperrings. A non trivial example is provided giving a model for such hyperrings; also a characterization theorem is proved.

Rosaria Rota

Papers

Codes and groups

Balanced and strongly balanced Pk-designs

Strongly distributive multiplicative hyperrings

Perfect octagon quadrangle systems

Hyperaffine planes over hyperrings