Enumerating and decoding perfect linear Lee codes

Using group theory approach, we determine all numbers q for which there exists a linear 1-error correcting perfect Lee code of block length n over Z _q , and then we enumerate those codes. At the same time this approach allows us to design a linear time decoding algorithm.      

Group codes over fields are asymptotically good

Group codes are right or left ideals in a group algebra of a finite group over a finite field. Following the ideas of a paper on binary group codes by Bazzi and Mitter in 2006, we prove that group codes over finite fields of any characteristic are asymptotically good.     

On self-orthogonal group ring codes

We obtain structural results about group ring codes over F[G], where F is a finite field of characteristic p > 0 and the Sylow p-subgroup of the Abelian group G is cyclic. As a special case, we characterize cyclic codes over finite fields in the case the length of the code is divisible by the characteristic of the field. By the same approach we study cyclic codes of length m over the ring R = F _q [u], u ^r  = 0 with r  > 0, gcd(m, q) = 1. Finally, we give a construction of quasi-cyclic codes over finite fields.      

测试模糊码的一种算法

Abstract. How to verify that a given fuzzy set A∈F(X ) is a fuzzy code? In this paper, an al-gorithm of test has been introduced and studied with the example of test. The measure notionfor a fuzzy code and a precise formulation of fuzzy codes and words have been discussed.     

A linear algorithm for minimum 1-identifying codes in oriented trees

Consider an oriented graph G=(V,A), a subset of vertices C⊆V, and an integer r?1; for any vertex v∈V, let denote the set of all vertices x such that there exists a path from x to v with at most r arcs. If for all vertices v∈V, the sets are all nonempty and different, then we call C an r-identifying code. We describe a linear algorithm which gives a minimum 1-identifying code in any oriented tree.     

An intrinsical description of group codes

A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism which maps G to the standard basis of . Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space , which does not assume an “a priori” group algebra structure on . As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed–Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes. Research supported by D.G.I. of Spain and Fundación Séneca of Murcia.     

Linear error-block codes

A linear error-block code is a natural generalization of the classical error-correcting code and has applications in experimental design, high-dimensional numerical integration and cryptography. This article formulates the concept of a linear error-block code and derives basic results for this kind of code by direct analogy to the classical case. Some problems for further research are raised.     

Malnormal subgroups of lattices and the Pukánszky invariant in group factors

Let G be a connected semisimple real algebraic group. Assume that G(R) has no compact factors and let Γ be a torsion-free uniform lattice subgroup of G(R). Then Γ contains a malnormal abelian subgroup A. This implies that the II₁ factor VN(Γ) contains a masa A with Pukánszky invariant {∞}.     

Tame group actions on central simple algebras

We study actions of linear algebraic groups on finite-dimensional central simple algebras. We describe the fixed algebra for a broad class of such actions.     

ContributionsOn binary k-paving matroids and Reed-Muller codes

A matroid M of rank r k is k-paving if all of its circuits have cardinality exceeding r − k. In this paper, we develop some basic results concerning k-paving matroids and their connections with codes. Also, we determine all binary 2-paving matroids.     

A survey on proper codes

The performance of a linear t-error correcting code over a q-ary symmetric memoryless channel with symbol error probability ε is characterized by the probability that a transmission error will remain undetected. This probability is a function of ε involving the code weight distribution and the weight distribution of the cosets of minimum weight at most t. When the undetectable error probability is an increasing function of ε, the code is called t-proper.

The paper presents sufficient conditions for t-properness and a list of codes known to be proper, many of which have been studied by these sufficient conditions. Special attention is paid to error detecting codes of interest in modern communication.