Invertible matrices over a class of semirings

We characterize the invertible matrices over a class of semirings such that the set of additively invertible elements is equal to the set of nilpotent elements. We achieve this by studying the liftings of the orthogonal sums of elements that are “almost idempotent” to those that are idempotent. Finally, we show an application of the obtained results to calculate the diameter of the commuting graph of the group of invertible matrices over the semirings in question.

On Properties of Semipositive Cones and Simplicial Cones

For a given nonsingular $n\times n$ matrix $A$, the cone $S_{A}=\{x:Ax\geq 0\}$ , and its subcone $K_A$ lying on the positive orthant, called as semipositive cone, are considered. If the interior of the semipositive cone $K_A$ is not empty, then $A$ is named as semipositive matrix. It is known that $K_A$ is a proper polyhedral cone. In this paper, it is proved that $S_{A}$ is a simplicial cone and properties of its extremals are analyzed. An one-one relation between simplicial cones and invertible matrices is established. For a proper cone $K$ in $\mathbb{R}^n$, $\pi(K)$ denotes the collection of $n\times n$ matrices that leave $K$ invariant. For a given minimally semipositive matrix (no column-deleted submatrix is semipositive) $A$, it is shown that the invariant cone $\pi(K_A)$ is a simplicial cone.

-MINIMUM SPANNING LENGTHS AND AN EXTENSION TO BURNSIDE’S THEOREM ON IRREDUCIBILITY

We introduce the $\textbf{h}$ -minimum spanning length of a family ${\mathcal A}$ of $n\times n$ matrices over a field $\mathbb F$ , where $\textbf{h}$ is a p-tuple of positive integers, each no more than n. For an algebraically closed field $\mathbb F$ , Burnside’s theorem on irreducibility is essentially that the $(n,n,\ldots ,n)$ -minimum spanning length of ${\mathcal A}$ exists if ${\mathcal A}$ is irreducible. We show that the $\textbf{h}$ -minimum spanning length of ${\mathcal A}$ exists for every $\textbf{h}=(h_1,h_2,\ldots , h_p)$ if ${\mathcal A}$ is an irreducible family of invertible matrices with at least three elements. The $(1,1, \ldots ,1)$ -minimum spanning length is at most $4n\log _{2} 2n+8n-3$ . Several examples are given, including one giving a complete calculation of the $(p,q)$ -minimum spanning length of the ordered pair $(J^*,J)$ , where J is the Jordan matrix.

On nonsingularity and group invertibility of combinations of two group invertible matrices

Let A and B be two group invertible matrices, we study the rank, the nonsingularity and the group invertibility of A-B, AA#-BB#, c1A + c2B, c1A + c2B + c3AA#B where c1,c2 are nonzero complex numbers. Under some special conditions, the necessary and sufficient conditions of c1A + c2B + c3and c1A + c2B + c3+ c4BA to be nonsingular and group invertible are presented, which generalized some related results of Ben?tez, Liu, Koliha and Zuo [4, 17, 19, 25].

FRACTAL CLUSTERING OF GROUP CODES IN THE SYSTEM OF GALOIS FIELD

The permutation decoding (PD) of group systematic noise-immune codes is proved to be the most efficient method in using the redundant data entered to the code as against other methods of decoding digital data [1-5]. This opens up the opportunity of solving a complex computational problem of finding an equivalent code (EC), which is used to search for the error vector. The essence of this solution is that the computational procedure of real-time search for EC for each new combination of redundant code is replaced by a preliminary process of training the decoder to put in accordance with each new permutation of characters the generating matrix of EC parameters, which are recorded in the decoder’s memory card during training. Thus, such a memory card is called a cognitive card (CC). The article estimates the memory size of the CC, when using the block code (15,7,5), and shows the possibility of implementing a permutation decoder on basis of existing integrated circuits based on proven statements. For the first time, the apparatus of fractal partitioning of augmented binary Galois fields using the clustering of the common space of code vectors of a given code is used to prove the main statements. An efficient algorithm is presented to search for a set of invertible matrices of rearranged codes that do not provide an EC and for this reason should be primarily detected in the decoding procedure of the received code vectors.

Some consequences of the rank normal form of a matrix

If A is a rectangular matrix of rank r, then A may be written as PSQ where P and Q are invertible matrices and s = \left( {\matrix{ \hfill {{{\rm{I}}_{\rm{r}}}} & \hfill {\rm{O}} \cr \hfill {\rm{O}} & \hfill {\rm{O}} \cr } } \right) . This is the rank normal form of the matrix A. The purpose of this paper is to exhibit some consequences of this representation form.

A key exchange protocol using matrices over group ring

The first published solution to key distribution problem is due to Diffie–Hellman, which allows two parties that have never communicated earlier, to jointly establish a shared secret key over an insecure channel. In this paper, we propose a new key exchange protocol in a non-commutative semigroup over group ring whose security relies on the hardness of Factorization with Discrete Logarithm Problem (FDLP). We have also provided its security and complexity analysis. We then propose a ElGamal cryptosystem based on FDLP using the group of invertible matrices over group rings.

New LCD MDS codes constructed from generalized Reed–Solomon codes

Maximum distance separable codes with complementary duals (LCD MDS codes) are very important in coding theory and practice, and have attracted a lot of attention. In this paper, we focus on LCD MDS codes constructed from generalized Reed–Solomon (GRS) codes over a finite field with odd characteristic. We detail two constructions of new LCD MDS codes, using invertible matrices and the roots of three classes of polynomials, respectively.