The geometry of Hermitian self-orthogonal codes

We prove that if $$n >k^2$$ n > k 2 then a k-dimensional linear code of length n over $${\mathbb F}_{q^2}$$ F q 2 has a truncation which is linearly equivalent to a Hermitian self-orthogonal linear code. In the contrary case we prove that truncations of linear codes to codes equivalent to Hermitian self-orthogonal linear codes occur when the columns of a generator matrix of the code do not impose independent conditions on the space of Hermitian forms. In the case that there are more than n common zeros to the set of Hermitian forms which are zero on the columns of a generator matrix of the code, the additional zeros give the extension of the code to a code that has a truncation which is equivalent to a Hermitian self-orthogonal code.

Pembentukan Grup Matriks Singular 2×2

The study aims to determine the set of the singular matrix 2×2 that forms the group and describes its properties. The type of research was used exploratory research. Using diagonalization of the singular matrix S, whereas a generator matrix, pseudo-identity, and pseudo-inverse methods, we obtained a group singular matrix 2×2 with standard multiplication operations on the matrix, with conditions namely: (1) closed, (2) associative, (3) there was an element of identity, (4) inverse, there was (A)-1 so A x (A)-1 = (A)-1 x A = Is. The group was the abelian group (commutative group). In addition, in the group, Gs satisfied that if Ɐ A, X, Y element Gs was such that A x X = A x Y then X = Y and X x A = Y x A then X = Y. This show that the group can be applied the cancellation properties like the case in nonsingular matrix group. This research provides further research opportunities on the formation of singular matrix groups 3×3 or higher order.

Projective MDS Codes Over GF(27)‎

MDS code is a linear code that achieves equality in the Singleton bound, and projective MDS (PG-MDS) is MDS code with independents property of any two columns of its generator matrix. In this paper, elementary methods for modifying a PG-MDS code of dimensions 2, 3, as extending and lengthening, in order to find new incomplete PG-MDS codes have been used over . Also, two complete PG-MDS codes over of length and 28 have been found.

Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

Our aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

Modular Irreducible Representations of the FpW4-Submodules ,()pFNof the Modules ,()pFMas Linear Codes, where W4is the Weyl Group of Type B4

The modular representations of the FpWn-Specht modules( , )KSas linear codes is given in our paper [6], and the modular irreducible representations of the FpW4-submodules( , )pFNof the Specht modules pFS ( , )as linear codes where W4is the Weyl group of type B4is given in our paper [5]. In this paper we are concerning of finding the linear codes of the representations of the irreducible FpW4-submodules( , )pFNof the FpW4-modules( , )pFMfor each pair of partitions( , )of a positive integer n4, where FpGF(p) is the Galois field (finite field) of order p, and pis a prime number greater than or equal to 3. We will find in this paper a generator matrix of a subspace((2,1),(1))()pU representing the irreducible FpW4-submodules((2,1),(1))pFNof the FpW4-modules((2,1),(1))pF Mand give the linear code of ((2,1),(1))()pU for each prime number p greater than or equal to 3. Then we will give the linear codes of all the subspaces( , )()pUfor all pair of partitions( , )of a positive integer n4, and for each prime number p greater than or equal to 3.We mention that some of the ideas of this work in this paper have been influenced by that of Adalbert Kerber and Axel Kohnert [13], even though that their paper is about the symmetric group and this paper is about the Weyl groups of type Bn

Group matrix ring codes and constructions of self-dual codes

In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring $$M_k(R)$$ M k ( R ) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring $$M_k(R)$$ M k ( R ) are one sided ideals in the group matrix ring $$M_k(R)G$$ M k ( R ) G and the corresponding codes over the ring R are $$G^k$$ G k -codes of length kn. Additionally, we give a generator matrix for self-dual codes, which consist of the mentioned above matrix construction. We employ this generator matrix to search for binary self-dual codes with parameters [72, 36, 12] and find new singly-even and doubly-even codes of this type. In particular, we construct 16 new Type I and 4 new Type II binary [72, 36, 12] self-dual codes.

Matrix-Product Codes over Commutative Rings and Constructions Arising from σ , δ -Codes

A well-known lower bound (over finite fields and some special finite commutative rings) on the Hamming distance of a matrix-product code (MPC) is shown to remain valid over any commutative ring R . A sufficient condition is given, as well, for such a bound to be sharp. It is also shown that an MPC is free when its input codes are all free, in which case a generator matrix is given. If R is finite, a sufficient condition is provided for the dual of an MPC to be an MPC, a generator matrix for such a dual is given, and characterizations of LCD, self-dual, and self-orthogonal MPCs are presented. Finally, the results of this paper are used along with previous results of the authors to construct novel MPCs arising from σ , δ -codes. Some properties of such constructions are also studied.