The method and simulation model for the synthesis of barker-like code sequences

Noise immunity is one of the essential characteristics of modern wireless data reception/transmission systems. In wireless systems such as Wi-Fi, HiperLan, or Bluetooth, the signal is propagated by electromagnetic oscillations in the environment. However, unlike wiring systems, these oscillations are unprotected from external influences. Noise immunity is one of the essential characteristics of modern wireless data reception/transmission systems. Moreover, if several wireless systems work close enough to each other, there is a possibility of overlapping waves, which will damage the information signal. It is determined that for the tasks of control of unmanned aerial vehicles and mobile robotic complexes increasing the noise immunity of data transmission channels is an urgent problem. It has been investigated that Barker-like code sequences based on ideal ring bundles provide an increase in the power of the obtained sequences by optimizing the parameters of the ideal ring bundles used. It is determined that the increase of noise immunity during data reception and transmission is achieved by choosing the optimal ratios of the parameters of the ideal ring bundle. It is shown that the advantages of Barker-like code sequences such as the optimal ratio between the length of the sequence and its correcting ability, the ability to change the length of the sequence depending on the level of interference are widely used in modern wireless communication and telecommunications systems. The method of synthesis of Barker-like code sequences with the use of ideal ring bundles has been improved, which, by taking into account the ratios of the parameters of ideal ring bundles, provides the choice of the minimum bit code sequence that takes into account the level of interference. A simulation model of synthesis of Barker-like code sequences, noise generation, and error correction has been developed on the basis of the improved method of synthesis of Barker-like code sequences. The developed simulation model is used to study the processes of coding, decoding, detection, and correction of errors in the obtained Barker-like code sequences. It has been investigated that the use of synthesized Barker-like code sequences based on ideal ring bundles provides data recovery of damaged no more than 25 % of the bits of each codeword, and detects up to 50 % of damaged bits in each codeword. Keywords: Barker-like code sequence; ideal ring bundle; noise-tolerant coding; simulation model.

Investigation of Ring and Star Polymers in Confined Geometries: Theory and Simulations

The calculations of the dimensionless layer monomer density profiles for a dilute solution of phantom ideal ring polymer chains and star polymers with f=4 arms in a Θ-solvent confined in a slit geometry of two parallel walls with repulsive surfaces and for the mixed case of one repulsive and the other inert surface were performed. Furthermore, taking into account the Derjaguin approximation, the dimensionless layer monomer density profiles for phantom ideal ring polymer chains and star polymers immersed in a solution of big colloidal particles with different adsorbing or repelling properties with respect to polymers were calculated. The density-force relation for the above-mentioned cases was analyzed, and the universal amplitude ratio B was obtained. Taking into account the small sphere expansion allowed obtaining the monomer density profiles for a dilute solution of phantom ideal ring polymers immersed in a solution of small spherical particles, or nano-particles of finite size, which are much smaller than the polymer size and the other characteristic mesoscopic length of the system. We performed molecular dynamics simulations of a dilute solution of linear, ring, and star-shaped polymers with N=300, 300 (360), and 1201 (4 × 300 + 1-star polymer with four arms) beads accordingly. The obtained analytical and numerical results for phantom ring and star polymers are compared with the results for linear polymer chains in confined geometries.

SYNTHESIS OF BARKER-LIKE SEQUENCES WITH ADAPTATION TO THE SIZE OF THE INTERFERENCE

The method of synthesis of noise-resistant barker-like code sequences with the use of ideal ring bundles has been improved. The method for fast finding of such noise-like noise-resistant code sequences, which are able to find and correct errors in accordance with the length of the obtained code sequence, has been improved. An algorithm is implemented to quickly find such noise-resistant barker-like code sequences that are able to find and correct errors in accordance with the length of the obtained code sequence. A simulation model of noise-tolerant barker-like coding with the use of ideal ring bundles has been developed. The possibility of reducing the redundancy of noise-tolerant code sequences by cutting code sequences by a certain number of bits without losing the regenerative capacity of noise-tolerant codes has been investigated. Theoretical analysis of the possibilities of this approach and its effectiveness is performed. Several series of experimental studies of the reliability of the described method on different data samples were performed and its functional efficiency was confirmed. The analysis of the obtained data and identification of key factors influencing the result is carried out. The practical software implementation of the simulation model of noise-tolerant barker-like coding for finding and correcting errors in the obtained noise-tolerant barker-like code sequences is carried out. The used methods and algorithms of data processing, the main components for message processing and their purpose are described. The possibility of reducing the redundancy of noise-tolerant code sequences by reducing the code sequences by a certain number of bits without losing the reproducibility of noise-tolerant codes has been investigated. Theoretical analysis of the possibilities of this approach and its effectiveness is performed. Several series of experimental studies of the reliability of the described method on different data samples were performed and its functional efficiency was confirmed. The analysis of the obtained results is performed and the main factors influencing the obtained result are determined. The proposed noise-tolerant barker-like code sequences have practical value, because with the help of the obtained barker-like code sequence it is quite simple and fast to find up to 50 % and correct up to 25 % of distorted characters from the length of noise-tolerant barker-like code sequence.

When Intersection Ideals of Graphs of Rings are a Divisor graph

Let R be a commutative principal ideal ring with unity. In this paper, we classify when the intersectiongraphs of ideals of a ring R G(R), is a divisor graph. We prove that the intersection graphs of ideals of a ring RG(R), is a divisor graph if and only if R is a local ring or it is a product of two local rings with each of them hasone chain of ideals. We also prove that G(R), is a divisor graph if it is a product of two local rings one of themhas at most two non-trivial ideals with empty intersection.

On projective intersection graph of ideals of commutative rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] the set of all nontrivial proper ideals of [Formula: see text]. The intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as the set [Formula: see text], and, for any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring [Formula: see text], [Formula: see text] is a projective graph if and only if [Formula: see text] where [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] of nilpotency three and [Formula: see text] is a field. Furthermore, it is shown that for an Artinian ring [Formula: see text] [Formula: see text] if and only if [Formula: see text] where each [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] such that [Formula: see text]

When is every module with essential socle a direct sum of automorphism-invariant modules?

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if [Formula: see text] is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then [Formula: see text] is right Noetherian. Also, we show a von Neumann regular (semiregular) ring [Formula: see text] with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring.

On a Class of Constacyclic Codes of Length 4ps over Fpm[u]〈 ua 〉

For any odd prime p such that pm ≡ 3 (mod 4), consider all units Λ of the finite commutative chain ring [Formula: see text] that have the form Λ = Λ0 + uΛ1 + ⋯ + ua−1 Λa−1, where Λ0, Λ1, …, Λa−1 ∊ 𝔽pm, Λ0 ≠ 0, Λ1 ≠ 0. The class of Λ-constacyclic codes of length 4ps over ℛa is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4ps is expressed as a direct sum of a −λ-constacyclic code and a λ-constacyclic code of length 2ps. In the main case that the unit Λ is not a square, we prove that the polynomial x4 − λ0 can be decomposed as a product of two quadratic irreducible and monic coprime factors, where [Formula: see text]. From this, the ambient ring [Formula: see text] is proven to be a principal ideal ring, whose maximal ideals are ⟨x2 + 2ηx + 2η2⟩ and ⟨x2 − 2ηx + 2η2⟩, where λ0 = −4η4. We also give the unique self-dual Type 1 Λ-constacyclic codes of length 4ps over ℛa. Furthermore, conditions for a Type 1 Λ-constacyclic code to be self-orthogonal and dual-containing are provided.

On (α + uβ)-constacyclic codes of length 4ps over 𝔽pm + u𝔽pm∗

For any odd prime [Formula: see text] such that [Formula: see text], the structures of all [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text] [Formula: see text] are established in term of their generator polynomials. When the unit [Formula: see text] is a square, each [Formula: see text]-constacyclic code of length [Formula: see text] is expressed as a direct sum of two constacyclic codes of length [Formula: see text]. In the main case that the unit [Formula: see text] is not a square, it is shown that the ambient ring [Formula: see text] is a principal ideal ring. From that, the structure, number of codewords, duals of all such [Formula: see text]-constacyclic codes are obtained. As an application, we identify all self-orthogonal, dual-containing, and the unique self-dual [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text].