Perfect Binary Arrays Research Articles

Development of an error correction method using perfect binary arrays

The research focuses on an innovative error correction method that uses perfect binary arrays (PBAs), a powerful mathematical tool with unique properties that make it ideal for error correction. The research is aimed at studying the impact of uncorrelated mixed-type errors in the data exchange path, which allows using it in smart technologies with limited computing capabilities. The effectiveness of the approach is confirmed by simulation and comparison with other error correction methods. In order to further study the structural, cross-correlation and distance properties of orthogonal two-dimensional codes and the correcting capabilities of the proposed method, an information technology system for data transmission based on an equivalent class of perfect binary arrays has been developed. The proposed model evaluates the performance of the error correction code based on perfect binary arrays under various conditions, including correlated and uncorrelated interference and data exchange paths. A generator of PBA of equivalent classes has been built. An experimental evaluation of the correcting ability of the proposed two-dimensional codes was carried out by simulating various pre-code situations, including packet and random errors, for the cases of correlated and uncorrelated interference. Using a graphical interface, users will be able to enter the number and type of errors, determine whether they are random or packet errors, manually or automatically, move errors through the data packet, and view intermediate results. Thus, the complex nature of this study can be positioned as a promising approach and a reliable choice in the field of error correction

Generalized binary arrays from quasi-orthogonal cocycles

Generalized perfect binary arrays (GPBAs) were used by Jedwab to construct perfect binary arrays. A non-trivial GPBA can exist only if its energy is $2$ or a multiple of $4$. This paper introduces generalized optimal binary arrays (GOBAs) with even energy not divisible by $4$, as analogs of GPBAs. We give a procedure to construct GOBAs based on a characterization of the arrays in terms of $2$-cocycles. As a further application, we determine negaperiodic Golay pairs arising from generalized optimal binary sequences of small length.

Interrelation Between the Class of Bent-Sequences and the Class of Perfect Binary Arrays

Interrelation Between the Class of Bent-Sequences and the Class of Perfect Binary Arrays

THE CLASS OF PERFECT TERNARY ARRAYS

In recent decades, perfect algebraic constructions are successfully being use to signal systems synthesis, to construct block and stream cryptographic algorithms, to create pseudo-random sequence generators as well as in many other fields of science and technology. Among perfect algebraic constructions a significant place is occupied by bent-sequences and the class of perfect binary arrays associated with them. Bent-sequences are used for development of modern cryptographic primitives, as well as for constructing constant amplitude codes (C-codes) used in code division multiple access technology. In turn, perfect binary arrays are used for constructing correction codes, systems of biphase phase- shifted signals and multi-level cryptographic systems. The development of methods of many-valued logic in modern information and communication systems has attracted the attention of researchers to the improvement of methods for synthesizing many-valued bent-sequences for cryptography and information transmission tasks. The new results obtained in the field of the synthesis of ternary bent-sequences, make actual the problem of researching the class of perfect ternary arrays. In this paper we consider the problem of extending the definition of perfect binary arrays to three-valued logic case, as a result of which the definition of a perfect ternary array was introduced on the basis of the determination of the unbalance of the ternary function. A complete class of perfect ternary arrays of the third order is obtained by a regular method, bypassing the search. Thus, it is established that the class of perfect ternary arrays is a union of four subclasses, in each of which the corresponding methods of reproduction are determined. The paper establishes the relationship between the class of ternary bent-sequences and the class of perfect ternary arrays. The obtained results are the basis for the introduction of perfect ternary arrays into modern cryptographic and telecommunication algorithms.

Digital modulation method based on perfect binary arrays

Digital modulation method based on perfect binary arrays

Composite matrix cipher based on perfect binary arrays

A composite matrix cipher consisting of four partial ciphers and based on perfect binary arrays has been proposed. This cipher possesses an easily controlled level of data protection from unauthorized access and other practically acceptable computational properties.

Class of minimax error-corecting codes based on perfect binary arrays

Aregular synthesis method has been developed for synthesizing infinite families of minimax M(N)-classes of perfect binary arrays having order N = 2k, where k is arbitrary natural number. This method enabled us to construct a new class of four-parameter minimax error-correcting M(n)-codes having length n = N2 that possess many practically attractive properties; in this case the task of decoding is reduced to the effective procedure of majority estimation of parameters of the received code words

Erratum to: “Methods of Improving Information Security by Integrating Multiplexing, Ciphering and Channel Encoding Operations”

A regular rule for constructing infinite families of orthogonal Walsh W(n) and orthogonal C(n)systems is suggested, which possesses the property of two-loop cyclic shift and is based on complete classes of perfect binary arrays (PBA) H(N), where N = 2k and N = 3 × 2k is PBA order, n = N2 is orthogonal systems order, and k is an arbitrary integer. Methods that increase information security by integrating multiplexing, ciphering and channel encoding operation are developed on the basis of families of C(n)orthogonal systems, which also allow solving system tradeoffs in CDMA technologies.

Family of two-dimensional correcting codes on a basis of perfect binary array

We consider a family of two-dimensional correcting E(N)-codes on a basis of perfect binary arrays H(N), and we show correcting possibilities of E(N)-codes in comparison with corresponding BCH-codes with maximal length are essentially better with regard to correction of package (correlated) errors, at that, uncorrelated errors are corrected identically.

Complete class of perfect binary arrays for 144 elements

A method has been proposed for building the complete class of perfect binary arrays having the size of 12×12 and its cardinality has been estimated.

Information security method based on perfect binary arrays

It has been shown that algebraic structures in the form of perfect binary arrays (PBA) can be successfully used for building noise-immune radio communications systems. In addition, a PBA-based information security method was proposed against unauthorized access.

Operation algorithms and block diagrams of codecs of orthogonal codes based on perfect binary arrays

Algorithms and efficient block diagrams of coder and decoders of sliding correlative decoding with regard to regular structural properties of double-loop cyclic shift of orthogonal codes based on perfect binary arrays are offered.

Classes of minimax bi-phase signals based on perfect binary arrays

Based on the full class of perfect binary arrays of the order N = 8 and of the volume J = 688128, an ensemble of minimax bi-phase signals of the length n = N 2 = 64, with their correlation parameter R min max = 6, is set up. It is established that the correlation properties of normal systems (64×64) of bi-phase signals based on perfect binary arrays are best from the viewpoint of minimax criterion, as compared with known derived Walsh–Hadamard systems used in the CDMA standard technologies.

The properties of the full class of perfect binary arrays for 36 elements

A new rule is suggested for construction of the Π(6)-class of perfect binary arrays (PBA) for 36 elements used for generation of the full U (6)-class of PBA. The properties of PBA of minimax M (6)-classes are defined. Also, the properties of the correlation functions of systems of orthogonal signals based on PBA of cyclic E (6)-classes are investigated.

Two-dimensional perfect quaternary arrays

We study two-dimensional (2-D) arrays of fourth roots of unity which have all out-of-phase periodic autocorrelations equal to zero. Generalizing the concept of a perfect binary array, we call these arrays perfect quaternary arrays. We establish connections with combinatorial design theory and exhibit large families of such arrays. Increasing the alphabet from size two to size four greatly increases the flexibility one has in choosing the dimensions for 2-D arrays with perfect periodic autocorrelation. For example, we show that the number of entries in a 2-D perfect quaternary array may be divisible by any Mersenne prime and indeed by many other primes, whereas 2-D perfect binary arrays are only known to exist with size equal to a power of two times a power of three.

Non-splitting Abelian (4 t, 2, 4 t, 2 t) Relative Difference Sets and Hadamard Cocycles

Using cohomology we show that in studying the existence of an abelian non-splitting (4 t, 2, 4 t, 2 t) relative difference set, D, we can assume the groups in question have a certain simple form. We obtain an explicit constructive equivalence between generalized perfect binary arrays and cocycles that define Hadamard matrices and thereby show directly that the existence of D corresponds to that of a symmetric Hadamard matrix of a certain form. This extends the well-known equivalence in the case of splitting relative difference sets.

A nonexistence result for abelian menon difference sets using perfect binary arrays

A Menon difference set has the parameters (4N2,2N2-N, N2-N). In the abelian case it is equivalent to a perfect binary array, which is a multi-dimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. Suppose that the abelian group\(H \times K \times Z_{p^\alpha }\) contains a Menon difference set, wherep is an odd prime, |K|=pα, andpj≡−1 (mod exp (H)) for somej. Using the viewpoint of perfect binary arrays we prove thatK must be cyclic. A corollary is that there exists a Menon difference set in the abelian group\(H \times K \times Z_{3^\alpha }\), where exp(H)=2 or 4 and |K|=3α, if and only ifK is cyclic.

Perfect binary arrays and difference sets

A perfect binary array is an r-dimensional array with elements ± 1 such that all out-of-phase periodic autocorrelation coefficients are zero. Such an array is equivalent to a Menon difference set in an abelian group. We give recursive constructions for four infinite families of two-dimensional perfect binary arrays, using only elementary methods. Brief outlines of the proofs were previously given by three of the authors. Although perfect binary arrays of the same sizes as two of the families were constructed earlier by Davis, the sizes of the other two families are new.

Nonexistence of certain perfect arrays

Perfect binary arrays are equivalent to Menon difference sets in certain groups. This note proves a theorem on the Menon difference set and interprets its consequences on the nonexistence of certain perfect arrays.

New constructions of menon difference sets

Menon difference sets have parameters (4 N 2, 2 N 2 − N, N 2 − N). These have been constructed for N = 2 a 3 b , 0 ⩽ a, b, but the only known constructions in abelian groups require that the Sylow 3-subgroup be elementary abelian (there are some nonabelian examples). This paper provides a construction of difference sets in higher exponent groups, and this provides new examples of perfect binary arrays.