FACTORIAL CODE WITH A GIVEN NUMBER OF INVERSIONS

Results


FCDR -Factorial Code with Data Recovery by Permutation;
FCGNI -Factorial Code with a Given Number of Inversions; DF -decision feedback; SCC -signal-code construction; ( ) A x -data word in a binary form; ( ) , M B q R -class of permutations with inversion numbers belonging to the class of residues q R ; ( ) j f i -number of i -bit errors that can transform the j - th permutation of SCC into any other permutation of the same construction; ( ) ( ) , M W q R -permutations class cardinality; ω -number of inversions in permutation.

INTRODUCTION
Rapid growth of modern society computerization level leads to the need to ensure the joint work of many computers including in open communication networks.Particularly acute is the need to ensure safe operation in open computer ПРОГРЕСИВНІ ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ networks in the exchange of financial information, as well as information of national importance, such as information from the state's law enforcement agencies.All this leads to the need for complex information security that provides for the following tasks: -the task of protection against information unauthorized reading (through its encryption); -the task of ensuring information integrity that includes: -information protection against errors due to noise in communication channels; -protection against obtrusion of false information.All these tasks are solved in existing computer and telecommunication systems and networks.However, each of them is solved independently from each other, by different means created at different times, and based on an individual mathematical basis.A consequence of this approach is an increase of the introduced redundancy.This entails a reduction of data channel bandwidth and leads to an increase of the load of computational basis used to realize the assigned tasks.
Therefore, the development of methods and tools for complex information security based on a single procedure for its processing that ensure a high level of protection and transmission reliability is an actual problem.
To date, a relatively large volume of extensive research on factorial coding methods has been performed from the viewpoint of providing complex information protection.The analysis of works [1][2][3][4][5][6][7][8][9] reveals the properties of factorial codes and allows planning further research in this scientific direction.In particular, the paper [5] is of exceptional interest.It examined the factorial code with data recovery by permutation (FCDR).
The most important property of FCDR is its ability to provide complex data protection at acceptable values of transmission rate loss.At the same time, FCDR does not detect an even number of bit errors that lead to the transformation of one permutation into another.In channels of acceptable quality (where 0 1 np λ = ≤ ), the probability of FCDR erroneous decoding is determined by the intensity of low-weight errors.As shown in [1], this probability decreases rapidly with increasing of error weight.Therefore, eliminating low-weight errors from the set of FCDR nondetectable errors will reduce the undetected error probability.The purpose of this work is to increase the reliability of information transmission when using FCDR.This purpose can be achieved by introducing additional redundancy by choosing a class of permutations that satisfy the given criterion.
1 PROBLEM STATEMENT Suppose that the information from the source goes to the encoder input in blocks ( ) A x of k bits.In this case, the cardinality of information words set is 2 k .FCDR realizes a bijective transformation of the set of information words ( ) into an allowed set of permutations of order M ( ) . Reducing the cardinality of information words of used permutations while preserving their order M makes it possible to choose from ! M permutations only those that satisfy the preassigned set of properties.
The task of developing a method of information coding based on factorial coding with data recovering by permutation consists in determining the effective criterion for choosing the class of used permutations, which allows one to vary flexibly the parameters of reliability of information transmission, and also in the analysis of the structure of allowed set of permutations and its influence on undetected error probability and relative transmission rate.

REVIEW OF THE LITERATURE
According to [5], the permutation π after the symbols encoding by a uniform binary code is represented as a data block ( )

R
x of FCDR r bits, where r l is a number of bits for encoding a single symbol of permutation.
For uniform binary code where a ⎡ ⎤ ⎢ ⎥ is the smallest integer greater than or equal to a (function 'ceiling' [10] of a ).
The length of FCDR code combination FCDR n corresponds to the number of bits in the representation of the permutation π of the order M in a binary form: It follows that -the maximum length of a binary information word to be converted into a permutation of the order M is given by [ ] -! k M − 2 permutations are not used by the source, and FCDR is a redundant code.Code redundancy factor (by cardinality) is defined in [6] as: When the value of the number of bits of the information word is chosen according to (3), the minimum redundancy factor (by cardinality) is provided for the given M .In this case, 1 2 < α < .
( 5 ) An error-detecting factorial code that satisfies the condition (5) As a result of research performed in [5,8], it is shown that FCDR provides protection against unauthorized reading of data if the law of transformation of source words into permutation is kept in secret.Meanwhile, the protection of information from errors in communication channel is provided by the properties of permutations.The most important of these properties is that the permutation π of order M is a sequence of symbols of the set The position of the symbols in permutation is determined by the information word.In addition, every symbol in the permutation is used only once.Checking the occurrence of each of the symbols only once at the receiving station ensures that all odd number of bit errors and part of even number of bit errors are detected in the received block [5].This check will be referred to as the data block validation.
The advantage of FCDR is also that it has the property of self-synchronization.This eliminates the need for a combination of frame synchronization to be entered in the data block.
The disadvantage of FCDR is that this code does not detect even number of bit errors, which lead to the transformation of one permutation into another.
In [6] it is shown that if for a given k the value of M is calculated according to the condition ( ) Then it is possible to insert additional check bits into information part before forming the checksum.This allows to increase the transmission reliability while maintaining the permutation order and the code rate.On the other hand, if the value of M is given in a data transmission system and [ ] , the transmission reliability can be increased due to artificially introduced redundancy.This redundancy is provided by reducing the size of the data block at the encoder input and introducing additional check bits.The results presented in [6] for 1024 k ≤ and 3 0 10 p − = indicate an increase in the energy gain due to the application of the proposed method by up to 1.6 dB.At the same time, the method proposed in [6] does not allow to detect some errors, including 2-bit errors, that lead to the transformation of one permutation into another (permutation of the permutation symbols).

MATERIALS AND METHODS
The proposed coding method provides the artificial redundancy by reducing the cardinality of used permutations.Consequently, the size of the source word also decreases (in comparison with its maximum possible value): We consider it obvious that an increase in redundancy should lead to an increase in the transmission reliability.
All the decisions made in this work are oriented to the creation of data transmission systems operating over a communication channel with independent bit errors.Primarily, this applies to block transmission systems where error correction is performed by retransmission of data in error (binary symmetric decision feedback (DF) systems).Such systems include most general and special purpose systems operating over wired telephone channels, radio relay, microwave, satellite and fiber-optic communication channels.
The main idea of the work is to use a number of inversions in permutation as a sign of its belonging to an allowed set.So, if we use only even (odd) permutations to transfer source words, then: -the cardinality of the allowed set of source words will be halved (exactly half of the permutations of the set of !M permutations is even (odd)); -the decoder will detect all the errors that lead to the transformation of a permutation into another permutation and change its parity.Accordingly, the decoder will not detect errors that lead to the transformation of a permutation into another permutation and do not change its parity.
Note that the transformation of one permutation into another is equivalent to a permutation of its symbols and can be represented as a product of transpositions.Therefore, the transformation of one FCDR codeword into another can be represented as a finite number of consecutive transpositions.Each transposition applied to a permutation changes its parity.Therefore, the consistent application of an odd number of transpositions changes the parity of a permutation.The consistent application of an even number of transpositions does not change the parity of a permutation.This means that the use of only even (odd) permutations allows to detect errors that lead to a transformation equivalent to an odd number of consecutive transpositions over the codeword.
Note also that if the source uses only even (odd) permutations, the decoder will detect all 2-bit errors in a FCDR codeword.This is because a 2-bit error is not detected by FCDR if and only if it generates a transposition of symbols in the codeword.Since the transposition changes the parity of a permutation, it will be detected by the decoder.In this case, an undetected decoding error can be estimated using the expression (2) from [5], where instead of the estimate ( ) Thus, it is possible to increase the transmission reliability of systems with FCDR by reducing the cardinality of permitted permutations and detecting their transpositions in the process of transportation.
In turn, reducing the cardinality of the set of permutations -information carriers -by half leads to a need to reduce the block length by one bit and, accordingly, to reduce the code rate.Therefore, the use of permutations for information transferring, in which a number of inversions satisfies the specified requirements, allows exchanging the code rate for the transmission reliability.

ПРОГРЕСИВНІ ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ
In view of the foregoing, the proposed code will be called a factorial code with a given number of inversions (FCGNI).We define the rule for choosing an allowed set of permutations based on the number of inversions.For this, we consider the theoretical basis for constructing FCGNI.
A number of inversions ( ) π in the permutation π of the order M satisfies the condition ( ) In addition, the frequency number It is shown in [11] that the distribution of inversions on all permutations of a fixed length coincides with the distribution of their major index.That is, the number of permutations of order M with ω inversions is the same as the number of permutations of order M with major index equal to ω.These numbers are known as Mahonian numbers.A stronger result is valid: the number of permutations of order M with major index k and ω inversions is the same as the number of permutations of order M with major index ω and k inversions, that is, the two statistics are equidistributed.
According to [12,13], the following recurrence relation is valid: ) In accordance with [12,14,15], the frequency numbers are coefficients in the expansion ( ) Encyclopedia of Integer Sequences (OEIS) [16] contains the sequence of McMahon numbers ( )

Properties of frequency numbers
( ) 1 − all the frequency numbers ( ) The symmetry property for 2 M ≥ : 6.The recurrence formula ( 6) can be reduced to the following form: to the expressions (I) from [12] and ( 9) from [17].We define the residue modulo q of a number of inversions in a permutation of the order M : q R = ω , where ( ) The set of various permutations π of the order M with a number of inversions of the residue class q R forms a subset (class) of permutations . Depending on values of q and R , cardinalities ( ) are calculated as follows: It was shown in [17] that if q M ≤ , then the cardinalities of classes ( ) B q R are invariant with respect to R and are equal to ( ) Note that the cardinalities ( ) B q R can differ significantly for the constant modulus q value.Therefore, in order to maximize the code rate, it is necessary to use the permutation classes of maximum cardinality for information transmission.In this case, the amount of information carried by one permutation will be equal to , , At the same time, it does not follow from this that the use of the permutation class ( ) , M B q R of maximum cardinality provides the greatest energy efficiency.By energy efficiency, we mean the difference in signal levels at the input of the FCGNI receiver and some other receiver of information, providing the same probability of error-free reception.
All classes ( ) ≤ have an equal cardinality.On other hand, the cardinalities of classes ( ) and should be subject to experimental evaluation.
The above theoretical basis of information factorial coding determines the way of FСGNI constructingselecting for information transmission only those permutations, which numbers of inversions belong to a selected class of residues q R .As shown above, if 2 q = , the code detects all 2-bit errors and a part of errors with higher weight.Selecting a modulus q other than two ( ) ( ) will result in a different distribution of detectable and undetectable errors.This requires an informed choice of class ( ) 0 q q = , the code will detect all odd permutations and a part of even permutations.
We will accept ( ) . Then the probability of undetected error Figure 1 -The structural scheme of the FCGNI encoder An integral indicator of data transmission quality is a relative transmission rate 0 ν [18].It is calculated for a DF system as follows: where 1 k n ν = .According to [18], for the simplest DF system , ud Q P FCGNI p ν = + , where ( ) The structural scheme of the FCGNI encoder is shown in Fig. 1.
From a data source, an information word of k bits is entered in the buffer register 1.Then a source word is transmitted to the block 2 for forming a permutation with a number of inversions of a given class of residues.This permutation, after encoding its symbols with a binary code, is transmitted through the modulator 3 to a receiving station via a communication channel.
Generation of a permutation with a number of inversions of a given class of residues can be performed, for example, by creating a table that consists of 2 k rows.In each of them one of the permutations with a number of inversions The number of different tables that can be constructed is equal to At the same time, it is created a table that links permutations with information words for decoder.If these tables are kept secret, then the prerequisites for information encryption are created, and the expression (15) determines the key space cardinality.With this approach, the transmitting part of the system with FCGNI is a ROM where tables are stored.A table is selected by a session key.A permutation is selected by a k -bit source word that defines an address of a cell storing this permutation.
The structural scheme of the FCGNI decoder is shown in Fig. 2. The demodulator 4 allocates data block (which can be affected by interference) from a received sequence (signal and noise mixture).Then the received data block comes to the block of permutation estimation 5.In block 5, the permutation correctness is evaluated -it is verified that each of the symbols { } 0,1, 2,..., 1 M − is contained only once and that the permutation belongs to the allowed part of permutation set.
If the correctness condition is not satisfied, the permutation estimation block 5 erases the received data block and, by the bus 2, generates a command of retransmission of data in error for the request generation block 8.The request generation block 8 transmits this command to the transmitting station via a feedback channel.
If the correctness condition is satisfied, the permutation estimation block 5, by the bus 1, delivers the received permutation to the block 6 of estimation a number of permutation inversions.If the number of inversions in the received permutation does not belong to a given class of residues, then the block 6 of estimation a number of permutation inversions, by the bus 2, generates a command of retransmission of data in error for the request generation block 8.
If the received permutation has retained the number of inversions belonging to a given class of residues, then, by bus 1 of the block 6 of estimation a number of permutation inversions, the received permutation is transmitted to the permutation decoding block 7.This block performs the inverse transformation of the permutation into an information word using the corresponding table.The decoded permutation from the output of the block 7 is given to a recipient.

EXPERIMENTS
In Table 1, for 8 M = ( ) B q R for a given q have an equal cardinality that is defined by (9).
We estimate the probability of undetected error (12), the code rate and the relative transmission rate (13).To do this, we randomly select from each class ( ) permutations that form the signal-code construction (SCC).

RESULTS
Fig. 3 shows the graph of dependence of the estimated probability of FCGNI undetected error on the modulus q for .In addition, it is presented the graph for the values of R from Table 1 providing the maximum cardinality of the class ( )

DISCUSSION
The graph of Fig. 6 indicates that for 8 M = and -by more than five orders in comparison with FCDR).This can be useful in systems with high requirements for the probability of "false alarm".
Note that the presented coding method has a disadvantage inherent in all block ciphers.Identical plaintext blocks are converted into identical ciphertext blocks.Therefore, in order to eliminate the statistical redundancy and reduce the probability of identical blocks appearance, it is advisable to compress the plaintext before transformation.

CONCLUSIONS
The problem of increasing the reliability of information transmission using FCDR is solved.
The scientific novelty of the work is as follows.The method of factorial coding with data recovery by permutation has been further developed by using in SCC permutations with a number of inversions from a given class of residues.This makes it possible to reduce the number of undetected errors and, accordingly, to increase the transmission reliability.
The practical significance of the obtained results lies in the experimental estimates of FCGNI parameters, as well as in the developed structural schemes of encoding and decoding devices, which make it possible to carry out their practical implementation.
It is shown that by choosing the appropriate class of residues for the number of inversions ω in a permutation π , one can select a desired value of energy gain in exchange for the loss of the relative transmission rate.For example, the choice of the class ( ) .At the same time, the task of developing the principles for choosing a permutations set of a given class that forms an optimal structure of SCC is the subject of further research.Метою цієї роботи є розробка та дослідження методу факторіального кодування з заданим числом інверсій, спрямованого на підвищення достовірності передавання інформації за рахунок введення додаткової надлишковості шляхом вибору класу перестановок, які відповідають заданому критерію.

amount of information carried bypν -relative transmission rate; 1 ν -code rate; 2 ν
-length of binary data word to be converted into a permutation of the class of bits for encoding a single symbol of permutation; M -permutation order; n -number of bits in a data block;( )MN ω -number of different permutations of the order with a number of inversions equal to ω ; -bit error probability;Qprobability of codeword error-free reception; codeword in a binary form; q R -residue class modulo q; FCDR r number of bits in FCDR codeword;α -code redundancy factor (by cardinality); 0 -dynamic component of transmission rate loss due to retransmission of data in error; the set of allowed permutations { } π is the condition of choosing an order M according to the inequality ! 2 k M for practical applications we shall proceed from the condition ! 2 k M > .
number ω of possible inversions corresponds to a frequency number ( ) M N ω .Due to the definition of frequency numbers, the next expression is valid:

Figure 2 -
Figure 2 -The structural scheme of the FCGNI decoder ; the value of R at which it is achieved; the value of The EC symbols mean that all the classes ( ), M

8 M
rate reaches the maximum values at the maximum values of the code rate.This is explained by the fact that q and R can vary. .It also follows from Figure6that FCDR exceeds FCGNI in the relative transmission rate for 8 = FCGNI has a gain in this indicator at 0 0.1 p > .At the same time, Figures3 and 4indicate that FCGNI is aimed at reducing the probability of undetected error (for example, for 2 q =