Recursive Construction of n -gonal Codes on the Basis of Block Design

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Introduction
In the progress of transmitting information on telecommunication networks there is a problem of error correction.Control of integrity of data and error correction -important tasks at many levels of work with information.One means of solving these problems is the use of error-correcting codes.
To date, developed many different error-correcting codes, which differ from each other by structure, redundancy, encoding and decoding algorithms ( [1] … [4]).On the basis of monotone Boolean functions constructed a number of cryptosystems [5 -8] with error correction.In particular, in the system based on triangular codes for construction codes great lengths used a recursive method [6].
However the recursive method used in [6] does not allow to build codes, if number of units in a code word more than 3.
The aim of this work is to develop a recursive method for constructing of error-correcting code great length, which has a maximum power and corrects any predetermined number of errors and has a high speed encoding and decoding.

Theoretical Foundations of n-gonal Codes
In this article as codewords we will consider vectors of length n ( ) 1 ,..., ,..., i n p p p p =  , which components accept values from a set { } 0,1 , and the number unit a component is equal in a vector to k.Such codewords have Hamming weight (number of units in a codeword) equal k.In Hamming distance (code distance) between two codewords p  and s  the number is called ( ) , p s ρ   equal to number a component in which they differ [1].Length of a code we will call length of a codeword.In total such codewords with weight of k can be k n C .Code distances between different words in this case can be 2, 4, 6, …, 2k.For correcting codes only codewords with distance 4, 6, …, 2k are applicable.The number of codewords is called as code power.Such codes with code distance 2k are uninviting, as their power is small and is always equal n k . In any such pair of codewords there are no conterminous single bits.Let's consider further the maximum codes with ( ) , as they correct the maximum number of errors.From determination of Hamming distance follows that in such codes for each pair of codewords of the general there can be only one unit.It is easy to present such codes in the form of monotonous Boolean functions of a rank n , weight 1 and power m , where m equally to code power.[6][7][8]] For research of such codes it is possible to use block designs.
In [9][10][11] the following definitions of the block design, BIB of the block design, the affine and projective planes are given.

Definition
Definition.The block design balanced with respect to pairs (elements , ,..., , ,..., , ∑ to blocks that: 1) i b blocks contains on k i < v of different elements at some i = l..., m; 2) each pair of elements appears together precisely in λ blocks.
Any block design or ВІВ the block design is a code.In this case numbers of single bits of codewords correspond to block elements, and codewords correspond to blocks in this block design.Thus, since.Definition.Correcting ability -the characteristic of a code which is equal to the maximum number of corrected errors in codewords.
Is defined as 1 2 , where d -the minimum distance between codewords.
As number of codewords, at which ( ) it is not enough, the greatest correcting ability of a code is reached,  What follows we consider only the block designs with 1 λ = , i.e.Steiner systems.
In the offered method creation of correcting codes the projective and affine planes will be used.

2,
PG n , having finite number of points is called as the projective plane of an order n , if satisfies to the following axioms: Through two different points of P and Q of the plane there passes a straight line, and, only one.
Any two straight lines have the general point.There are three points which are not lying on one straight line.
Each straight line contains not less than three points Generally projective plane of an order n has 2 1 n n + + points and as much straight lines.Each line contains 1 n + points, and each point belongs 1 n + straight line.

2,
EG n , having finite number of points is called as the affine plane of an order n , if satisfies to the following axioms: For any two different points there is only one straight line containing these points.
Crossing of two different straight lines contains exactly one point.
There is a set from four points, any three of which do not belong to one straight line.
Generally projective plane of an order n has 2 n points and 2  n n + straight lines.Each line contains n points, and each point belongs 1 n + straight line.The affine and projective planes are block designs in which each block consists of numbers of points being on some straight line.
Let's make definition n -gonal code.
Definition 1. n -gonal code is called the Steiner system in which each block contains n elements.
n -gonal codes are block and nonlinear as a difference of any codewords are not a code.
Before describing a recursive method of creation of codes on the basis of block designs we will enter the following definitions.
, constructed for each block from k of elements, and b2 of the affine planes constructed for each block from k + 1 element of Steiner system A0.

Recursive Methods of Constructing n-gonal Codes
Let's describe a recursive method of creation of block designs on an example of creation of a code in length 64 of a code in length 21.
For construction distribution from an example 1 on any Steiner systems previously we will prove 3 lemmas.

Lemma 1. Any not allocated block can enter only into one of Steiner systems Bi .
Proof.As as A0 the system, any 2 ai and aj blocks from A0 is considered Steiner either are not crossed, or have one general element.In the first case of a set of Ei and Ej have one general element equal to 0, so Steiner of system Bi and Вj, constructed of elements of these sets, have no general blocks.In the second case ai and aj blocks have the general element x, and sets of Ei and Ej have as the general elements 0 and pair (x,y) where x it is constant, and y can accept all admissible values.In Ei and Ej there is so much in total general elements, how many they contain in one block of Steiner systems Bi and Вj.All these general elements enter into one allocated block, the general for Steiner systems Bi and Вj.It proves that Вi and Вj have no general not allocated blocks.
Lemma 2. Any two distinct blocks of Steiner systems Bi and Вj do not contain the general pair of elements.
Proof.On construction the allocated blocks have no general pair and no pair from the allocated block can enter into not allocated block.Let's allow 2 not allocated blocks from Вi and Вj contain the general pair of elements.Then this pair is the general for sets of Ei and Ej, so according to a lemma 1 the general enters into the allocated block, for Вi and Вj.Then Вi contains the allocated and not allocated blocks with the general pair of elements.It contradicts that Вi is Steiner system.The lemma is proved.Lemma 3. Any Steiner system Bi obtained by renumbering the elements of Steiner systems B1.
Proof.Steiner systems B1 and Вi are under construction of a 1 and a i blocks of Steiner system A0.Let's unequivocally display a1 block elements on a i block elements.Thus the general element of 2 blocks (if it is) is mapping on itself.To elements of a 1 and a i blocks there correspond components x in pairs (x,y), belonging to sets to E1 and Ei.In such a mapping obtained uniquely renumbering of the elements of E1 to the elements of Ei.Thus from Steiner system B1 obtained the Steiner system Bi.
In the proof of lemmas 1-3 that in an example 1 in quality Вi and Вj the affine planes undertake is not used.Therefore lemmas 1-3 are fair, when Вi and Вj is Steiner systems Let's extend construction from an example 2 to a case of the any v and k .
The projective plane B0 exists for k -1 look i p , where psimple number.Steiner system B0 is symmetric.
Proof.According to a lemma 1 at b the projective planes Bi there are b • (k (k -1) + 1) -b • k of not allocated blocks.Adding to them v of the allocated blocks we will receive that the projective extension C0 of the block design of A0 contains b (k -1) 2 + v of blocks.In all of these blocks by construction includes (k -1) + 1 elements, as x in pairs (x,y) changes generally from 0 to v-1, and y from 0 to k -3.Each pair (x,y) is included into r • (k -1) not allocated blocks and in one allocated block.0 enters into v of the allocated blocks.As in any Steiner system is carried out r • (k-1) + 1 = v, each element of the projective C0 expansion enters into v of blocks, and pair of elements ((х 1 ,у 1 ), (х 2 ,у 2 )) according to a lemma 2 is included only into one block.In all b From the block 1 we will receive just as in an example the 2nd block design В1(13, 13, 4, 4).From it renumbering of elements we will receive block designs В2 -В13.As a result we will receive 117 not allocated and 13 allocated blocks of block design С0.
Theorem A. Exists a triangular code with words of code length k ( k is odd and ( ) Then there is a triangular code with words of code length 2 n k = power ( ) Theorem B. There are systems of triples of Steiner of orders of v 1 and v 2 : S 1 (v 1 , b 1 , r 1 ,3) and S 2 (v 2 , b 2 , r 2 ,3).Then there is Steiner a system ( ) Moore's theorem.Let there are systems of triples of Steiner of orders v 1 and v 2 : Let also either v 3 =1, or v 3 is order of system of triples of Steiner S 3 (v 3 , b 3 , r 3 ,3) which is a subsystem in S 2 .Then it is possible to construct system of triples of Steiner S(v, b, r, 3) with v = v 3 + v 1 (v 2 -v 3 ), containing v 1 of subsystems of an order of v 2 and at least on one subsystem of orders v1 and v 3 (if v 3 is not equal 1).
For a case 4 k = it is possible to prove the theorem similar to the theorem B: Theorem 4. Let there are systems of fours Steiner's of orders v 1 and v 2 : S 1 (v 1 , b 1 , r 1 ,4) and S 2 (v 2 , b 2 , r 2 ,4).Then there is Steiner a system ( ) ( ) Proof.We form of the elements S 1 and S 2 1 2 v v v = pairs.Of these pairs we will build the four of Steiner system S.At the first stage we take any element а i from S 1 and any block ) , that is equal to number a fours in Steiner system S.The number of repetitions of each element is equal in these four .The theorem is proved.
As the Hamming distance between any codewords is not less ( )  symbols of a transferred codeword.Codewords in the table of coding are ordered on increase.

Algorithms of Encoding and Decoding of n-gonal Codes
Decoding is carried out by means of the table (the two-dimensional massif) where in headings of lines and columns numbers of single bits, and in the table of number of codewords will be written down.We determine the transferred information message by numbers of two bits of a codeword.Thus numbers of columns correspond lsb, and numbers of lines to the msb of the transferred message.EXAMPLE 5. Detection of one error.Let's consider a code with 4 k = and length n , on the channel cryptosystem two messages 2 and 1 are transferred.These messages are coded by two codewords (according to table 2): 0000 … 01110001 and 0000 … 00001111, i.e. all it is transferred 2n bat.Let two codewords 0000 … 0110001 and 000 … 10001111 have been accepted.At reception numbers of bits of codewords for the 1st codeword are defined these are bits: 0, 4, 5, for the 2nd -0, 1, 2, 3, 7. Let's apply the decoding table (table 3) to definition of the devoted message corresponding to the first codeword.Number of lowest single bit of the transferred word is equal 0, and number of the highest is 5. On crossing of the corresponding line and a column is 2. It means that the message 2 is transferred.The error of transfer for the 1st codeword is corrected.
Let's consider, how there is a decoding in case instead of four single bits 5 single bits were accepted, on an example of the second codeword.It is possible to form 10 pairs of single bits of the 2nd codeword: (0; 1), (0; 2), (0; 3), (1; 2), (1; 3), (2; 3), (0; 7), (1; 7), (2; 7), (3; 7).To these pairs in the table of decoding there correspond the transferred messages: 1, 1, 1, 1, 1, 1, 4, 6, 9, 10.Six of the messages found in the table are identical and equal 1.It means that 6 first pairs of bits form the second codeword, i.e. the codeword 0000 … 00001111 has been transferred.The error of transfer for the 2nd codeword is corrected.It is possible to show that in case of reception of 5 bits instead of 4 it is enough to find in tables for the received pairs two conterminous messages, i.e. in the given example instead of 6 pairs was enough to check only pairs (0; 1) and (0; 2).At worst it is enough to check 6 pairs that demands 6 readings from table 2.
It is simple to count up, how many on the average it is necessary to check pairs in this case.Let's designate through x a random variable -number of checked pairs, pprobabilities of values x.Values p i we will find on a formula  Mathematical expectation mх = 3.14.Therefore, in this case on the average it is necessary to check 3 pairs of bits.
Example 6.Detection of two errors.In a considered example chances when: 1. Two units, other zero are accepted.
2. Four units one of which is not on the place are accepted.
3. Six units are accepted.
In the first case at once by two units we determine a transferred codeword.
In the second case we have 3 true (correspond to one transferred message) and 3 incorrect pairs (correspond to different transferred messages).Then it is enough to find to two pair one transferred message.For finding of two pairs from one transferred message it is necessary to check at least 2 and at most 5 pairs.Let's count up, how many on the average it is necessary to check pairs of bits.Let x a random variable -number of checked pairs, p -probabilities of values x.Values p i we will find on a formula  Mathematical expectation mх = 3.5.Therefore, in this case on the average it is necessary to check 3-4 pairs of bits.
In the third case we have 6 true (correspond to one transferred message) and 9 incorrect pairs (correspond to different transferred messages).For finding of two pairs from one transferred message it is necessary to check at least 2 and at most 11 pairs.Let's count up, how many on the average pairs of bits in 338 Recursive Construction of n-gonal Codes on the Basis of Block Design this case it is necessary to check.Similarly previous, the distribution law for this random variable: Mathematical expectation mх = 4.57.Therefore, in the third case on the average it is necessary to check 4-5 pairs of bits.

Conclusions
In conclusion, we note the following.The main result is the development of a universal recursive method of constructing codes great length on the basis of Steiner systems.These codes have a maximum power and maximum correcting ability among the codes with a given number of units in the codeword.Is proved a number of properties of such codes, not previously described in the literature.Constructed error-correcting code with simple encoding and decoding algorithms and the ability to change the code for one permutation.

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The balanced incomplete block design (or simply the block design) B ( ) , , , , v b r k λ is called such placement of v of different elements on b to blocks that each block contains precisely k of the different elements, each element appears precisely in r different blocks and each pair of different elements i b , j b appears precisely in λ Recursive Construction of n-gonal Codes on the Basis of Block constructed count have the general pairs of elements.It is obvious that number λ corresponds to Hamming distance, namely: ( )( )

Theorem 1 .
If there is Steiner system A0(v, b, r, k) and there is an affine plane k-1st order B0((k -1)2 , k (k -1), k, k -1) = EG (2, k -1), exists the affine expansion C0(v (k -2) +1, b • k (k -2) + v, v, k -1)of Steiner system A0.The affine planes Bi exist for k -1 look i p , where p -simple number.Proof.According to a lemma 1 at b the affine planes Bi there are b • k (k -1) -b • k of not allocated blocks.Adding to them v of the allocated blocks we will receive that the affine expansion C0 of Steiner system A0 contains b•k (k -2) + v of blocks.Into all these blocks on construction enter v(k -2) +1 elements, as x in pairs (x,y) changes generally from 0 to v -1, and y from 0 to k -3.Each pair (x,y) is included into r•(k-1) not allocated blocks and in one allocated block.0 enters into v of the allocated blocks.As in any Steiner system is carried out r•(k-1)+1= v, each element of the affine expansion C0 enters into v of blocks, and pair of elements ((х 1 ,у 1 ), (х 2 ,у 2 )) according to a lemma 2 is included only into one block.b p 1 and p 2 -simple numbers.
c a d a e , such the fours exists 1 2 v b .Also at the second stage we take any block ( ) , , , a b c e from S 1 and any element d from S 2 .We form the four of pairs d c d e d .Such the fours exists 2 1 v b .At the third stage we take any block from ( ) , , , a b c d S 1 and any block ( ) , , , e f g h from S 2 .It is possible to form 24 four of these pairs of 2 blocks.Let's order pairs in each of 24 fours on four elements ( ) , , , a b c d .Then the second elements of pairs in each of 24 fours represent shift of elements of the 12 b v b v b b b = + + blocks.For any block designs formulas are carried out: bk vr =

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We have the probability distribution of this random variable: •(k -2)/2, i.e. to number of pairs, which form v(k -2) + 1 elements.It proves that C0 Steiner system.If there is Steiner system A0(v, b, r, k) and there is a projective plane 2, i.e. to number of pairs, which form v (k -1) + 1 elements.It proves that C0 Steiner system.

Table 1 .
Distribution law of the random variable

Table 2 .
Coding table

Table 4 .
Distribution law of the random variable

Table 5 .
Distribution law of the random variable