Page 130 - ISCI’2017

(m 1 a b −  ) +
0 ≤ y ≤ i   ,
 m  (4)
)
− 
y − i int gere , i∈  0, , (n 2.

For compensating of distortions of arbitrary initial code word of a source x in result of summing
with elements of a vector of Gaussian random variables Ξ for each of numbers of the code word

x , i ∈ i * [0, ,n − ] 1 will be entered couple of auxiliary non-negative variables w 2i 1+ ,w 2i 2+ .

This pair characterizes possible double-sided deviations of number x i * which are a consequence

of action of a vector of a noise Ξ. One variable from this pair enters in calculation with sign "+", it

means that it is added to the number distorted by a noise x i * , other–with sign «-», it means that it is

subtracted from x i * . Then for observed code word X = { x , ,x * n-1 } system of equations is
*
*
0
formed:

x =  x − * w + w ;
 0 0 1 2
 x = x − * 0 w + 3 w ;
1
4
 (5)
 
 *
 x n1 − = x n1 − − w 2n1 − + w .
2n
In each of the equations (5) one of pair of auxiliary variables w with an even or odd index will
equal zero because the deviation from action of a noise can be only towards reduction, or towards

increase of true number. At the same time variables x i must satisfy to inequality 0 x≤ i ≤ (m1− ) . For

decision of the task of decoding is planned to use the linear programming (LP), then the left inequality

of this restriction (non negativity support) is automatically executed according to terms of the
canonical task LP.

Issue LP-in a canonical form requires of representation of all restrictions of area of admissible
decisions in the form of equalities. Therefore, for changeover from the right inequality to equality

non-negative integer auxiliary variables x , x   n n 1+ , ,x  2n 1− is entered:

−−
  x = (m 1 x * ) w − w;
+
 n 0 * 1 2
−−
+
 x  n1 + = (m 1 x 1 ) w − w;
4
3
 (6)
x  2n 1 − =  m1 x * n 1 −  + w 2n 1 − − w .
−−
2n
   2  

In case of execution of restrictions (6) is reached execution of the following system of equalities:
x +  x =  m 1;−
 x + 0 x  n = m 1;−
  1 n1 (7)
+
 x n1 + x  2n1 = m1.
−

−
−
On the basis of (3) and (4) are defined values y i corresponding to multipliers of equivalent
algebraic representation of operation of computation of the module m:
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