Coding with the using of cyclic codes
If is k given - placing inbound combination m, so n – placing coding word u of cyclic code with generating polynomial g(x) you can take by two methods.
First input k – placing combination expressed in the form of polynomial
m(x) = mk - 1 xk - 1+…+ m1 x + m0 of (k – 1) degree, is multiplied to generating polynomial g(x) of (n – k) degree:
u(x) = m(x) g(x).
Received by such method code looses property of system. I.e it isn’t possible to indicate where are the information symbols, and where the checked. Decoding of such code presents determined difficulties.
The second, a polynomial procedure of dividing is used and computation of excess. In correspondence with the rules of polynomial dividing for every pair of polynomial С(х) and g(x), g(x) 0 exists only one pair of polynomial q(x) —particular and (х) — excess, such as
С(х) = q(x) g(x) (х) ,
We multiply input polynomial m(x) to xn - k. received polynomial will have (k – 1) + (n – k) = (n - 1) degree. Let’s imagine it in the form of
m(x) xn - k = q(x) g(x) (х),
where q(x) – particular from dividing m(x) xn - k to generating polynomial g(x), а (х) – excess from dividing. As far as g(x) degree is equal to (n - k), then (x) degree must be (n - k - 1) or less, and polynomial (x) will be in the form of
(x)= n - k - 1 xn - k - 1 +...+ 2 x2+ 1 x +0.
With account of arithmetic rule in GF(2) выражение m(x)xn - k = q(x)g(x) (х) expression you may rewrite by the following form:
(x) xn - k m(x) = q(x) g(x)
where it can be seen that polynomial (x) xn - km(x) is g(x) - tuple and has n - 1 degree or less. Hence it corresponds to the properties of coding combination of cyclic codes and it represents a password coding informational consequence.
Opening the last expression we receive
(x m(x).) m(x)xn - k = mk - 1 xn - 1+…+ m1 xn - k+1 + m0 xn - k + n - k - 1 xn - k - 1 +…+ 1 x + 0 ,
That corresponds to a password u = (mk - 1 … m1 m0 n - k - 1 1 ... 0).
Therefore a password of cyclic code consists of unchangeable informational part m and (n - k) checking symbols. Checking symbols are coefficients of polynomial (x), i.e of excess of dividing m(x) xn - k to generating polynomial g(x).
Example. With using of code, setting by generating polynomial g(x) = x3+x+1,we encode random consequence, e.g m=(1001).
Sequence m =(1001) corresponds to m (x)= x3+ 1 polynomial.
We multiply m(x) to xn - k :
m(x) xn - k =m(x) x3 =( x3+ 1) x3 = x6 + x3.
We divide m(x) xn - k to generating polynomial g(x) :
X4
X4 + 0 + X2 + X
X2+ X=(x).
In such away coding polynomial corresponds to informational consequence m = (1001), will have the following form .
U(x) = 1*X6+ 0*X5+ 0*X4 + 1*X3 + 1*X2+ 1*X1 + 0*X0,
And corresponding to coding word u = (1001110).
2.The Procedure.
2.1. Research of data transform with binary symmetric channel and bch code with fixing length of coding combination..
For carrying out of research it is necessary to compose a module of communication channel, carrying out the following operations :
• using a command commlibv1, to call to the screen structural scheme of communication channel and to activate a block of information coding Error control coding;
• on appearing screen Demo Error - Control Coding/Decoding Library to choose window BCH codec;
• on new window to choose a system with sequence transformation of binary elements of code combination BCH Binary sequence BCH encode;
• activate a window with green color which allows to receive etalone module for BCH codes БЧХ Sequence codec demo;
• on line of the main menu of the new window to activate a word File and in appeared menu click New, further on the right menu click Model, appears a free window;
• using a method of dragging to gather a model by sample shown in fig 1.
Figure
1. Model for testing channel with BCH code
Before starting out a model for collection of statistic data it is necessary to install parameters for each block of modeling system.
Thereby conform interaction of blocks of models by impulse are reached by this.
For carrying out of given procedure it is necessary to click twice with left mouse in the field of interested block and then to install the necessary parameters on the opening panel
Tuning of block parameter we consider on the example of the source of discrete signals. The form of panel for installation of parameters only for one period is shown in Figure 1
On the first line a period signal sequence in seconds is installed and a number of active output. For example installation of value 5,1 means that 20 period of sequence of the source combination of information contains 5 second, and active output of block is output with number1
The second line indicates timing frequency of reading of information signals from sensor of accidental number. Usual this frequency is installed in correspondence with the relationship 1/k, where is k – a number of information symbols in coding combination.
On the third line a cycle of sequence of accidental code combination is installed. On installation 1 a cycle is repeated before finishing of modeling time. On installation of “0” cycle is repeated only once. В indicated processes is necessary for debugging of modeling.
On the forth line an active output of block is determined.