Laboratory work 2 Cyclic codes correcting capability research

Objective: To study the properties of the cyclic code (15, 11); to build the characteristics that determine the corrective properties of this code.

Laboratory emulator:

After opening the file “CK15_11.exe” you will propose to enter initial data from the table 3.2, where p₀ – error probability, L – amount of codewords.

Table 3.2 – Data for laboratory experiment

p0	L	i1	i2	k0	kd	W
0,05	106
0,03	106
0,01	106
0,005	5,0·106
0,003	5,0·106
0,001	5,0·106

In the result of the experiment, you get next parameters:

• i1 – amount of errors at the input of the decoder;

• i2 – amount of errors at the output of the decoder;

• k₀ – errors coefficient in the discrete symmetric channel (DSC);

• k_d – errors coefficient in the output of communication system;

• W – coding gain.

Parameters of the cyclic code (15, 11) k_0, W and k_d calculated by formula

k₀ =i1/Ln,

k_d =i2/Lk.

W=p₀/k_d.

With a large value L error coefficients lose their random nature, approaching a constant value close to the error probability, which is formally given by

Therefore, we can assume that for sufficiently large values L and i1> 100 error coefficients k₀ and k_d are almost equal to probabilities p₀ and p_d. With the help of probabilistic characteristic of decoding (PCD) p_d = f(p₀) you can make a conclusion about correction ability of the code.

Laboratory task:

1. To run the program “CK15_11.exe”.

2. To enter initial data from the table 3.2.

3. To fill the table 3.2. For getting the results for each line you need to rerun the program “CK15_11.exe”.

4. To build PCD for the code (15, 11) as graph k_d = f(p₀) using graph from home task.

Home task:

1. To learn item 1.2 of this teaching manual.

2. To write down the answers to the general questions.

3. To build graph of the theoretical PCD p_d = f(p₀) for the code (15, 11) using table 3.3 and application A.

Table 3.3 – Output calculation values p_er

p0	0,05	0,03	0,01	0,005	0,003	0,001
pd	2,6·10–2	1,0·10–2	1,3·10–3	3,4·10–4	1,2·10–4	1,4·10–5

4. To prepare table 3.2 in the protocol.

General questions:

1. What is the syndrome of errors?

2. How is the place of error with the help of syndrome determined?

3. Explain how the syndrome S(x) is defined by the codeword F(x) of cyclic code.

4. Describe disadvantages of the syndrome decoding.

5. What is errors coefficient?

6. How errors coefficient is error probability corresponds to?

7. Explain the meaning of PCD of the correcting code.

8. How is code gain W determined with the help of graph PCD knowing p₀?

Protocol content:

1. Subject and objective.

2. Executed home task.

3. Graph and table 3.2 according to laboratory task.

4. Conclusion about equivalence of experimental and theoretical graphs of PCD.