3 Questions

3.1 Specify the methods of error-control codes decoding which are known by you.

3.2 What is syndrome of code word and what purpose is it used for?

3.3 What (n, k) block error-control codes the syndrome decoding is mainly used for?

3.4 What function is executed by the analyzer of syndrome in the decoder of error-control code?

3.5 How errors are corrected in code words of error-control code?

3.6 How is the syndrome of code words calculated for cyclic codes?

3.7 That does determine CG of error-control code?

3.8 What parameters of error-control code does CG depend on?

4 Home task

4.1 In the workbook, prepare a graph р = f ( ) for BPSK and QPSK signals (redraw from figure 2 or calculate).

4.2 Use the graph to calculate the CG, provided by error-control block (n, k) code, at a given probability p_d according to table 1.

4.3 Prepare to the discussion of questions.

Table 1 – Error-control code parameters for home task calculations

Brigade number	Error probability at the decoder output pout	Error probability at the decoder input pin	Code rate Rcode = k/n
1	10–4	5∙10–2	0,5
2	10–5	2∙10–2	0,6
3	10–4	10–2	0,7
4	10–5	5∙10–3	0,8
5	10–4	2∙10–3	0,9

5 Laboratory task

5.1 Acquaintance with the computer model of decoder. For this purpose to start the program, using an icon “Laboratory works” on a desktop, and then folder of TT‑2. To master a conducting method research of correcting ability of (n, k) block code, that by introduction of basic data, start of the program, reading of results. To bring the chart of researches to the LR report.

Table 2 – Parameters entered in the study

Denotation	Comments
n	Code word length (bit)
k	Number of information bits
dmin	Code distance
g(x)	Generate polynomial
prob_err	p – decoder input error probability
num_rand	N – number of code words

At each new dimension prob_err and num_rand must be changed.

5.2 Experimental research of correcting ability of decoder. The experimentally determined values р and р_d and recorded in the table 4. Research is conducted for codes (n, k):

- brigade №1: (31, 26), (23, 12) codes;

- brigade №2: (30, 25), (22, 11) codes;

- brigade №3: (29, 24), (21, 10) codes;

- brigade №4: (28, 23), (20, 9) codes;

- brigade №5: (27, 22), (19, 8) codes.

On the teacher task can be researched other (n, k) codes for n  32.

Generator polynomials for researches get out from table 3.

Table 3 – Generator polynomials of cyclic codes (n, k)

Order n – k	Generator polynomials
4	x4 + x3 + 1; x4 + x + 1
5	x5 + x2 + 1; x5 + x3 + 1;
11	x11 + x10 + x6 + x5 + x4 + x2 + 1; x11 + x9 + x7 + x6 + x5 + x + 1
Example of polynomial input: х^4+х^3+1

During research to set error probability at the decoder input р < 0,01 on diminishing (5–6 points) until error probability at the decoder output р_d will not attain a value, near by 10^–5.

Table 4 – The results of measurements (specify code)

Measu-ring number	Code words number N	Decoder input		Decoder output
		Ner. in	p	Ner. out	pd	Fault in decoding
1
2
3
4
5

Symbol error probability at the decoder input and output are calculated on formulas:

p = and p_d = , (4)

where N_{er in} and N_{er out} – number of word errors at the decoder input and output in times of supervision.

According to Table. 4 shall be calculated dependence р = f ( ), then the CG is determined at levels p_d = 10^–4 and p_d = 10^–5.