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
d min	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
	x 4 + x 3 + 1; x 4 + x + 1
	x 5 + x 2 + 1; x 5 + x 3 + 1;
	x 11 + x 10 + x 6 + x 5 + x 4 + x 2 + 1; x 11 + x 9 + x 7 + x 6 + x 5 + 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
N er. in	p	N er. out	p d	Fault in decoding

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.

6 Description of the computer program of (n, k) code correcting ability research

6.1 Decoder. For researches the realized programmatic model of Meggitt decoder of cyclic (n, k) code is for n ≤ 32, the block diagram of which is resulted in a figure 3. A Meggitt decoder is a tabular decoder in which entered the following of decoding [8]:

1. Syndrome analyzer includes pre-computing “Table of syndromes” that contains all errors configuration that can be corrected by this (n, k) code and their corresponding syndromes.

2. For the accepted code word a concrete syndrome is calculated by ordinary rule is dividing of the accepted code word on generator polynomial. Calculator syndrome is based on an (n – k)-bit shift register. Syndrome is the remainder of the division.

3. The "Syndrome analyzer" is performed by a search in the tables of the calculated concrete syndrome, reading of errors configuration and presentation of the proper sequence of "1" to "Errors corrector" for the errors correction in the accepted code word.

6.2 Research of correcting ability of (n, k) code. Conducted on a block diagram, resulted in a figure 4.

As a generator of errors symbols is used generator of pseudo noises numbers 0 and 1, in which probability 1 equals probability of error of input symbols р.

Meters fix:

- number of code words which are analyzed in times of supervision – decod_suc, let's designate N;

- number of code word errors at the decoder input – input_err, let's designate N _{er. in};

- number of code word errors at the decoder output – output_err, let's designate N _{er. out};

- number of decoding fault (the calculated syndrome is not found in tables) – decod_err.