Questions

3.1 What codes are named as error-control codes?

3.2 Explain error-control code encoder and decoder destination.

3.3 What is redundancy and code rate?

3.4 What is the Hamming distance between code words, code distance, and error order?

3.5 How to explain detecting and control code ability?

3.6 Explain error detecting and error correction general principles.

3.7 What codes are named cyclic?

3.8 How to write code word as polynomials?

3.9 Explain cyclic codes encoding and decoding principles.

4 Home task

4.1 Study "Error-control codes" from the compendium of lectures and literature [5, p. 137...150; 2, p. 287...297].

4.2 Write down the number of brigade N + 8 in the binary number. Considering these number as information word, length k = 5 on the encoder input, form cyclic code (10, 5) code word, using generate polynomial g(x) = x⁵ + x⁴ + x² + 1.

4.3 For three errors words е₁(х), е₂(х) and е₃(х), given in table 3, calculate syndromes, and then, using table 2, define decoder work result. If the calculated syndrome is wrote in table 2, a decoder inverts code word symbol which is considered as error. If the calculated syndrome is absent in table 2, a decoder does not change code word symbols which are decoded. Code word on the output of decoder is code word on decoder input first k symbols.

Table 3 – Polynomials of errors for the home task

Number of work place N	e1(x) – single error	e2(x) – doubled error	e3(x) – triple error
1, 11	x9	x9 + 1	x7 +x6 + 1
2, 12	x8	x6 + x8	x9 + x8 + x4
3	x7	x6 + x	x9 + x8 + x2
4	x6	x9 + x	x7 + x6 + x
5	x5	x7 + x6	x8 + x7 + x2
6	x4	x7 + x4	x8 + x7 + x3
7	x3	x9 + x2	x8 + x7 + x
8	x2	x6 + x3	x8 + x6 + x
9	x	x7 + x2	x9 + x6 + x4
10	1	x8 + x	x8 + x6 + x4

4.4 Draw a shame error-control code encoder and decoder including in the digital communication channel structure.

4.5 Be ready to discuss key questions.

5 Laboratory task

5.1 Acquaintance with a virtual model. Start the program 4.2, using the icon TT (English) on a desktop. It is necessary to study the structure of a virtual model using its description in part 6 of this LW. Coordinate the plan of performance of the laboratory task the teacher.

5.2 Encoding process research. The cyclic code (10, 5) from home task is researched. For this purpose:

- choose in a menu "That do we research?" point "Encoding";

- choose a code (10, 5) and set a proper generate polynomial;

- enter information code word determined in home task.

Run the program and compare the code word on the encoder output with calculated in the home task.

5.3 Transmission by a communication channel process research. For this purpose:

- choose in a menu "That do we research?" point "Transmission by a channel";

- enter zero error word, length п.

Run the program at the same settings (from item 5.2). Make sure, that in the case of zero error code word on the output "Telecommunication channel" coincides with code word on the input.

Set error word, which corresponds to the single error е₁(х) from Table 2 for your variant. Run the program, compare code words on the telecommunication channel input and output and make sure in the telecommunication channel with errors correct work.

5.4 Research of process of decoding. For this purpose:

- choose in a menu "That do we research?" point "Decoding";

- enter zero error word.

Run the program at the same settings. Make sure, that a syndrome is zeroes, and code word on the decoder output coincides with code word on the encoder input.

Set errors word which corresponds to the single error е₁(х) from the home task. Run the program at the same settings in encoder. Make a table with the decoding results according to the sample Tabl. 4.

Table 4 – Cyclic code (10, 5) for the code word N + 8 = 22 (encoder input – 10110, encoder output - 1011001101) researches results

Errors word е(х)	Encoder input	Encoder output	Syndrome s(x)	Error symbol number, which decoder define
x9	0011001101	10110	х4 + х3 + 1	x9
…	…	…	…	…

Repeat research of decoding at double е₂(х) and triple е₃(х) errors from the home task. Compare got results with calculations in the home task results.

Repeat research of decoding at the arbitrary fourfold error е₄(х). For certain error words, for example, е₄(х) = х⁶ + х⁵ + х³ + х syndrome is zeroes, that confirms that a code (10, 5) has code distance d_min = 4.

5.5 Other cyclic code (n, k) research.

Teacher gives a cyclic code (n, k). Repeat research item 5.4.