Main principles. The systematic code is error-control code, which code word contain k information bits and r = n – k checking symbols (checking symbols are linear combination

The systematic code is error-control code, which code word contain k information bits and r = n – k checking symbols (checking symbols are linear combination of information bits). Systematic codes are denoted as (n, k) or (n, k, d _min). In this work code (7, 4) or (7, 4, 3) is studied.

Error-control codes with code distance d _min = 3, allowing to correct first order errors at decoding, name as Hamming codes [5, p. 149]. We will determine connection between error-control code parameters n and k. It is known that for any natural number r the Hamming code of lengths n = 2 ^r – 1 or k + r = 2 ^r – 1 exists [5, p. 149]. These equalities can be used and as inequalities k £ 2 ^r ‑ r ‑ 1. The last expression allows choosing n and r at given k.

The matrix method of linear block codes coding and decoding processes description is most useful (see Sections 5, 6). So, coding by systematic code (n, k) consist in addition to code words checking symbols and can be described by matrix equality

A×G = B, (1)

where A = (b ₁ b ₂ … b_k) is row matrix size k, correspond to information code word;

B = (b ₁ b ₂ … b _k b _k+1 … b _n) is row matrix size n, correspond to error-control code word.

G – generator matrix k ´ n, the elements of which g_ij take on values 1 or 0.

The G matrix rows must satisfy next conditions [6, p. 86…88].

1 Distance between any two rows must not be less d _min.

2 Every row must contain no less then d _min units.

3 All rows must be linearly independent, i.e. none of rows can be got by adding (XOR) of some other rows.

For example, for a code (7, 4) generator matrix looks like formulas (4.4), (4.7).

The coder operation algorithm:

1 k information bits in a parallel code or in a series code (in last case a shift register is needed) on the coder input.

2 The checking symbols r = n – k by adders with r calculates.

3 k information bits and r checking symbols in a parallel or series code (in last case the converter of parallel in series code is needed) on the coder output.

On the figure 5.1 the code (7, 4) with generator matrix (4.4) encoder functional diagram is resulted. Input and output code words are represented in a parallel code.

The process of decoding includes the syndrome calculation. In matrix form is written down as:

S = H× , (2)

where H is check matrix r ´ n.

is matrix-column, size n, correspond to the code word on the decoder input;

S is syndrome, matrix-column, size r.

The decoder algorithm is following:

1 n symbols of the code word come on the decoder input.

2 Using (6) a syndrome calculates.

3 Syndromes analyzer, built on the syndromes table basis, forms signals for error symbols correction.

4 Error symbols correction consist in their inverting: XOR for error symbol and unit (lets is some symbol, then ).

5 After correction of error the information code word of k symbols come on the decoder output.

On the figure 5.2 the code (7, 4) decoder functional diagram is resulted. Input and output code words are represented in a parallel code.

A code, encoder and decoder of which, is built on multiinput adders (with XOR operation), name as Hamming code or even parity check code. Using the indicated principles it is possible to build encoder and decoder for different values of n and k.