Syndrome decoding of the block codes

The pprinciple of syndrome decoding we will consider on an example of simple block code.

Example 5.3. The syndrome decoder of systematic code (7, 4).

According to a rule (5.8) for realization of the syndrome decoder it is necessary to form the transposed parity check matrix of a code (7, 4). The parity check matrix of this code looks like (5.5). Applying to it a rule of a transposition of matrixes it is received:

; . (5.9)

It is convenient to note the single errors in transmission channel so:

e ₁ = (100…0), e ₂ = (010…0), e ₃ = (001..0), …, e _n = (000..1). (5.10)

In such form the error vector e _i represents a symbol set from n elements in which on a place with number i the symbol of an error 1 (at the left) is arranged and on remaining places zero symbols are arranged. Error vectors can be presented in the form of an identity matrix:

, (5.11)

which each row is the single error vector. Using properties of identity matrixes, it is easy to show, that the matrix of syndromes coincides with the transposed parity check matrix of this code (5.9) is:

S = E · H ^T = I _n· H ^T = H ^T. (5.12)

By the syndrome decoding of a block code the matrix of syndromes S coincides withthe transposed parity check matrix of a code H ^T. It is the foundation for tabling of syndromes. The more low reduced table 5.1 of syndromes for a code (7,4) is made according to rows of the transposed parity check matrix (5.9). In the table to each vector of an error there corresponds the vector of the syndromespecifying a location of an error symbol in the received code word.

table 5.1 – syndromes for decoding of the code (7,4)

syndromes
Errors	e1	e2	e3	e4	e5	e6	e7

It allows to formulate of syndrome decoding algorithm:

1 Forming of the transposed parity check matrix of a code H ^T.

2 Tabling of syndromes for decoding of (n, k) code.

3 An evaluation of syndromes (as table 5.1) on structure of code transposed parity check matrix H ^T and error symbols vector of a decoded codeword by rule (5.12).

4 Forming of a vector of an error e _i on the basis of the syndromes table.

5 Error correction in the received code word by a rule: .

The structure of syndrome decoder of code (7,4) realizing this algorithm is reduced on figure 5.2. According to rule (5.12) received channel symbols move to modulo-2 adders. The connections with lines of channel symbols are available there where in rows of transposed parity check matrix the symbol 1 is arranged. In the scheme of syndrome analyzer with according to given table 5.1 there is transformation of syndrome vectors S = (s ₀, s ₁,..., s_{n–k –1}) in the corresponding error vectors e which then move to the error corrector.