6.3 Complexity of coding and decoding algorithms

The using of code control ability depends on decoding algorithm. By full decoding use all possibilities to correct errors following from properties of a code. According to Shannon fundamental theorem the error-control codes used for correction of channel errors should get out long enough. However with growth of a code word length n increases complexity of realization procedures both of encoding and decoding that causes difficulty of practical realization of codecs. In the coding theory of along with estimations of code error-control ability of can estimate complexity of realization of encoding/decoding procedures which can be realized by software or hardware. Thus as argument of complexity function the length of a codeword n should act.

coding complexity of a block codes C_cod with use of generator matrix (n, k) code with a size nk = n²(1 – R_code) usually estimate in the value which is proportional to number of elements of the generator matrix

C_cod = nk = n²(1 – R_code). (6.3)

The decoding algorithms appear more difficult. Among them it is considered to be the most difficult full-search algorithm according to which the decoder by the full searching compares the received code word with the set of all possible words and the decision on that transmitted from the allowed word which appears on the minimum distance from the received word (decoding by a minimum distance) passes. It is considered to be complexity of full-search decoding algorithm proportional to quantity of all possible code words to volume of full search:

C_decod = mⁿ = 2ⁿ. (6.4)

It is said that complexity of full-search decoding increases «as an exponent» with growth of code length. Clearly, that full-search decoding algorithms are practically difficult for realising for long codes.

Questions

6.1What is practical significance of use of Hamming upper bound and Varshamov-Gilbert lower bound for an estimation of block error-control codes performances?

6.2 to what bound (upper or lower) is it necessary to aspire by elaborating of new block codes?

7 Important classes of block codes

The big number of codes, various on structure, construction principles and control ability is known. In this Chapter the classes of effective block codes with simple decoding algorithms are considered.

7.1 Hamming codes

Hamming codes (by R. Hamming) – systematic block codes with parameters:

– code word length n = 2^r – 1;

– Quantity of information symbols k = 2^r – r – 1; (7.1)

– Number of additional symbols r = n – k;

– Minimum distance d_min = 3, r = 2, 3, 4.

Hamming codes – perfect codes which correct single errors.

By parameters choice r = 2, 3, 4 according to formulas (7.1) it is possible to set all known binary Hamming codes. For example, at r = 3 the parameters of a code (7, 4) will be the following:

– Length of the code word n = 3;

– Quantity of information symbols k = 4;

– Minimum distance d_min = 3;

– Code rate R_code = (2^r – r – 1)/(2^r – 1) = 4/7.

Generator and check matrixes of this code have been considered earlier, in section 4.1 (formulas (4.4) and (4.7)). As it has been noted earlier, this code allows to detect double errors also. Structures of encoder and syndrome decoder of Hamming code have been considered earlier in section 5.1 (figures 5.1, 5.2). According to the formula (3.5) transposed parity check matrix of this code looks like:

. (7.2)