66806

General description Encoding and decoding

An (n,k) lincar codc is called a cyclie codę if a cyclic shift of any codcword is also a codcword. The shift may hc leftwards or rightwards. by any nunibcrof bits._

The cyclic codę naturę can be utilized when encoding and decoding, using linear-feedback shift registers. Hardware implementations are not discussed.

Consider the Hamming codę (7,4), defined by parity-check sums (6.4.7). Its codewords are presented in Table 6.4.2. Thcn. consider the codcword w, = [11010011. I(s cyclic shift, rightwards by one bit: v = [1110100] is not the codeword. as its syndrome s = (101) * [(XX)]. Thcrcforc. such Hamming codc is not a cyclic codc.

Consider an other codc. obtaincd from the Hamming codc by permutation of columns. such that systematic codę is obtained. The codę is defined by the following parity-check niatrix and parity-check equations:

*y*l

1 1	1 1	0	1	0	0
0 1	1 1	1	0	1	0
1 1	1 0	1	0	0	1

H

Pl = ®'2

p₂ = i₂ ® i j ® i₄ p₃=i_l ®/₂ ©i₄

Ali codę words are presented in Table 7.1.1. As can be obscrved. the obtained codę is the cyclic one, there are four cycles in this codę. Assignment of codewords to cycles is outlined in the last column of this Table. All-0 word gives the 1* cycle, all-1 word gives the 2^nd cycle, cycles III and IV are shown in Fig.7.1.1.