BCH code: Difference between revisions

Revision as of 08:05, 14 December 2019

Revision as of 20:31, 10 April 2020

Line 427:

==== Decoding with unreadable characters ====

Suppose the same scenario, but the received word has two unreadable characters [ 1 {{color|red|0}} 0 ? 1 1 ? 0 0 {{color|red|1}} 1 0 1 0 0 ]. We replace the unreadable characters by zeros while creating the polynompolynomial reflecting their positions <math>\Gamma(x) = \left(\alpha^8x - 1\right)\left(\alpha^{11}x - 1\right).</math> We compute the syndromes <math>s_1=\alpha^{-7}, s_2=\alpha^{1}, s_3=\alpha^{4}, s_4=\alpha^{2}, s_5=\alpha^{5},</math> and <math>s_6=\alpha^{-7}.</math> (Using log notation which is independent on GF(2⁴) isomorphisms. For computation checking we can use the same representation for addition as was used in previous example. Hexadecimal description of the powers of <math>\alpha</math> are consecutively 1,2,4,8,3,6,C,B,5,A,7,E,F,D,9 with the addition based on bitwise xor.)

Let us make syndrome polynomial

Line 498:

Let us show the algorithm behaviour for the case with small number of errors. Let the received word is [ 1 {{color|red|0}} 0 ? 1 1 ? 0 0 0 1 0 1 0 0 ].

Again, replace the unreadable characters by zeros while creating the polynompolynomial reflecting their positions <math>\Gamma(x) = \left(\alpha^{8}x - 1\right)\left(\alpha^{11}x - 1\right).</math>

Compute the syndromes <math>s_1 = \alpha^{4}, s_2 = \alpha^{-7}, s_3 = \alpha^{1}, s_4 = \alpha^{1}, s_5 = \alpha^{0},</math> and <math>s_6 = \alpha^{2}.</math>

Create syndrome polynomial