Letm≥3 be an integer and lettbe a positive integer (i.e. the error correction capabilityof the code). Then, binary BCH codes exist for (n, k) wheren= 2m−1 ;n−mt≤k < n.(74)Generator polynomials for a large set of binary BCH codes can be found in a number ofdigital communication texts (e.g. Proakis ).Example 7.14:In Table 7.10-1 of Proakis , find the generator polynomialfor the (15,7) binary BCH code. What is the codeword for information vectorXm= .From Proakis , Table 7.10-1, the generator polynomial coefficients are givenby the octal number7 2 1 = [1 1 1 0 1 0 0 0 1].The polynomial has degreen−k= 8. It isg(p) =p8+p7+p6+p4+ 1.[1 1 1 0 1 0 0 0 1](75)[1 1 1 1 0 0 0]Cm= [0 0 0,1 0 1 1,1 0 1 1 1,1 1 1]
Kevin Buckley - 20101407.5.9Other Linear Block CodesIn this Subsection, we overviewed a number of important binary linear block codes. Theseare important in their own right, and they are also often used as building blocks for othercodes, such as concatenated and product codes, which we will investigate later. Descriptionsof other useful codes, such as Euclidean geometry, projective geometry, quadratic residueand fire codes, can be found in Lin & Costello’s text .We have considered binary linear block codes, there performance parametersdminand theweighting functionwnj, and some encoder structures. We now turn to decoding algorithmsfor and performance analysis of these codes.
Kevin Buckley - 20101417.6Binary Linear Block Code Decoding & Performance AnalysisIn a digital communication system, we receive noisy symbols.For a system employingblock coding and a given modulation scheme, we will receive a sequence of waveforms, thatrepresent a block codewordCm, which is corrupted by noise. Under the assumptions that:•the sequence of information bits is statistically independent,•a memoryless modulation scheme is employed,•channel is memoryless; and•the noise is white,the optimum receiver front end will consist of a filter matched to the symbol waveformfollowed by a symbol-rate sampler. This receiver structure is illustrated in Figure 60. Thematched filter output vectorrcontains the samples corresponding to a transmitted codewordCm. That is, it contains thenmatched filter output samples corresponding to thensymbolsrepresentingCm. Under the assumptions stated above, this vector can be processed (withoutany information of the received data for other codewords) to optimally estimateCm. Theexact statistical characteristicsr, and this the communications system performance, willdepend on the modulation scheme employed.linear binaryblock codeencodermodulatorCmXmcommunicationchannelquantizerdecoderMLrXm^Cm^Xm^Cm^Xm^Cm^soft decisiondecoderdecisionshardhard decisiondecodern TcRYmatched filterr (t)r (t)Figure 60: Digital communication system with block channel encoding.