Let m 3 be an integer and let t be a positive integer ie the error correction

# Let m 3 be an integer and let t be a positive integer

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Let m 3 be an integer and let t be a positive integer (i.e. the error correction capability of the code). Then, binary BCH codes exist for ( n, k ) where n = 2 m 1 ; n mt k < n . (74) Generator polynomials for a large set of binary BCH codes can be found in a number of digital communication texts (e.g. Proakis [2]). Example 7.14: In Table 7.10-1 of Proakis [2], find the generator polynomial for the (15 , 7) binary BCH code. What is the codeword for information vector X m = [0001111]. From Proakis [2], Table 7.10-1, the generator polynomial coefficients are given by the octal number 7 2 1 = [1 1 1 0 1 0 0 0 1] . The polynomial has degree n k = 8. It is g ( p ) = p 8 + p 7 + p 6 + p 4 + 1 . [1 1 1 0 1 0 0 0 1] (75) [1 1 1 1 0 0 0] C m = [0 0 0 , 1 0 1 1 , 1 0 1 1 1 , 1 1 1]
Kevin Buckley - 2010 140 7.5.9 Other Linear Block Codes In this Subsection, we overviewed a number of important binary linear block codes. These are important in their own right, and they are also often used as building blocks for other codes, such as concatenated and product codes, which we will investigate later. Descriptions of other useful codes, such as Euclidean geometry, projective geometry, quadratic residue and fire codes, can be found in Lin & Costello’s text [1]. We have considered binary linear block codes, there performance parameters d min and the weighting function wn j , and some encoder structures. We now turn to decoding algorithms for and performance analysis of these codes.
Kevin Buckley - 2010 141 7.6 Binary Linear Block Code Decoding & Performance Analysis In a digital communication system, we receive noisy symbols. For a system employing block coding and a given modulation scheme, we will receive a sequence of waveforms, that represent a block codeword C m , 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 waveform followed by a symbol-rate sampler. This receiver structure is illustrated in Figure 60. The matched filter output vector r contains the samples corresponding to a transmitted codeword C m . That is, it contains the n matched filter output samples corresponding to the n symbols representing C m . Under the assumptions stated above, this vector can be processed (without any information of the received data for other codewords) to optimally estimate C m . The exact statistical characteristics r , and this the communications system performance, will depend on the modulation scheme employed. linear binary block code encoder modulator C m X m communication channel quantizer decoder ML r X m ^ C m ^ X m ^ C m ^ X m ^ C m ^ soft decision decoder decisions hard hard decision decoder n T c R Y matched filter r (t) r (t) Figure 60: Digital communication system with block channel encoding.