Reliability Level List Based Direct Target Codeword Identification Algorithm for Binary BCH Codes

Reliability Level List Based Direct Target Codeword Identification Algorithm for Binary BCH Codes B.YAMUNA, T.R.PADMANABHAN 2 Department of ECE, 2 Department of IT Amrita Vishwa Vidyapeetham. Amrita School of Engineering Amrita Nagar, Coimbatore- 2 INDIA b_yamuna@cb.amrita.edu Abstract: - In BCH coded schemes the reliability information available with the demodulated bits can be effectively used for soft decision decoding (SDD) to improve signal to noise ratio performance. Chase algorithms, their adaptations, and modifications available for SDD trade complexity for performance to different levels. A new iterative algorithm Reliability Level List based Direct Target Codeword Identification Algorithm (DTCI) - is proposed in the paper; the algorithm yields the best that is possible with SDD. The concept of reliability level list (RLL) introduced in the paper is central to the application of the algorithm. At every stage of the iterative process followed, the algorithm uses the reliability information of the bits and identifies the next most likely candidate word to be examined. This ensures that the correct decoded codeword is identified through the shortest number of steps. Detailed simulation studies with different BCH codes amply bring out the effectiveness and superiority of the algorithm. Key-Words: - BCH codes, Reliability based decoding, Soft decision decoding, Probability of error. Introduction Channel coding techniques which combat channel impairments are an integral part of digital wireless communication systems. The fact that Hard Decision Decoding (HDD) discards information regarding the reliability of the received bit has been its bane. If the number of errors exceeds d min, the decoding is not unique. Approaches where we choose reliability information in the received bit sequence to go beyond HDD and identify codewords in a more meaningful manner comprehensively, constitute Soft Decision Decoding (SDD). Chase and Forney [-2] are the forerunners of reliability based SDD algorithms which use reliability information along with the hard decision decoded sequence to refine the decoding process. Fossorier et al. proposed a soft decoding scheme based on processing of the most reliable positions [3]. Of these, Chase algorithms have been of great interest to many researchers; its adaptations are available for binary linear error-correcting block codes []. These refinements have basically been multidimensional in nature in the sense that they exploit available computing power to improve performance, allow transmission through feeble channels (channels with low SNR values), exploit channel capacity to the utmost subject to specific performance criteria, and so on. The class of Chase algorithms decodes using the reliability values obtained from the demodulator and running subsequent trials of codeword estimation using suitable test patterns. The trials are run on a conventional (algebraic) decoding algorithm. These algorithms are all based on processing of the least reliable bits so that in each trial, different combinations of the least reliable received bits are processed and the decoder output is the candidate word with the best soft decision metric. These algorithms have a decoding radius of (d min -), d min being the minimum Hamming distance. Variations of Chase algorithms essentially focus on reducing the number of trials and hence the complexity involved in decoding to the codeword. Chana et al. have proposed an interesting SDD algorithm for binary cyclic codes which performs permutation decoding over a fixed number of least reliable positions of the received word []. A reliability based soft decision decoding algorithm for binary cyclic block codes that identifies the codeword with minimal effort has been proposed recently []. In this paper, the reliability based soft decision decoding algorithm has been extended for decoding binary BCH codes over the Additive E-ISSN: 222-28 28 Issue, Volume 2, June 203

White Gaussian Noise (AWGN) channel with Binary Phase-Shift Keying (BPSK) signaling. Performance wise the proposed algorithm has an edge over the class of Chase algorithms. In the proposed algorithm the reliability of information obtained from the demodulator is used in computing the error probability and hence estimating all possible patterns of (d min -) errors and even beyond. The error probability calculated from the reliability information is used to decide on the occurrence of single error, double errors, triple errors etc., and hence decode from amongst the 2 n possible words. The procedure is structured to yield the target codeword with minimal number of trials. The rest of the paper is organized as follows: Section 2 discusses the preliminaries of soft decision decoding and the Chase algorithms, Section 3 presents the proposed Direct Target Codeword Identification (DTCI) algorithm, Section discusses the simulation results and Section forms the conclusion. 2 Preliminaries For an (n,k) binary BCH code the message sequence of length k is encoded into an n-bit length codeword C using conventional encoding techniques. The code bits C i = C C 2..C n are fed to data modulator which converts the bit stream to the appropriate wave form (BPSK modulated) for transmission through the channel. This signal may be corrupted by the channel. Let the received signal be represented as r = (r 0, r, r 2.r n- ). From the r i values, a binary sequence Z i is generated based on the hard-decision rule: Z i = 0 for r i < 0 for r i 0 () The magnitude of r i can be used as a reliability measure of the hard-decision decoded bit and decoding decision regarding the error position could be made. The larger the r i the more reliable the hard decision regarding the bit value and less likely that the bit is in error. The Chase algorithms use a test vector T which is added to the received word to form the distorted word and seek the codeword by scanning around the distorted word within a radius of d min [2]. The three Chase algorithms differ in the number and pattern of test vectors T (and hence the number of trials) needed to decode to the closest codeword. Hence the scope of Chase algorithms is limited to a Hamming sphere of radius d min - about the received word with the exception of Chase III which decodes some patterns of errors beyond d min -. For Chase I algorithm the set T consists of all binary vectors of length n which contain exactly d min ones. (i.e., all possible patterns of errors up to d min -). For Chase II algorithm the set T consists of every combination of s, which are located in the d min least reliable positions. For Chase III algorithm the set T consists of all binary vectors of length n which contain ones in the i least reliable positions and zeros elsewhere, where i = 0, 2,,..., d min -, if d min is odd and i =0,, 3,,..., d min -,if d min is even. The main drawback of Chase I algorithm is the large number of test patterns involved in the trials. n C d min 2 numbers of test patterns which run through the entire length of n bits are invoked in the algorithm. The test pattern generation does not use the reliability information; the latter is used to derive an analog weight which decides the selection of the error pattern from amongst the set of error patterns obtained. Chase II algorithm generates distorted words with min 2 d test patterns all of them focused on the least reliable end. Chase III algorithm uses d min + test patterns for the search. But with these two unlike with Chase I only a restricted number of error patterns up to d min - are decoded. The decoding limit of Chase algorithms for a specific case of (,) binary BCH code with d min = is given in Figure. From Figure it is seen that in Chase I all the distorted words - C 2 in numberlie within a Hamming sphere of radius d min = 2. These on decoding using conventional algebraic decoding method identify all the codewords within a Hamming sphere of radius d min - around Z i (Figure a), these being the candidate codewords. All possible 0 to error patterns are included here. It is seen that Chase II uses four test patterns ( min 2 d ) - all zero pattern and three patterns with all combinations of s in the least two d min reliable positions. All the candidate codewords lie within Hamming spheres of radius d min - here d min = 2 - around the corresponding distorted words - that is within the envelope of these spheres as represented in Figure E-ISSN: 222-28 288 Issue, Volume 2, June 203

b. It is evident from the figure that all these candidate codewords lie within a Hamming sphere of radius d min - and form a subset of those in Chase I. It is pertinent to point out here that the simplification from Chase I to Chase II has left out some of the candidate codewords within the Hamming sphere of radius d min - (Figure b). If one of the left out candidate words were to have a lower value of the metric, Chase II would have decoded wrongly. One can see from the figure that the search is restricted to patterns of 2 errors, all 3 error cases where one of the errors is in any one of the two least reliable bit positions, and all error cases where two of the errors are in the two least reliable bit positions in the received word. It is seen that Chase III uses three test patterns ( d min + ) - all zero pattern, a pattern of two s and another of four s from the least reliable end. All the candidate codewords lie within Hamming spheres of radius d min - here d min = 2 - around the corresponding distorted words- that is within the envelope of these 3 spheres as shown in Figure c. It is evident from the figure that all these candidate codewords do not lie within a Hamming sphere of radius d min -; in this respect Chase III stands apart from both Chase I and Chase II. From the figure we find that the search is restricted to all patterns of 2 errors, all error cases when two of the errors are in the two least reliable bit positions, all error cases where four of the errors are in the four least reliable bit positions in the received word. Though it corrects up to d min - errors in any of the positions, Chase I algorithm is of limited interest due to the extensive search involved. Chase II and Chase III algorithms are simpler by one or two orders. The simplicity is achieved by a trade-off in the error pattern decoded as has been explained above. The same is true of Generalized Minimum Distance (GMD) [] (Essentially Chase III is the same as GMD vis-à-vis binary codes) as well as a number of other modifications [8] and generalizations [9] for bounded distance decoding. Hamming sphere of radius 2 around Z i (a) Chase I (b) Chase II Legend Received word Hamming sphere of radius around Z i Envelopes of Hamming spheres of radii 2 ( c ) Chase III Distorted word at d = from received word Distorted word at d = 2 from received word Distorted word at d = from received word Figure Decoding Bounds of Chase Decoding Algorithms: In each case the enclosures in thick lines represent the scan range. While decoding, with these classes of algorithms the focus is on optimizing a metric in terms of bits at the least reliable end; others optimize a metric defined in terms of bits at the most reliable end [0] to identify the target codeword. By providing different trade-offs between error performance and complexity these algorithms play a significant role in soft decision decoding. However the fact remains that there is a clear need for an algorithm which can use the reliability information in a structured manner and with minimal effort identify the most reliable codeword. Such an algorithm is evolved in the sequel. E-ISSN: 222-28 289 Issue, Volume 2, June 203

3 Direct target codeword identification algorithm A new class of algorithm that uses soft information from the channel directly as an index for estimating the error positions is proposed here. The novelty of the algorithm lies in the following: All the patterns of errors that are guaranteed to be decoded by Chase algorithm I are successfully decoded by using an approach that eliminates the need for running repeated trials on conventional decoder. This approach is much simpler as brought out through the case studies. The algorithm decodes all patterns of errors up to d min - which are beyond the decoding limits of Chase II and Chase III algorithms. There exists the possibility of Chase III erroneously declaring a less reliable candidate codeword as the target codeword which is avoided here. This has been brought out through an illustrative example. The algorithm also has an edge over the class of Chase algorithms by being able to decode beyond their upper bound of d min -. Every hard decision represented by () has a probability associated with it. For the AWGN channel it is given by mi Q = m i σ e 2π 2 x 2 dx (2) σ 2 being the variance of the noise and r i = m i is the reliability value of bit i. In the proposed algorithm the received word is decoded by successive scanning done from the least reliable end. At every step the probability of error as given by (2) is used as the index for selecting the next candidate word that is to be examined. The steps involved in the algorithm are explained through an illustrative example of a (, ) binary BCH code. Example - I 30eh is the transmitted code word. With σ = 0.8, the received word obtained is 8cch. There are errors at bit positions,,,. As before let r i = m i be the reliability value of bit i. The bits are assigned integer reliability indices k from to in the ascending order of magnitude m i as in Table. Chase I decodes the codeword with an extensive search as explained earlier. Chase II fails to decode this error pattern. The possible distorted words are 8cch, 80ch, 88ch, and 8ch. Algebraic decoding of these four distorted words results in decoding failure. With Chase III the three possible distorted words are 8cch, 80ch and 8eh. Algebraic decoding of these four distorted words also results in decoding failure. If Table Reliability magnitude and index for Example - I i m i k 0.003 0 0.09 3 2.93 3 0.982 8 0.09 2.38 0.0992 0.02 2 8 2.08 9.90 3 0.000 9 0.392 2.3200 2 3 0.088 0.329 mi Q = m i σ e 2π 2 x 2 dx represents the probability that the i th bit with k as the reliability index is received wrongly then let mi q i[ = Q (3) E-ISSN: 222-28 290 Issue, Volume 2, June 203

mi pi[ = Q () where p i [ is the probability that the i th bit with k as the reliability index is received correctly. The index i representing the bit position being superfluous here has been left out in the rest of the discussion. All the 2 terms in the expansion of the product k= ( p [ + q[ ) together represent the probability space of the candidate word of bits. These 2 elements of this space have their own associated probability of being the target codeword. All the possible 2 codewords are included in this set of 2 each with its own probability. The list formed with all the terms in the expansion of the above expression arranged in descending order of magnitude is called as Reliability Level List (RLL) here. For any given k value q[ 0. and hence p[ 0.. Hence the term p[ k= representing the probability that all bits are received without error has the largest magnitude and is at the top of RLL. k= p[ = (-0.002) (-0.009) (-0.03) (-0.08) (-0.09) (-0.0832) (-0.023) (-0.083) (-0.2929) (-0.293) (-0.308) (-0.309) (-0.300) (-0.) (- 0.22) =0.0229 The corresponding word being the most reliable of all the words deserves examination first; the word here is the received word (8cch) itself. The examination can be carried out directly by dividing the word by the generator polynomial of the code. A zero remainder implies the word to be a codeword and the search stops here. This not being the case here the search has to be continued with the next entry in the RLL. q [] being the maximum amongst all the q [ values, the product q [] [ k = 2 p = 2.2 0 - has the second largest magnitude; it is the second entry in the RLL indicating that the word with the bit for which k = that is b being in error is the next one to be examined. The word is 88ch which is not a codeword. The third entry in the RLL is the product q [2] p [] p [ =. 0 - k = 3 The corresponding word to be examined is 8ch taking b (since i = for k =2) to be in error. Since this is not a codeword, the fourth entry in RLL is to be examined: It is decided by the larger of the two of the products:. q [3]p[2]p[] the probability that bit for k = 3 (b )is in error. 2. p [3]q[2]q[] the probability that bits for k = 2 and k = (b and b ) are in error. If q[3]p[2]p[] > p[3]q[2]q[], this entry in RLL is q[3]p[2]p[] p [, else it is k = p[3]q[2]q[] p [. k = The dual process of identifying the next most probable word to be examined that is the next entry in the RLL and dividing it by the generator polynomial to ascertain whether it is a codeword is continued until the codeword the target codeword itself is identified. This completes successful decoding and RLL formation need not be continued further. The progress of the decoding process is summarized in Table 2. The salient features of the proposed algorithm can be summarized as follows:. Once a codeword has been identified further scan is not needed because subsequent codewords if identified will be less reliable than the earlier one. 2. The search has shown the bits b, b, b and b to be in error and the identified target codeword - 30eh - is the transmitted codeword itself. 3. The RLL entry at any stage can be represented k= as the product f p[. The term p[ in k= this product being a constant, one need to compute only the factor f at each stage to decide the next entry in RLL and the corresponding word to be examined; use of the factor f in place of the full probability value simplifies the computations [].. In fact the magnitude of f associated with a word decides the position of the word in the RLL. For a candidate codeword the f value represents its metric. In turn from amongst a set of candidate codewords the one with the maximum value of f represents the target codeword. The values of f are also given in Table 2.. The candidate codewords identified by E-ISSN: 222-28 29 Issue, Volume 2, June 203

applying Chase I algorithm are given in Table 3 along with their respective f values. Despite the extensive search, Chase I also zeros on to 30eh as the target codeword, its value of the metric being the smallest.. Though Chase I will successfully decode here, Chase II and Chase III fail. S.N o Table 2 RLL Decoding Sequence index for Example - I Bits error in K Error Probability None 22.9 0 - f 8,, 22.289 x 0-0 0.3320 9, 2, 0. x 0-0 0.3308 20, 2, 3.33 x 0-0 0.33 2, 3,. x 0-0 0.3323 22,3 3,. x 0-0 0.3308 23, 3,.3 x 0-0 0.39 2, 3,.99 x 0-0 0.38 2 2.2 0-0.828 3 2. 0-0.802 3 3.2 0-0. 2,,,2, 2,,3,2, 0.3 x 0-0 0.209 x 0-0 0.2993 9 0.290023,,2 99. 0-0.23298 2,,,2, 3 x 0-0.223 8,,3.9 0-0.23023 3 28,,,2, 9. x 0-0.2999 9, 2,3 3.0 0-0.03 8,,,2, 3 8.9 0-0.999 3 9 8.2 0-0.08 0 3 3. 0-0.38. 0-0. 2 0.3 0-0.0 3,, 0.229 0-0.3399,3, 0.0829 0-0.32028 3, 2, 0.0293302 0-0.339,3 2, 0.0038 0-0.3 29,,,3, 30,,3,3, 3,, 2,3, 32,,3 2,3, 33,,,3, 3,,,3, 3,, 2,3, 3,, 2,3, 2. x 0-0.23 2.8 x 0-0.238 9. x 0-0.289.9 x 0-0.202.328 x 0-0.2029 0.3888 x 0-0.2982 0.0882 x 0-0.2283 0.0222 x 0-0.2222,, 38. 0-0 0.388 38 3,,,,2, 3, 0.0090 x 0-0.220308 9 E-ISSN: 222-28 292 Issue, Volume 2, June 203

38,,,3,2, 3, 39,,,,2, 3, 0,,,,2, 3, 0.000 x 0-0.00022 x 0-0.2939 0.20822. x 0-20 0.208208 39,3, 0.8998 x 0-20 0.932 2,, 0. x 0-3,, 0.030 x 0-3,, 0.008 x 0-3,, 0.0002 x 0 -,, 0.0008 x 0 -,,3,, 8,,3 2,, 9,,,, 0,,,.,3,,, 2,3,,, 3,, 2,,,, 2,,,3, 2,, 20 20 20 20 20 0.82 0.8382 0.83 0.828 0.9 2.832920 x 0.98 0-2 0.392 x 0-2 0.03 0.08 x 0-2 0.00 x 0-2 0.0033 x 0-2 0.202 0.8 0.28 23. x 0-30 0.09 3. x 0-30 0.93 0.0393 x 0-30 0.2 0.039 x 0-30 0.3,,3 3,, 0.008 x 0-0.83,,,, 8,, 2,, 9,, 3,, 0,, 3,,,3, 3,, 2,3, 3,, 3,, 3,,,,,3,2,,,,,,2,,,,,,2,, 30 0.00 x 0-30 0.389 0.0002x 0.392 0-30 3.0 x 0-3 0.3892 0. x 0-3` 0.3898 0.00 x 0-3` 0.009 x 0-3` 0.009 x 0-3` 0.382 0.3989 0.32 0.00030 x 0.28008 0-3` 9.33 x 0-0 0.29 0 0.9890 x 0-0 3 8 0.022 x 0-8,,3, 9,,3,,2,,,2,, 0,,,3 2,3,,,,,,2,, 0 0.00293 x 0-0 0.280 9 0.2 0.22988 2 0.0003x 0-0 0.200. x 0-0.988 3 0.80 x 0-0.38 0 2 0 9 0.0 x0-0.03 3,,, 2,3,,,,, 2,3,, 0.0002 x 0-0.0009 x 0-0.9 0.23 E-ISSN: 222-28 293 Issue, Volume 2, June 203

,,3,,,3,,,, 2,3,, 2,3,, 2,3,,.2 x 0-0 0.08 3 0.8329 x 0-0 0.0 0.0882 x0-0 0.033 9 Direct Target Codeword Identification Algorithm (DTCI) The procedure evolved through the illustrative example above can be cast as a structured algorithm; we call this the Direct Target Codeword Identification Algorithm (DTCI). The step by step procedure is as follows:. Let {r i }, {m i }, and {b i } represent the sets of received signal values, their magnitudes, and the corresponding bit values. 2. Compute the set {Q(m i )} where m i σ mi 2 x 2 Q = e dx 2π is the probability of the i th bit being in error. m 3. Arrange the set { Q i } in descending order of magnitude and assign integer values k to them such that k = for the bit with the largest value of m Q i, k =2, for the next and so on until k = n, for the bit with the smallest value of Q.. Let {q[} be the rearranged set of numbers in step (3) above. Compute s[ as q[ s[ = q[ and form the set {s[}.. The first three entries of the Reliability Level List (RLL) are, s[], and s[2] respectively in that order. The corresponding candidate words to be examined are {b i }, {b i } with b i[] complemented, and {b i } with b i[2] complemented respectively. Examine each in the same order; if any of them is a codeword, it is the target codeword. This completes decoding. Note that the indices i[] and i[2] here stand for the bit indices corresponding to k = and k =2.. If none of the above is the target codeword, the next entry in the RLL is to be decided; it is s[3] if s[3]>s[]s[2] and the candidate word is {b i } with b i[3] complemented; else the RLL entry is s[]s[2] m i and the candidate word is {b i } with b i[] and b i[2] complemented. Examine the candidate word; if it is a codeword, it is the target codeword and decoding is complete.. If decoding is not completed in step (), the next entry in RLL is to be identified; If the previous entry was s[]s[2], this one is s[3] and the candidate word is {b i } with b i[3] complemented; else it is the larger of s[] and s[]s[2]. Correspondingly the candidate word is {b i } with b i[] complemented or {b i } with b i[] and b i[2] complemented as the case may be. Examine the candidate word; if it is a codeword, it is the target codeword and decoding is complete. Proceed as above always deciding the next entry in RLL, forming the candidate word by complementing the selected bits and checking whether it is a codeword the target codeword the one with the maximum value of f the metric. Candidate code words Table 3 Chase I Decoding results for Example - I Hamming distance from 8cch f Analog metric value ac8h 2 0.002 2.8938 8cfh 3 0.030033.82 d8ch 3 0.000 3.3 30eh 0.0339.00 cdh 0.00390 2.3800 09ch 0.008 2.29839 9cch 0.000093.2329 A few illustrative examples are considered here to bring out the effectiveness of the algorithm. Example - II This is a hypothetical case of a (, ) code with 30eh as the transmitted code word and 3eh at a Hamming distance of 3 from it as the received word; the errors are in b, b, and b 8. The bit indices and the corresponding reliability indices assigned to each of them are given in Table. E-ISSN: 222-28 29 Issue, Volume 2, June 203

Table Reliability magnitude and index for Example II Bit index (i) Reliability index (k) 3 2 3 2 0 9 0 8 9 8 3 2 3 2 0 Chase I decodes this properly but Chase II fails. Chase III uses the three T vectors- the received word, the received word with b i[] and b i[2] complemented, and the received word with b i[], b i[2], b i[3], and b i[] complemented. The Hamming spheres around the corresponding distorted words 3eh, 3dh, and 39h are scanned to identify candidate codewords; the word 39h is identified as the only candidate codeword and erroneously returned as the target codeword. The s values for the first few least reliable bits are reproduced in Table ; selected and relevant segments of the RLL are shown in Table. Applying the proposed algorithm, the word 30eh - is the first candidate codeword identified and it is returned as the target codeword. Incidentally the codeword 39h with f = 0.883 is not the target codeword, since its reliability is less than that of 30eh for which f = 0.9023. S.No S.No Table s values for Example - II Bit index i k s [ 0 0.9 2 2 0.90 3 2 3 0.99 0 0.98 0.9 0.9 8 0.9 Table Partial RLL for Example - II Bits in error 0 0.9 2 2 0.90 3 2 3 0.99............ 0,,8,, 0.903290 2 2,,8 3,, 0.903292 3,,8,, 0.9023 Example - III A (3, ) binary BCH code with d min = and t = d min = 3 is taken. The message ab30h has been encoded as 98ad2h and transmitted over an AWGN channel with σ = 0.8. Details of an erroneously received word are reproduced in Table. k f E-ISSN: 222-28 29 Issue, Volume 2, June 203

Table Reliability magnitude, index and s values for Example - III S.No Bit index i m i k s [ 9 0.38 0.280 2 0.332 2 0.2938 3 0.3220 3 0.822 2 0.3893 0.98 8 0.3880 0.83 0.02893 0.3 0.080 0.382 8 22 0.339 8 0.399 9 2 0.8 9 0.3902 0 0.80 0 0.388 3 0.93 0.3 2 2 0.02 2 0.328 3 20 0.929 3 0.3898 9 0.32 0.2020 2 0.8980 0.209.09982 0.0923 28.890 0.08938 8.099 8 0.08333 9.39908 9 0.083 20 2.303 20 0.03830 2.2 2 0.0399 22 3.0 22 0.02 23.932 23 0.0233 2 8.3232 2 0.0209 2 2.0 2 0.092 2 23.0 2 0.0092 2 30. 2 0.0382 28 2.8080 28 0.0203 29 0.889 29 0.009323 30 0.998 30 0.003 3 29 2.802 3 0.0003 S.No Table 8 Partial RLL for Example -III Bits in error 9 0.280 2 2 0.2938 3 3 0.822............ 333 9,,,2, 0.020 33 9,,28,2, 0.0229 33,,,20 3,,0,3 0.0232898 Selected and relevant segments of the RLL are shown in Table 8. Applying the proposed algorithm, the word 98ad2h with f = 0.0232898 is the first candidate codeword identified and it is returned as the target codeword. This is returned after 33 entries in the RLL have been checked. Example - IV Similar results were obtained with (2, ) code, once again bringing out the efficacy of the proposed method. A (2, ) binary BCH code with d min = 2 and t = d min = 0 is taken. The message 00320023h has been encoded as {32a90,8009a2b,8e22e3,bae9ch} and transmitted over an AWGN channel with σ = 0.. k f E-ISSN: 222-28 29 Issue, Volume 2, June 203

Details of the bits in an erroneously received word are reproduced in Table 9. The received word Z = {22a90,a0380e2b,8e222,f2a9ch}; the error pattern e = {00000,20300,80,0800h}; K values in error:,2,3,2,8,2,,,3,,30,,2. Table 9 Reliability magnitude, index and s values for Example - IV S.No Bit index i m i K s [ proposed algorithm. Plots of Block Error Rate with conventional hard decision decoding, RLL decoding and Chase-2 decoding for about 000 transmissions for (,) and (3,) BCH codes are given in Figure 2 and Figure 3 respectively. For RLL the reliability index k is used as a threshold for error correction. In Figure 2 and Figure 3, a threshold value of and ( = d min -) are used. However the performance of RLL can be improved by increasing the threshold (the error correcting capability). The same is shown in Figure as plots of BLER versus threshold values (>) for (,) code for some representative SNR values. SNR in db 0 2 3 8 0-0. 0 0.8 0.2 2 0.2 2 0.833 BLER - -. -2-2. HDD RLL Chase-2 3 22 0.2 3 0.2932 32 0.8 2 0.02 3 0.2 8 0.023 0.2 2 0.383 0. 0.23 8 80 0. 0.0989-3 -3. Figure 2. Comparision of Block Error Rate between RLL, HDD, Chase-2 for (,) BCH code The numbers in the y axis represent the respective log value. SNR in db 0 2 3 8 0 9 8 0. 3 0.0989 0 8 0. 0.992 93 0.3 30 0.2000 2 0.08 0.8338 BLER -0. - -. -2-2. -3-3. HDD RLL Chase-2 3 20 0.2 2 0.32 Chase II and Chase III algorithms fail to decode this received word and the proposed Direct Target Codeword Identification Algorithm identifies the target codeword with the f value as 0.0003999 which is the transmitted codeword. Simulation Results Extensive simulations were carried out with (, ), (3, ), and (2, ) BCH codes using the Figure 3. Comparision of Block Error Rate between RLL, HDD, Chase-2 for (3,) BCH code The numbers in the y axis represent the respective log value. min/ 2 The implementation of Chase-2 calls for 2 d [] number of algebraic decodings for all values of min/ 2 SNR. With 2 d algebraic decodings, the complexity of Chase-2 decoding is made orders higher by the Berlekamp Massey algorithm having multiplicative complexity O(t 2 ), and Chien search E-ISSN: 222-28 29 Issue, Volume 2, June 203

requiring O(tn) multiplications [2]. BLER 0-0. - -. -2-2. Threshold 9 3 SNR = 0 db SNR =. db SNR = 2.0 db SNR =. db Average Decoding Complexity 200 000 800 00 00 200 SNR = 0 db SNR =. db SNR = 2.0 db SNR =. db -3-3. 0 9 3 Threshold Figure. Plot of Block Error Rate versus Threshold value (>) for (,) code With RLL, the algebraic decoding is simplified to division by the generator polynomial. Since the evaluation of RLL entry differs with the noise level the RLL complexity varies with SNR. With this as the basis for complexity measure, the plot of average decoding complexity for (,) code for a representative threshold value of is given in Figure. It can be noted that complexity here is substantially lower than that of implementation of algebraic decoding using say Berlekamp algorithm and Chien search [2]. A similar study has been carried out for different threshold values with other codes as well. The study shows that with increase in the threshold value for performance improvement, the complexity increases significantly at very low noise levels. Plots of average decoding complexity versus threshold (>) for the (,) code at representative SNR values are shown in Figure. Average Decoding Complexity 2 0 8 2 0 0 2 8 SNR in db RLL Complexity Figure. Complexity curve of (,) BCH code for a threshold value of Figure. Average decoding complexity versus threshold (>) for (,) BCH code. Conclusion A new algorithm for soft decoding of binary codes is evolved in the paper. It uses soft information and achieves extended error correcting radius. The superiority of the algorithm lies in the number of errors and patterns of errors corrected even under low SNR conditions. The concept of the structured RLL that has been introduced in the paper is central to the proposed soft decoding algorithm; it enables identification of the target codeword with minimal search. The need for identifying a number of candidate codewords and selecting the most reliable one from amongst them as with the class of Chase algorithms and their modified versions is obviated. Detailed simulations carried out bring out the effectiveness of the approach. Significantly the algorithm yields the best that is possible with SDD based approaches. The simulation results have been summarized in the paper and a few representative cases presented to illustrate the superiority of the approach. All soft decision algorithms are equally effective at high SNR conditions and the decisive factor in the preference of one or another is the effectiveness under low SNR conditions. Communication schemes of recent interest which target low SNR channels can benefit from algorithms of the type presented here. Non-binary BCH codes especially Reed Solomon Codes are in wide use either in stand alone or in concatenated schemes in various applications. After Guruswami and Sudan s novel algorithm [3] of list decoding of Reed Solomon codes many E-ISSN: 222-28 298 Issue, Volume 2, June 203

significant developments [] have come up. The seminal work of Kotter [] has spurred interest in soft decision decoding strategies for decoding Reed Solomon Codes beyond the conventional decoding radius. In this scenario the extension of the concept of RLL and its use for soft decoding of non-binary codes can be of real potential. References: [] G. D. Forney, Jr., Generalized Minimum Distance Decoding, IEEE Transactions on Information Theory, Vol. 2, 9, pp. 2-3. [2] David Chase, A Class of Algorithms for Decoding Block Codes With Channel Measurement Information, IEEE Transactions on Information Theory, Vol. 8, 92, pp.0-82. [3] Marc P. C. Fossorier, and Shu Lin, Soft- Decision Decoding of Linear Block Codes based on Ordered Statistics, IEEE Transactions on Information Theory, Vol., 99, pp. 39-9. [] Jos.H.Weber, Low Complexity Chase-Like Bounded-Distance Decoding Algorithms GLOBECOM, 2003, pp.08-2. [] Idriss Chana, Hamid Allouch, and Mostafa Belkasmi, An Efficient New Soft Decision Decoding Algorithm for Binary Cyclic Codes, International Conference on Multimedia Computing and Systems, 20. [] B.Yamuna, T.R.Padmanabhan, A Reliability Level List based SDD algorithm for binary Cyclic Block codes, International Journal of Computers, Communication and Control, Vol., No.2, 202, pp. 388-39. [] Eran Fishler, Ofer Amrani, and Yair Be ery, Geometrical and Performance Analysis of GMD and Chase Decoding Algorithms, IEEE Transactions on Information Theory, Vol., 999, pp.0-22. [8] Jos H. Weber, Marc P.C. Fossorier, Limited- Trial Chase-Like Algorithms achieving Bounded-Distance Decoding, IEEE Transactions on Information Theory, Vol. 0, 200, pp. 338-3323. [9] Marc P. C. Fossorier, and Shu Lin, Chase-Type and GMD Coset Decodings, IEEE Transactions on Commuications, Vol. 8, 2000, pp.3-30. [0] WenyiJin and Marc.P.C.Fossorier, Reliability-Based Soft-Decision Decoding with Multiple Biase, IEEE Transactions on Information Theory, Vol. 3, 200, pp. 0-20. [] Toshimitsu Kaneko, Toshihisa Nishijima,Hiroshige Inazumi and Shigeichi Hirasawa, An efficient Maximum- likelihood- Decoding algorithm for linear Block codes with algebraic decoder, IEEE Transactions on Information Theory, Vol. 0,No.2, 99, pp. 320-32. [2] Schipani D, Elia M, Rosenthal J, On the decoding complexity of cyclic codes up to BCH bound, IEEE International Symposium on Information Theory Proceedings (ISIT), 20. pp. 83-839. [3] Robert J. McEliece, The Guruswami-Sudan decoding algorithm for Reed-Solomon codes, Interplanetary Network Progress Report (NASA), 2003, pp. 2 3. [] Jing Jiang, Krishna R. Narayanan, Algebraic Soft-Decision Decoding of Reed Solomon Codes Using Bit-Level Soft Information, IEEE Transactions on Information Theory, Vol., pp, 2008, 390-3928. [] Ralf Koetter, Alexander Vardy, Algebraic Soft-Decision Decoding of Reed Solomon Codes, IEEE Transactions on Information Theory, Vol. 9, 2003, pp. 2809-282. E-ISSN: 222-28 299 Issue, Volume 2, June 203