T.Y. Hwang ,"Decoding Linear Block Codes for Minimizing Word Error Rate," IEEE Transactions on Information Theory, Vol.25, No.6, pp. 733-737, November 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Ontario (CITO) This work is a continuation of [1] words of the dual code in the decoding process. This method results in a lower complexity as compared to an exhaustive search if the dual code has a smaller number of code words. Two modifications of the basic exhaustive method is presented in [3]. It gives a set of necessary and sufficient conditions for achieving minimum symbol error probability decoding and uses these conditions to derive a non exhaustive optimum decoding algorithm of a reduced complexity. Bit by bit soft decision decoding of binary cyclic codes is considered in [4] ....

T.Y. Hwang ,"Decoding Linear Block Codes for Minimizing Word Error Rate," IEEE Transactions on Information Theory, Vol.25, No.6, pp. 733-737, November 1979.

....which is still exhaustive, but uses the set of code words of the dual code in the decoding process. This method results in a lower complexity as compared to an exhaustive search if the dual code has a smaller number of code words. Two modifi cations of the basic exhaustive method are presented in [2]. It gives a set of necessary and suiicient conditions for achieving minimum symbol error probability decoding and uses these conditions to derive a non exhaustive optimum decoding algorithm of a reduced complexity. Bit by bit soft decision decoding of binary cyclic codes is considered in [3] ....

T.Y. Hwang ,"Decoding Linear Block Codes for Minimizing Word Error Rate," IEEE Transactions on Information Theorg, Vol.25, No.6, pp. 733-737, November 1979.

....if the dual code has a smaller 1 This work is financially supported by Natural Sciences and Engineering Research Council of Canada (NSERC) and by Communications and Information Technology Ontario (CITO) number of code words. Two modifications of the basic exhaus tive method are presented in [2]. It gives a set of necessary and sufficient conditions for achieving minimum symbol error probability decoding and uses these conditions to derive a non exhaustive optimum decoding algorithm of a reduced complexity. Bit by bit soft decision decoding of binary cyclic codes is considered in [3] ....

T.Y. Hwang ,"Decoding Linear Block Codes for Minimizing Word Error Rate," IEEE Transactions on Information Theory, Vol.25, No.6, pp. 733-737, November 1979.

....c = I whose leftmost nonzero coordinate is fixed (say, to 1) we can make a bijection between codewords and supports and, thus, also speak of minimal codewords. For binary codes, there is no difference between minimal codewords and minimal supports. Minimal supports in linear codes were studied in [2] in connection with a maximum likelihood decoding algorithm and recently in [3] 5] for the cryptographical problem of constructing secret sharing schemes. The set of minimal supports of a code C will be denoted by P(C) or simply P. Supported in part by the International Science Foundation ....

T.-Y. Hwang, "Decoding linear block codes for minimizing word error rate," IEEE Trans. Inf. Theory, IT-25, No. 6 (November 1979), 733--737.

....Besides being a valuable theoretical instrument, the Voronoi regions in a code can be employed in the decoding process itself. This was suggested by Landau [16] who also pointed out the significance of the number of facets of the Voronoi regions. An iterative algorithm was developed by Hwang [17] and investigated in more detail by Butovitsch [18, pts. D E] The idea, which has also been considered for other point sets and in other applications [19] 20] is basically as follows: i) Select a codeword c and compute its distance d to the input x . ii) Compute the distance d from a new ....

....rectangle. v) Two codewords forming the diagonal of a rectangle are nonneighbors. Two codewords that do not form any diagonal are neighbors. Again regarding codewords as strings of bits, the last of these statements can be translated into the following important rule, which was first given in [17], though not in Voronoi terminology. To show that N C 0 ( is equal to the projecting set of [17] compare Corollary 1 in [12] with Definition 2 in [17] C rule: A codeword is a 0 neighbor if and only if it covers 5 no other nonzero codeword. To establish whether a given codeword c C is a ....

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T.-Y. Hwang, "Decoding linear block codes for minimizing word error rate," IEEE Trans. Inform. Theory, vol. IT-25, no. 6, pp. 733--737, Nov. 1979.

....if its support does not contain the support of any other nonzero codeword as true subset. The support of a minimal codeword is called minimal with respect to C. The sets of minimal codewords of linear codes were considered in connection with constructing a decoding algorithm (Tai Yang Hwang [4]) For the Euclidean space, this connection was addressed also in [1] Later the sets of minimal codewords of linear codes were used in a series of papers sparked by [6] to describe minimal access structure in linear secret sharing schemes Ashikhmin and Barg [2] have determined the set of minimal ....

Tai-Yang Hwang, Decoding linear block codes for minimizing word error rate, IEEE Trans. on Information Theory, IT-25, 1979, 6, 733-737.

....under both models. Known decoding methods with complexity asymptotically less than that of exhaustive search can be roughly divided into three groups, namely, gradient like methods, syndrome decoding, and information set decoding. Methods in the first class include minimal vectors algorithm [22] and zero neighbors algorithm [26] The common feature of these methods is the use of a certain fixed (precomputed) set of codewords in order to successively improve the current decision. The underlying idea bears similarity with methods of steepest descent in continuous spaces. As for syndrome ....

....to decoding in the sphere of radius In the rare case that the algorithms find no codeword at all, we can take any codeword as the decoding result. Asymptotically this will not affect the decoding error rate. We wish to underline that these algorithms, in contrast to gradient like methods [22], 26] perform complete maximum likelihood decoding only in the limit as Their error probability as a function of approaches the error probability of complete maximumlikelihood decoding (the limit of their quotient is one) Recently, Dumer [16] 17] extended both results to the case of much more ....

T.-Y. Hwang, "Decoding linear block codes for minimizing word error rate," IEEE Trans. Inform. Theory, vol. IT-25, pp. 733--737, Nov. 1979.

....apply the strategy to the Golay code and the Leech lattice. III. Fast decoding algorithms for the Golay code The direct search algorithm (see (2.1) for the [24,12] Golay code 24 takes about 24 . 2 12 = 98304 steps, while the trellis decoding method of (2.5) is even slower. Hwang s algorithm [44], 45] for the shortened Golay code of length 23 takes roughly 24 . 1376 = 33024 steps. In this section we give two algorithms that are much faster. The first is based on the subcode (7) takes about 1584 steps, and is described in detail. The second is based on the subcode (4) takes about 1728 ....

T. Y. Hwang, Decoding linear block codes for minimizing word error rate, IEEE Trans. Information Theory, IT-25 (1979), 733-737.

....subsets fl 2 G Gamma . Thus, a natural question 1 is to study which access structures are defined by well known linear codes. Massey [6] and Blakley and Kabatianskii [7] suggest this question as a research problem. Interestingly, the set of minimal codewords has been studied already in [13] (for decoding purposes) There this set was called the projecting set of a code. For a code C, we denote its set of minimal codewords by P(C) or simply by P . We assume that 0 62 P and suggest a notation C Theta Delta = C Gamma f0g. There are many binary codes with P(C) C Theta . They ....

....codewords with a nonzero first coordinate define minimal authorized coalitions. In Sec. 4, we explore properties of minimal codewords and find their number in typical linear codes. The subsequent asymptotic analysis enables us to assess, from the complexity point of view, the decoding algorithm in [13]. We also find P(C) for C the Hamming code and secondorder Reed Muller code. In Sec. 5, we extend our study to the case of S a Galois ring, when the corresponding secret sharing schemes turn out to be nonperfect. It also turns out that, especially in connection with rings, it is more accurate to ....

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T.-Y. Hwang, "Decoding linear block codes for minimizing word error rate," IEEE Trans. Inf. Theory, IT-25, No. 6 (November 1979), 733--737.

....of codes, including the Hamming codes and 2nd order Reed Muller codes. Further, we extend the concept of minimal vectors to codes over rings and compute them for several examples. Turning to applications, we introduce a general gradient like decoding algorithm of which minimal vectors decoding [14] is an example. The complexity of minimal vectors decoding for long codes is determined by the size of the set of minimal vectors. Therefore we compute this size for long randomly chosen codes. Another example of algorithms in this class is given by zero neighbours decoding [15] We discuss ....

....2C 375, Murray Hill, NJ 07974. A substantial part of this research done while with Department of Mathematics and Computing Science, Technical University of Eindhoven, Eindhoven, The Netherlands. combinatorics (cycles in linear matroids) In the coding context, minimal vectors were introduced in [14] where they were used to construct a minimum distance decoding algorithm of linear codes (see Sect. 4) For the Euclidean space, this connection was again addressed in [1] Recently the interest in this subject has been renewed in a series of works sparked by [17] where it was observed that ....

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T.-Y. Hwang, "Decoding linear block codes for minimizing word error rate," IEEE Trans. Inf. Theory, IT-25, no. 6 (1979), 733--737.

....in discrete space. There have been numerous attempts to reduce the decoding complexity by embedding E n q into R n and applying classical gradient methods. However, in this case local optima hinder the successful decoding and any significant results still remain out of reach. Hwang [86], 87] was the first to introduce minimal words in binary codes in the decoding context. Moreover, he showed that minimal codewords can be used in the more general setting of soft decision decoding. This is related to an observation of Agrell [1] which states that if a binary code is considered ....

T.-Y. Hwang, "Decoding linear block codes for minimizing word error rate," IEEE Trans. Inform. Theory, IT-25 (6) (1979), 733-737.

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