IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 3, MARCH 2000 345 Transactions Letters________________________________________________________________ Chase-Type and GMD Coset Decodings Marc P. C. Fossorier, Member, IEEE, and Shu Lin, Fellow, IEEE Abstract—In this letter, Chase decoding algorithms are gener- alized into a family of bounded distance decoding algorithms, so that the conventional Chase algorithm-2 and Chase algorithm-3 become the two extremes of this family. Consequently, more flex- ibility in the tradeoffs between error performance and decoding complexity is provided by this generalization, especially for codes with large minimum distance. Finally, this approach is extended to decoding with erasures. Index Terms—Block code, Chase decoding reliability-based de- coding, GMD decoding, soft-decision decoding. I. INTRODUCTION R ECENTLY, Chase algorithm-2 [1] has received renewed interests due to its use in several iterative soft-decision de- coding algorithms for binary linear block codes [2]–[5]. This algorithm performs algebraic decodings of a fixed number of sequences obtained by adding to the bit-by-bit hard-decision received sequence all the test patterns belonging to a predeter- mined test set. For each algebraically decoded candidate code- word, the corresponding correlation metric with the received sequence is evaluated, and the candidate codeword with the largest correlation metric becomes the decoding solution. We refer to this class of decoding algorithms as Chase-type algo- rithms. Three different algorithms of this class are presented in [1]. For an code of length and minimum Ham- ming distance the numbers of test patterns considered in [1] are for algorithm-1, for algorithm-2, and for algorithm-3. Due to the large number of error patterns to be tested, algorithm-1 is of little interest in practice. Also, the exponential decoding complexity associated with al- gorithm-2 prevents the use of this algorithm for decoding pow- erful codes with large Methods for reducing the size of the set of test patterns for Chase algorithm-2 have been proposed in [3], [4], and [6], but in the worst case, only a reduction by a factor of 1/2 is achieved. Finally, algorithm-3 seems interesting for powerful codes since its complexity increases linearly with Paper approved by S. B. Wicker, the Editor for Coding Theory and Tech- niques of the IEEE Communications Society. Manuscript received September 15, 1998; revised April 15, 1999. This work was supported by the National Sci- ence Foundation under Grant NCR-94-15374 and Grant CCR-97-32959, and NASA Grant NAG 5-931. This paper was presented in part at the 1998 Interna- tional Symposium on Information Theory and Its Applications (ISITA), Mexico City, Mexico, October 1998. The authors are with the Department of Electrical Engineering, University of Hawaii, Honolulu, HI 96822 USA. Publisher Item Identifier S 0090-6778(00)02269-8. Unfortunately, at practical bit-error rates (BER’s), the im- provement provided by Chase algorithm-3 upon algebraic de- coding decreases significantly with as suggested from the results of [7]. In this letter, a new generalization of Chase-type decoding is proposed. For an code, a family of bounded-distance decoding algorithms is introduced. For algorithm of this family processes test patterns, so that the extremes and simply correspond to algorithm-2 and algorithm-3 of [1] with minimum test pattern set, respectively. For powerful codes, this family of decoding algorithms allows to trade off error performance and decoding complexity. Also, it presents interesting options to decode long powerful codes with the iterative algorithms of [2]–[5], for which Chase algo- rithm-2 requires too many test patterns and Chase algorithm-3 performs poorly. Finally, the extension of this approach to erasure decoding as introduced in [8] is discussed. It is shown that each Chase-type decoding algorithm of the proposed family can be extended to erasure decoding and still achieve bounded-distance decoding. These two families of bounded distance decoding algorithms can be viewed as the combination of coset decoding with Chase algorithm-3 and GMD [8], respectively. II. GENERALIZED CHASE DECODING ALGORITHMS A. Algorithm Description Assume binary phase-shift keying (BPSK) transmission nor- malized to 1 over the additive white Gaussian noise (AWGN) channel. The codeword with is mapped into the BPSK sequence with At the receiver, the noisy received sequence is first decoded into the bit-by-bit hard-de- cision sequence For simplicity and without loss of generality, assume that these sequences are indexed in decreasing values of reliability, such that for A Chase-type decoder evaluates the correlation metrics associ- ated with the codewords delivered by an algebraic decoder cor- responding to the inputs where the vector is referred to as a test pattern in a test set. It is well known that if the minimum Hamming distance of the code considered is even, then all error patterns of Hamming weight with a given po- sition are also correctable [9]. Although any position can be chosen, error patterns of Hamming weight with 0090–6778/00$10.00 © 2000 IEEE