Subcodes of Reed-Solomon Codes Suitable for Soft Decoding
Reed-Solomon (RS) codes over GF(2 m ) have traditionally been the most popular non-binary codes in almost all practical applications. The distance properties of RS codes result in excellent performance under hard-decision bounded-distance decoding. In thi
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Abstract. Reed-Solomon (RS) codes over GF(2m ) have traditionally been the most popular non-binary codes in almost all practical applications. The distance properties of RS codes result in excellent performance under hard-decision bounded-distance decoding. In this work, we consider certain subcodes of RS codes over GF(q m ) whose q-ary traces are BCH codes over GF(q). The properties of these subcodes are studied and low-complexity hard-decision and soft-decision decoders are proposed. The decoders are analyzed, and their performance is compared with that of comparable RS codes. Our results suggest that these subcodes of RS codes could have some advantages when compared to RS codes.
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Introduction
Reed-Solomon (RS) codes [1] are the most prevalent and commonly used codes today with applications ranging from satellite communications to computer drives. RS codes are popular, in theory, for their elegant algebraic construction. In practice, RS codes can be encoded and decoded with manageable complexity and high speed. RS codes continue to remain objects of active research with most recent interest being in list and soft-decision decoding [2][3]. Efficient soft decoding of RS codes has traditionally been a problem of importance. Early methods for soft decoding of RS codes included Chase decoding and Generalized Minimum Distance (GMD) decoding [4]. Other methods for soft decoding RS codes include [5][6]. Recently, the Koetter-Vardy algorithm [3] and belief-propagation-based iterative algorithm [7] have been proposed. Common themes in the above methods include (1) a coding gain of around 1dB, (2) an increase in complexity with size of the field, and (3) an increase in complexity for higher coding gain. As a result, efficient soft decoders are not readily available for high rate RS codes over large fields. In this work, we study certain subcodes of q m -ary RS codes that are more amenable to efficient decoding. Specifically, we consider subcodes whose traces are q-ary BCH codes. Suitable non-consecutive zeros are added to the set of zeros of a parent RS code to enable the trace to be a BCH code. Though the subcode is not typically maximum-distance-separable (MDS), our analysis shows that a large fraction of errors beyond minimum distance are correctable. Hence, S. Bozta¸s and H.F. Lu (Eds.): AAECC 2007, LNCS 4851, pp. 217–226, 2007. c Springer-Verlag Berlin Heidelberg 2007
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S.J. Raj and A. Thangaraj
the performance of these subcodes of RS codes is comparable to that of a MDS RS code at the same rate. We refer to these select subcodes of RS codes as sub Reed-Solomon (SRS) codes in the rest of this article. Because of the trace structure, the SRS codes are amenable to efficient softdecision decoding. Since the image of a q m -ary code is a concatenation of its q-ary trace, a soft decoder for the trace can be efficiently used to process soft input for the image. Using this idea, we propose simple soft decoders for SRS codes. Our simulations show that the proposed soft decoders for high-rate (> 0.9) SRS codes over large fields (G
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