Low-complexity Chase Research Articles

High-Throughput Interpolator Architecture for Low-Complexity Chase Decoding of RS Codes

In this paper, a high-throughput interpolator architecture for soft-decision decoding of Reed-Solomon (RS) codes based on low-complexity chase (LCC) decoding is presented. We have formulated a modified form of the Nielson's interpolation algorithm, using some typical features of LCC decoding. The proposed algorithm works with a different scheduling, takes care of the limited growth of the polynomials, and shares the common interpolation points, for reducing the latency of interpolation. Based on the proposed modified Nielson's algorithm we have derived a low-latency architecture to reduce the overall latency of the whole LCC decoder. An efficiency of at least 39%, in terms of area-delay product, has been achieved by an LCC decoder, by using the proposed interpolator architecture, over the best of the previously reported architectures for an RS(255,239) code with eight test vectors. We have implemented the proposed interpolator in a Virtex-II FPGA device, which provides 914 Mb/s of throughput using 806 slices.

High-Speed RS(255, 239) Decoder Based on LCC Decoding

Algebraic Soft-Decision Decoding (ASD) of Reed–Solomon (RS) codes provides higher coding gain over the conventional hard-decision decoding (HDD), but involves high computational complexity. Among the existing ASD methods, the Low Complexity Chase (LCC) decoding is the one with the lowest implementation cost. LCC decoding is based on generating 2 η test vectors, where η symbols are selected as the least reliable symbols for which hard-decision or the second more reliable decision are employed. Previous decoding algorithms for LCC decoders are based on interpolation and re-encoding techniques. On the other hand, HDD algorithms such as the Berlekamp–Massey (BM) algorithm or the Euclidean algorithm, despite of their low computational complexity, are not considered suitable for LCC decoding. In this paper, we present a new approach to LCC decoding based on one of these HDD algorithms, the inversion-less Berlekamp–Massey (iBM) algorithm, where the test vectors are selected for correction during decoding on occurrence of hard-decision decoding failure. The proposed architecture when applied to a RS(255, 239) code with η=3, saves a 20.5% and 2% of area compared to the LCC with factorization and a factorization-free decoder, respectively. In both cases, the latency is reduced by 34.5%, which is an increase of throughput rate in the same percentage since the critical path is the same in all the competing architectures. So an efficiency of at least 56% in terms of area-delay product can be obtained, compared with previous works. A complete RS(255, 239) LCC decoder with η=3 has been coded in VHDL and synthesized for implementation in Vitex-5 FPGA device, and by using SAED 90 nm standard cell library as well, and find a decoding rate of 710 Mbps and 4.2 Gbps and area of 2527 slices and 0.36 mm2, respectively.

Algebraic Soft-Decision Decoder Architectures for Long Reed–Solomon Codes

Algebraic soft-decision (ASD) decoders of Reed-Solomon (RS) codes can achieve significant coding gain with polynomial complexity. Most prior work on ASD decoder architecture design is for relatively short RS codes. However, one major application of RS codes, magnetic recording, usually requires a code length of 4Kbits or longer. For long RS codes, the low-complexity Chase (LCC) ASD decoding needs to interpolate over a large number of test vectors, which leads to long latency. This brief proposes a unified backward-forward interpolation scheme and a corresponding architecture for the LCC decoding. The proposed architecture can achieve almost twice the speed with only 40% area overhead. Another contribution of this brief is that the hardware complexity analysis for different ASD decoders is provided for the first time. For a (458, 410) RS code over GF(210), the proposed LCC decoder can achieve much higher efficiency in terms of speed-over-area ratio than other ASD decoders with similar error-correcting performance.

Modified Low-Complexity Chase Soft-Decision Decoder of Reed–Solomon Codes

Reed---Solomon (RS) codes are widely used as error-correcting codes in digital communication and storage systems. Algebraic soft-decision decoding (ASD) of RS codes can achieve substantial coding gain with polynomial complexity. Among practical ASD algorithms, the low-complexity chase (LCC) algorithm that tests 2 ? vectors can achieve similar or higher coding gain with lower complexity. For applications such as magnetic recording, the performance of the LCC decoding is degraded by the inter-symbol interference from the channel. Improving the performance of the LCC decoding requires larger ?, which leads to higher complexity. In this paper, a modified LCC (MLCC) decoding is proposed by adding erasures to the test vectors. With the same ?, the proposed algorithm can achieve much better performance than the original LCC decoding. One major step of the LCC and MLCC decoding is the interpolation. To reduce the complexity of the interpolation, this paper also proposed a prioritized interpolation scheme to test a small proportion of the vectors at a time, starting with the ones with higher reliabilities. For a (458, 410) RS code, by testing 1/8 of the vectors at a time, the area requirement of the MLCC decoder with ??=?8 can be reduced to 57%, and the average decoding latency is reduced to 73%.

Backward Interpolation Architecture for Algebraic Soft-Decision Reed–Solomon Decoding

Recently developed algebraic soft-decision (ASD) decoding of Reed-Solomon (RS) codes have attracted much interest due to the fact that they can achieve significant coding gain with polynomial complexity. One major step of ASD decoding is the interpolation. Available interpolation algorithms can only add interpolation points or increase interpolation multiplicities. However, backward interpolation, which eliminates interpolation points or reduces interpolation multiplicities, is indispensable to enable the reusing of interpolation results in the following two scenarios: 1) interpolation needs to be carried out on multiple test vectors, which share common entries and 2) iterative ASD decoding where interpolation points have decreasing multiplicities. Examples for these cases are the low-complexity chase (LCC) decoding and bit-level generalized minimum distance (BGMD) decoding. With lower complexity, these algorithms can achieve similar or higher coding gain than other practical ASD algorithms. In this paper, we propose novel backward interpolation schemes and corresponding efficient implementation architectures for LCC and BGMD decoding through constructing equivalent GrOumlbner bases. The proposed architectures share computational units with forward interpolation architectures. Hence, the area overhead for incorporating the backward interpolation is very small. Substantial area saving or speedup can be achieved by using the backward interpolation. When the proposed architecture is applied to the LCC decoding of a (255, 239) RS code with eta = 3, the area is reduced to 39% of those required by prior architectures. In terms of speed/area ratio, the proposed architecture is 48% more efficient than the best available architecture. For the BGMD decoding of the same code, the proposed architecture can achieve around 20% higher efficiency.

Performance of a Chase-type decoding algorithm for Reed–Solomon codes on perpendicular magnetic recording channels

Algebraic soft-decision Reed–Solomon (RS) decoding algorithms with improved error-correcting capability and comparable complexity to standard algebraic hard-decision algorithms could be very attractive for possible implementation in the next generation of read channels. In this work, we investigate the performance of a low-complexity Chase (LCC)-type soft-decision RS decoding algorithm, recently proposed by Bellorado and Kavčić, on perpendicular magnetic recording channels for sector-long RS codes of practical interest. Previous results for additive white Gaussian noise channels have shown that for a moderately long high-rate code, the LCC algorithm can achieve a coding gain comparable to the Koetter–Vardy algorithm with much lower complexity. We present a set of numerical results that show that this algorithm provides small coding gains, on the order of a fraction of a dB, with similar complexity to the hard-decision algorithms currently used, and that larger coding gains can be obtained if we use more test patterns, which significantly increases its computational complexity.