Yan-Haw Chen

Algebraic decoding of (103, 52, 19) and (113, 57, 15) quadratic residue codes

In this paper, two algebraic decoders for the (103, 52, 19) and (113, 57, 15) quadratic residue codes, which have lengths greater than 100, are presented. The results have been verified by software simulation that programs in C++ language have been executed to check possible error patterns of both quadratic residue codes.

Algebraic Decoding of Quadratic Residue Codes Using Berlekamp-Massey Algorithm *

In this paper, an algebraic decoding method is proposed for the quadratic residue codes that utilize the Berlekamp-Massey algorithm. By a modification of the technique developed by He et al., one can express the unknown syndromes as functions of the known syndromes. The unknown syndromes are determined by an efficient algorithm also developed in this paper. With the appearance of unknown syndromes, one obtains the consecutive syndromes that are needed for the application of the Berlekamp-Massey algorithm. The decoding scheme, developed here, is easier to implement than the previous decoding algorithm developed for the Golay code and the (47, 24, 11) QR code. Moreover, it can be extended to decode all codes of the family of binary quadratic residue codes with irreducible generating polynomials.

The diversity study of AES on FPGA application

In the applications of AES, the long-term robustness/reliability during the period of operation should be taken into serious considerations. From such considerations, one may initiate the requirements of the design for diversity against break through from outside. In system design, the use of reconfigurable FPGA can provide higher level of flexibility. In this paper, the proposed system uses different generators, various transforms, modules and algorithms to enhance the randomization of the ciphertext. It is also a challenge to improve the system flexibility and to get a more secure design in the AES system. Several reconfigurable modules are developed on our integrated test-bench.

The design of RS decoder with a small core for portable communication

A new VLSI architecture using free discrepancy Berlekamp-Massey (FDBM) algorithm is proposed for wireless applications. Firstly, this project uses the FDBM algorithm to reduce the path delay. A method of module reuse is applied to reduce the overall core size successfully. Using single system clock, it is easy to integrate the core into SoC. As a result, this RS decoder has reduced core size and performs in low power with simple system integration.

Fast algorithm for decoding of systematic quadratic residue codes

A general algorithm for decoding the binary systematic quadratic residue (QR) codes with lookup tables is presented in this study. The algorithm can be applied in decoding the QR codes with either reducible or irreducible generator polynomials. If the generator polynomial of the QR codes is reducible, the number of elements in the Galois field is less than the sum of all correctable error patterns. In other words, the mapping between elements of syndrome set and all correctable error patterns is not one to one. The key idea of decoding based on the mapping between the ordered q-tuples of the primary known syndrome and error patterns is one to one. In addition, the algorithm directly determines the error locations by lookup tables without the operations of multiplication over a finite field. According to the simulation result, the new lookup table decoding algorithm for the (31, 16, 7) QR code and the (73, 37, 13) QR code dramatically reduces the memory required by approximately 90 and 92%, respectively. Moreover, the high speed of decoding procedure could be utilised in modern communication system.

Efficient Decoding of Systematic (41, 21, 9) Quadratic Residue Code

A new effective lookup table for decoding the binary systematic (41, 21, 9) quadratic residue (QR) code up to 4 errors is presented in this paper. The key ideas behind this decoding technique are based on one to one mapping between the syndromes ldquoS1rdquo and the error correctable patterns. Such an algorithm determines the error locations directly by lookup tables without the operations of multiplication over a finite field. Moreover, the methods to dramatically reduce the memory requirement are given. The new algorithm has been verified through a software simulation by C language. The new approach is modular, regular and naturally suitable for SOC software implementation.

Decoding of binary quadratic residue codes with hash table

An efficient decoding of quadratic residue (QR) codes utilising hashing search to find error patterns is presented in this study. The key idea behind the proposed decoding method is theoretically based on the existence of a one-to-one mapping between the single primary known syndrome and correctable error patterns. Compared with the binary search time approach, one of the advantages of utilising this method presented in this study is that the hashing search time can be reduced by a factor of two. This method would help reduce the binary search time for finding error patterns when decoding the (23, 12, 7), (41, 21, 9), and (47, 24, 11) QR codes. Furthermore, it would reduce the decoding time by 45% using the fast lookup table decoding method to decode the (71, 36, 11) QR code. Ultimately, the proposed decoding algorithm for QR codes can be made regular, simple, and suitable for software implementations.

A nearly constant delay and low power VLSI design of a pipeline Reed–Solomon encoder for storage and communication systems

Linear feedback shift register (LFSR) is a common method used in the design of Reed–Solomon (RS) encoders. Most previous LFSR-based articles have focused on the design of finite field multipliers and the throughput improvement of the RS encoders. However, these articles have some problems. First, the fan-out of the feedback register grows with respect to the increase of the error-correction capability, leading to longer encoding time. Second, in applications that require high-throughput, parallel implementations are usually adopted, which result in huge chip area. This article presents a new pipelined encoder architecture for systematic RS code, which has a nearly constant delay with less than 10% increase in cycle time for higher error-correction capability. The key technique behind this encoding technique is based on the modified polynomial of the Lagrange interpolation formula. It not only reduces the number of finite field multipliers, but also increases performance. Moreover, we propose a method of low computation complexity for the implementation of the constant multipliers in order to save area and power. Compared to the conventional LFSR architectures over the (204, 188, t = 8) RS code, the proposed encoder design reduces the critical path delay by 71% and the power consumption by 44%. Because of the high-throughput and low latency, the new design is readily adaptable for use in many applications such as RAID 6, CD, VCD, DVD, and high definition television, especially for storage and communication applications.

More on general error locator polynomials for a class of binary cyclic codes

Recently, the general error locator polynomials have been widely used in the algebraic decoding of binary cyclic codes. This paper utilizes the proposed general error locator polynomial to develop an algebraic decoding algorithm for a class of the binary cyclic codes. This general error locator polynomial differs greatly from the previous general error locator polynomial. Each coefficient of the proposed general error locator polynomial is expressed as a binary polynomial in the single syndrome and the degrees of nonzero terms in the binary polynomial satisfy at least one congruence relation.

Algebraic decoding of (79,40,15) quadratic residue code using inverse-free Berlekamp-Massey algorithm

An algebraic decoding method is proposed for the quadratic residue codes that utilize the Berlekamp-Massey (BM) algorithm. By applying a technique developed by R. He et al. (see IEEE Trans. Inf. Theory, vol.47, p.1181-6, 2001), one can express unknown syndromes as functions of known syndromes. An efficient algorithm is also developed to determine the unknown syndromes. With the appearance of unknown syndromes, one obtains the consecutive syndromes that are needed for the application of the inverse-free BM algorithm. The new decoding scheme can be used to implement the (79,40,15) quadratic residue (QR) code which has not been treated so far. It is verified by a computer program that uses the C++ language.