Quadratic Residue Code

The algebraic decoding of binary quadratic residue codes can be performed using the Peterson or the Berlekamp-Massey algorithm once certain unknown syndromes are determined or eliminated. The technique of determining unknown syndromes is applied to the nonbinary case to decode the expurgated ternary quadratic residue code of length 23

On formulas for decoding binary cyclic codes

We adress the problem of the algebraic decoding of any cyclic code up to the true minimum distance. For this, we use the classical formulation of the problem, which is to find the error locator polynomial in terms of the syndroms of the received word. This is usually done with the Berlekamp-Massey algorithm in the case of BCH codes and related codes, but for the general case, there is no generic algorithm to decode cyclic codes. Even in the case of the quadratic residue codes, which are good codes with a very strong algebraic structure, there is no available general decoding algorithm. For this particular case of quadratic residue codes, several authors have worked out, by hand, formulas for the coefficients of the locator polynomial in terms of the syndroms, using the Newton identities. This work has to be done for each particular quadratic residue code, and is more and more difficult as the length is growing. Furthermore, it is error-prone. We propose to automate these computations, using elimination theory and Grbner bases. We prove that, by computing appropriate Grbner bases, one automatically recovers formulas for the coefficients of the locator polynomial, in terms of the syndroms

On the Decoding Complexity of Cyclic Codes Up to the BCH Bound

The standard algebraic decoding algorithm of cyclic codes $[n,k,d]$ up to the BCH bound $t$ is very efficient and practical for relatively small $n$ while it becomes unpractical for large $n$ as its computational complexity is $O(nt)$. Aim of this paper is to show how to make this algebraic decoding computationally more efficient: in the case of binary codes, for example, the complexity of the syndrome computation drops from $O(nt)$ to $O(t\sqrt n)$, and that of the error location from $O(nt)$ to at most $\max \{O(t\sqrt n), O(t^2\log(t)\log(n))\}$.Comment: accepted for publication in Proceedings ISIT 2011. IEEE copyrigh

An efficient combination between Berlekamp-Massey and Hartmann Rudolph algorithms to decode BCH codes

In digital communication and storage systems, the exchange of data is achieved using a communication channel which is not completely reliable. Therefore, detection and correction of possible errors are required by adding redundant bits to information data. Several algebraic and heuristic decoders were designed to detect and correct errors. The Hartmann Rudolph (HR) algorithm enables to decode a sequence symbol by symbol. The HR algorithm has a high complexity, that's why we suggest using it partially with the algebraic hard decision decoder Berlekamp-Massey (BM). In this work, we propose a concatenation of Partial Hartmann Rudolph (PHR) algorithm and Berlekamp-Massey decoder to decode BCH (Bose-Chaudhuri-Hocquenghem) codes. Very satisfying results are obtained. For example, we have used only 0.54% of the dual space size for the BCH code (63,39,9) while maintaining very good decoding quality. To judge our results, we compare them with other decoders

Decoding Generalized Reed-Solomon Codes and Its Application to RLCE Encryption Schemes

This paper compares the efficiency of various algorithms for implementing quantum resistant public key encryption scheme RLCE on 64-bit CPUs. By optimizing various algorithms for polynomial and matrix operations over finite fields, we obtained several interesting (or even surprising) results. For example, it is well known (e.g., Moenck 1976 \cite{moenck1976practical}) that Karatsuba's algorithm outperforms classical polynomial multiplication algorithm from the degree 15 and above (practically, Karatsuba's algorithm only outperforms classical polynomial multiplication algorithm from the degree 35 and above ). Our experiments show that 64-bit optimized Karatsuba's algorithm will only outperform 64-bit optimized classical polynomial multiplication algorithm for polynomials of degree 115 and above over finite field $GF(2^{10})$. The second interesting (surprising) result shows that 64-bit optimized Chien's search algorithm ourperforms all other 64-bit optimized polynomial root finding algorithms such as BTA and FFT for polynomials of all degrees over finite field $GF(2^{10})$. The third interesting (surprising) result shows that 64-bit optimized Strassen matrix multiplication algorithm only outperforms 64-bit optimized classical matrix multiplication algorithm for matrices of dimension 750 and above over finite field $GF(2^{10})$. It should be noted that existing literatures and practices recommend Strassen matrix multiplication algorithm for matrices of dimension 40 and above. All our experiments are done on a 64-bit MacBook Pro with i7 CPU and single thread C codes. It should be noted that the reported results should be appliable to 64 or larger bits CPU architectures. For 32 or smaller bits CPUs, these results may not be applicable. The source code and library for the algorithms covered in this paper are available at http://quantumca.org/

Efficient Decoding of Gabidulin Codes over Galois Rings

This paper presents the first decoding algorithm for Gabidulin codes over Galois rings with provable quadratic complexity. The new method consists of two steps: (1) solving a syndrome-based key equation to obtain the annihilator polynomial of the error and therefore the column space of the error, (2) solving a key equation based on the received word in order to reconstruct the error vector. This two-step approach became necessary since standard solutions as the Euclidean algorithm do not properly work over rings

Complexity Analysis of Reed-Solomon Decoding over GF(2^m) Without Using Syndromes

For the majority of the applications of Reed-Solomon (RS) codes, hard decision decoding is based on syndromes. Recently, there has been renewed interest in decoding RS codes without using syndromes. In this paper, we investigate the complexity of syndromeless decoding for RS codes, and compare it to that of syndrome-based decoding. Aiming to provide guidelines to practical applications, our complexity analysis differs in several aspects from existing asymptotic complexity analysis, which is typically based on multiplicative fast Fourier transform (FFT) techniques and is usually in big O notation. First, we focus on RS codes over characteristic-2 fields, over which some multiplicative FFT techniques are not applicable. Secondly, due to moderate block lengths of RS codes in practice, our analysis is complete since all terms in the complexities are accounted for. Finally, in addition to fast implementation using additive FFT techniques, we also consider direct implementation, which is still relevant for RS codes with moderate lengths. Comparing the complexities of both syndromeless and syndrome-based decoding algorithms based on direct and fast implementations, we show that syndromeless decoding algorithms have higher complexities than syndrome-based ones for high rate RS codes regardless of the implementation. Both errors-only and errors-and-erasures decoding are considered in this paper. We also derive tighter bounds on the complexities of fast polynomial multiplications based on Cantor's approach and the fast extended Euclidean algorithm.Comment: 11 pages, submitted to EURASIP Journal on Wireless Communications and Networkin