Berlekamp—Massey Algorithm, Continued Fractions, Pade Approximations, and Orthogonal Polynomials

In 1969, J. Massey published a now-famous paper showing, among other things, that an iterative algorithm introduced by Berlekamp for decoding BCH codes also solved the problem of finding a shortest-length feedback shift register circuit for generating a given finite sequence of digits. This nice physical interpretation opened the door to connections with many other problems, including the minimal partial realization problems of linear system theory, Pade approximations and continued fractions, the fast algorithms of Levinson and Schur for Toeplitz matrices, inverse scattering, VLSI implementations, etc. This paper is an informal account of some of the different contexts in which the Berlekamp-Massey algorithm have been encountered in the work of the author and his students.

This paper presents an efficient algorithm for encoding and decoding Shortened Reed-Solomon (SRS) codes based on evaluation. The encoding and syndrome computations are performed using polynomial evaluation and the decoding is performed using a recursive extension of the original algorithm. The proposed decoding algorithm initially calculates the error locator polynomial by using the Berlekamp Massey (BM) algorithm and with the recursive computation of discrepancies, it decodes directly the message polynomial. Compared to the traditional decoding algorithm using BM algorithm, Chien search and Forney formula, the proposed algorithm results in area savings for hardware implementations due to hardware re-use and the reduced number of stages. Also, the savings in memory for the software implementation makes it suitable for memory constrained applications such as Wireless Sensor Networks (WSN) applications. For the hardware implementations, the number of clock cycles required are lower for the proposed scheme, which results in time savings. Also, software implementation results of RS (32, 24) for resource constrained WSN application on ATMEL Atmega 128 microcontroller show that the proposed encoder and decoder is only about 52% and 72% of the memory footprint of a traditional encoder and decoder respectively.

An algorithm for shifted continued fraction expansions in parallel linear time

The Berlekamp-Massey algorithm [L. S. De Jong, Numerical aspects of realization algorithms in linearsystems theory, Ph.D. thesis, T. H. Eindhoven, the Netherlands, 1975] and [J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Information Theory, IT-15 (1969), pp. 122–127] for minimal realization problems is a special case of the Padé approximation problem. As a matter of fact, it computes among other polynomials the denominators of the elements of the Padé table that are on the descending diagonal $\{ [0/1],[1/2], \cdots ,[k/k + 1], \cdots \} $ as far as they exist and this algorithm works for nonnormal Padé tables too. This algorithm does not seem to be well-known in Padé approximation literature. It is not very difficult to generalize this algorithm so as to compute the other Padé approximants of a nonnormal table. Some variants will lead to a generalization of the algorithm of Brezinski [Computation of Padé approximants continued fractions, J. Comput. Appl. Math., 2 (1976), pp. 113–123], which computes the descending diagonals of a normal Padé table, and of the algorithm of Watson [D. Bussonnais, “Tous” les algorithmes de calcul par recurrence des approximants de Padé d’une serie, Construction de fractions continues correspondantes, Séminaire d’Analyse Numérique, No. 293, Grenoble,1978], [G. Claessens, new look at the Padé table and the different methods for computing its elements, J. Comput. Appl. Math., 1 (1975), pp. 141–152], [P. J. S. Watson, Algorithms for differentiation and integration, Padé Approximants and their Applications, P. R. Groves-Morris, ed., Academic Press London, 1973, pp. 93–98] that computes the descending staircases of normal tables. This work is the dual of Cordellier’s [algorithmes de calcul recursif des elements d’une table de Padé non normale, Conference on Padé approximation, Lille, France, 1978], and of McEliece and Shearer [A property of Euclid’s algorithm and an application to Padé approximation, SIAM J. Appl. Math., 34 (1978), pp. 611–616] deriving similar results for ascending diagonals and staircases. A more continued fraction-like approach would generalize the Thatcher algorithm and may be found in [J. A. Murphy and H. R. O’Donohoe, A class of algorithms for obtaining rational approximants to functions which are defined by power series, Z. Angew. Math. Phys., 28 (1977), pp. 1121–1131] and [A. Bultheel, Division algorithms for continued fractions and the Padé table, Applied Mathematics and Programming Division, Rep. TW41, Katholicke Universitiet Leuven, Heverlee, Belgium, August1978].

We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λt} of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λt}. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

Contiguous relations, basic hypergeometric functions, and orthogonal polynomials, I

Another viewpoint is presented on the derivation of the Berlekamp-Massey algorithm. Our approach differs from previous ones in the following manner. The properties of the shortest linear feedback shift register that generates a given sequence are first derived Without reference to the Berlekamp-Massey algorithm. The Berlekamp-Massey algorithm is then derived using these properties. Our approach has the advantage of being easier to understand.

Orthogonal Laurent polynomials and strong moment theory: a survey

As the technology moves into the nano-realm, traditional single-error-correcting, double-error-detecting (SEC-DED) codes are no longer sufficient for protecting memories against transient errors due to the increased multi-bit error rate. The well known double-error-correcting BCH codes and the classical decoding method for BCH codes based on Berlekamp-Massey algorithm and Chien search cannot be directly adopted to replace SEC-DED codes because of their much larger decoding latency. In this paper, we propose the hierarchical double-error-correcting (HDEC) code. The construction methods and the decoder architecture for the codes are described. The presented error correcting algorithm takes only 1 clock cycle to finish if no error or a single-bit error occurs. When there are multi-bit errors, the decoding latency is O(log2m) clock cycles for codes defined over GF(2m). This is much smaller than the latency for decoding BCH codes using Berlekamp Massey algorithm and Chien search, which is O(k) clock cycles -- k is the number of information bits for the code and m ~ O(log2k). Synthesis results show that the proposed (79, 64) HDEC code requires only 80% of the area and consumes < 70% of the power compared to the classical (78, 64) BCH code. For a large bit distortion rate (10−3 ~ 10−2), the average decoding latency for the (79, 64) HDEC code is only 36% ~ 60% of the latency for decoding the (78, 64) BCH code.

Rational functions over finite fields having continued fraction expansions with linear partial quotients

Algebraic feedback shift registers (AFSRs) [] are pseudo-random sequence generators that generalize linear feedback shift registers (LFSRs) and feedback with carry shift registers (FCSRs). With a general setting, AFSRs can result in sequences over an arbitrary finite field. It is well known that the sequences generated by LFSRs can be synthesized by either the Berlekamp-Massey algorithm or the extended Euclidean algorithm. There are three approaches to solving the synthesis problem for FCSRs, one based on the Euclidean algorithm [], one based on the theory of approximation lattices [] and Xu’s algorithm which is also used for some AFSRs []. Xu’s algorithm, an analog of the Berlekamp-Massey algorithm, was proposed by Xu and Klapper to solve the AFSR synthesis problem. In this paper we describe an approximation algorithm that solves the AFSR synthesis problem based on low-dimensional lattice basis reduction []. It works for AFSRs over quadratic integer rings \(\mathbb {Z}[\sqrt{D}]\) with quadratic time complexity. Given the first \(2\varphi _\pi (\mathbf {a})+c\) elements of a sequence \(\mathbf {a}\), it finds the smallest AFSR that generates \(\mathbf {a}\), where \(\varphi _\pi (\mathbf {a})\) is the \(\pi \)-adic complexity of \(\mathbf {a}\) and \(c\) is a constant.

On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials

PC-fractions and orthogonal Laurent polynomials for log-normal distributions

Sharp Estimates for Jacobi Matrices and Chain Sequences

Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. II. Associated big q-Jacobi polynomials