This paper proposes a novel algorithm for finding error-locators of algebraic-geometric codes that can eliminate the division-calculations of finite fields from the Berlekamp-Massey-Sakata algorithm. This inverse-free algorithm provides full performance in correcting a certain class of errors, generic errors, which includes most errors, and can decode codes on algebraic curves without the determination of unknown syndromes. Moreover, we propose three different kinds of architectures that our algorithm can be applied to, and we represent the control operation of shift-registers and switches at each clock-timing with numerical simulations. We estimate the performance in comparison of the total running time and the numbers of multipliers and shift-registers in three architectures with those of the conventional ones for codes on algebraic curves.
We propose a new algorithm for decoding Reed-Solomon codes (up to half the minimum distance) and for computing inverses in $F[x]/m(x)$. The proposed algorithm is similar in spirit and structure to the Berlekamp-Massey algorithm, but it works naturally for general $m(x)$.
The two primary decoding algorithms for Reed-Solomon codes are the Berlekamp-Massey algorithm and the Sugiyama et al. adaptation of the Euclidean algorithm, both designed to solve a key equation. In this article an alternative version of the key equation and a new way to use the Euclidean algorithm to solve it are presented, which yield the Berlekamp-Massey algorithm. This results in a new, simpler, and compacter presentation of the Berlekamp-Massey algorithm.
We present a recursive minimal polynomial theorem for finite sequences over a commutative integral domain $D$. This theorem is relative to any element of $D$. The ingredients are: the arithmetic of Laurent polynomials over $D$, a recursive 'index function' and simple mathematical induction. Taking reciprocals gives a 'Berlekamp-Massey theorem' i.e. a recursive construction of the polynomials arising in the Berlekamp-Massey algorithm, relative to any element of $D$. The recursive theorem readily yields the iterative minimal polynomial algorithm due to the author and a transparent derivation of …
The paper introduces the simultaneous partial-inverse problem (SPI) for polynomials and develops its application to decoding interleaved Reed--Solomon codes beyond half the minimum distance. While closely related both to standard key equations and to well-known Padé approximation problems, the SPI problem stands out in several respects. First, the SPI problem has a unique solution (up to a scale factor), which satisfies a natural degree bound. Second, the SPI problem can be transformed (monomialized) into an equivalent SPI problem where all moduli are monomials. Third, the SPI problem can be s…
In this work we will present algorithms for decoding rank metric codes. First we will look at a new decoding algorithm for Gabidulin codes using the property of Dickson matrices corresponding to linearized polynomials. We will be using a Berlekamp-Massey-like algorithm in the process. We will show the difference between our and existing algorithms. Apart from being a new algorithm, it is also interesting that it can be modified to get a decoding algorithm for general twisted Gabidulin codes.
For Reed-Solomon codes, the key equation relates the syndrome polynomial---computed from the parity check matrix and the received vector---to two unknown polynomials, the locator and the evaluator. The roots of the locator polynomial identify the error positions. The evaluator polynomial, along with the derivative of the locator polynomial, gives the error values via the Forney formula. The Berlekamp-Massey algorithm efficiently computes the two unknown polynomials. This chapter shows how the key equation, the Berlekamp-Massey algorithm, the Forney formula, and another formula for error evalua…
In this paper, we characterize the decoding region of algebraic soft decoding (ASD) of Reed-Solomon (RS) codes over erasure channels and binary symmetric channel (BSC). Optimal multiplicity assignment strategies (MAS) are investigated and tight bounds are derived to show the ASD can significantly outperform conventional Berlekamp Massey (BM) decoding over these channels for a wide code rate range. The analysis technique can also be extended to other channel models, e.g., RS coded modulation over erasure channels.
In this paper we present an interpolation-based decoding algorithm to decode a family of maximum rank distance codes proposed recently by Trombetti and Zhou. We employ the properties of the Dickson matrix associated with a linearized polynomial with a given rank and the modified Berlekamp-Massey algorithm in decoding. When the rank of the error vector attains the unique decoding radius, the problem is converted to solving a quadratic polynomial, which ensures that the proposed decoding algorithm has polynomial-time complexity.
The problem of finding the shortest linear shift-register capable of generating t finite length sequences over some field F is considered. A similar problem was already addressed by Feng and Tzeng. They presented an iterative algorithm for solving this multi-sequence shift-register synthesis problem, which can be considered as generalization of the well known Berlekamp-Massey algorithm. The Feng-Tzeng algorithm works indeed, if all t sequences have the same length. This paper focuses on multi-sequence shift-register synthesis for generating sequences of varying length. It is exposed, that the …
We derive the Wu list-decoding algorithm for Generalised Reed-Solomon (GRS) codes by using Gröbner bases over modules and the Euclidean algorithm (EA) as the initial algorithm instead of the Berlekamp-Massey algorithm (BMA). We present a novel method for constructing the interpolation polynomial fast. We give a new application of the Wu list decoder by decoding irreducible binary Goppa codes up to the binary Johnson radius. Finally, we point out a connection between the governing equations of the Wu algorithm and the Guruswami-Sudan algorithm (GSA), immediately leading to equality in the decod…
Analysis of the Berlekamp-Massey-Sakata algorithm for decoding one-point codes leads to two methods for improving code rate. One method, due to Feng and Rao, removes parity checks that may be recovered by their majority voting algorithm. The second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic. In this work, formulae are given for the redundancies of Hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance. The results proceed from an analysis of numerical…
A closer look at linear recurring sequences allowed us to define the multiplication of a univariate polynomial and a sequence, viewed as a power series with another variable, resulting in another sequence. Extending this operation, one gets the multiplication of matrices of multivariate polynomials and vectors of powers series. A dynamical system, according to U. Oberst is then the kernel of the linear mapping of modules defined by a polynomial matrix by this operation. Applying these tools in the decoding of the so-called one point algebraic-geometry codes, after showing that the syndrome arr…
In this paper, a new approach for decoding low-rate Reed-Solomon codes beyond half the minimum distance is considered and analyzed. Unlike the Sudan algorithm published in 1997, this new approach is based on multi-sequence shift-register synthesis, which makes it easy to understand and simple to implement. The computational complexity of this shift-register based algorithm is of the same order as the complexity of the well-known Berlekamp-Massey algorithm. Moreover, the error correcting radius coincides with the error correcting radius of the original Sudan algorithm, and the practical decodin…
An efficient procedure for error-value calculations based on fast discrete Fourier transforms (DFT) in conjunction with Berlekamp-Massey-Sakata algorithm for a class of affine variety codes is proposed. Our procedure is achieved by multidimensional DFT and linear recurrence relations from Grobner basis and is applied to erasure-and-error decoding and systematic encoding. The computational complexity of error-value calculations in our algorithm improves that in solving systems of linear equations from error correcting pairs in many cases. A motivating example of our algorithm in case of a Reed-…
Codes in the sum-rank metric have various applications in error control for multishot network coding, distributed storage and code-based cryptography. Linearized Reed-Solomon (LRS) codes contain Reed-Solomon and Gabidulin codes as subclasses and fulfill the Singleton-like bound in the sum-rank metric with equality. We propose the first known error-erasure decoder for LRS codes to unleash their full potential for multishot network coding. The presented syndrome-based Berlekamp-Massey-like error-erasure decoder can correct $t_F$ full errors, $t_R$ row erasures and $t_C$ column erasures up to $2t…
Algebraic decoding algorithms are commonly applied for the decoding of Reed-Solomon codes. Their main advantages are low computational complexity and predictable decoding capabilities. Many algorithms can be extended for correction of both errors and erasures. This enables the decoder to exploit binary quantized reliability information obtained from the transmission channel: Received symbols with high reliability are forwarded to the decoding algorithm while symbols with low reliability are erased. In this paper we investigate adaptive single-trial error/erasure decoding of Reed-Solomon codes,…
The performance of algebraic soft-decision decoding of Reed-Solomon codes using bit-level soft information is investigated. Optimal multiplicity assignment strategies of algebraic soft-decision decoding with infinite cost are first studied over erasure channels and the binary symmetric channel. The corresponding decoding radii are calculated in closed forms and tight bounds on the error probability are derived. The multiplicity assignment strategy and the corresponding performance analysis are then generalized to characterize the decoding region of algebraic softdecision decoding over a mixed …
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 thi…
Error Correcting Codes are required to have a reliable communication through a medium that has an unacceptable bit error rate and low signal to noise ratio. In IEEE 802.15.6 2.4GHz Wireless Body Area Network (WBAN), data gets corrupted during the transmission and reception due to noises and interferences. Ultra low power operation is crucial to prolong the life of implantable devices. Hence simple block codes like BCH (63, 51, 2) can be employed in the transceiver design of 802.15.6 Narrowband PHY. In this paper, implementation of BCH (63, 51, t = 2) Encoder and Decoder using VHDL is discussed…
In coding theory, Reed-Solomon codes are one of the most well-known and widely used classes of error-correcting codes. In this thesis we study and compare two major strategies known for their decoding procedure, the Peterson-Gorenstein-Zierler (PGZ) and the Berlekamp-Massey (BM) decoder, in order to improve existing decoding algorithms and propose faster new ones. In particular we study a modified version of the PGZ decoder, which we will call the fast Peterson-Gorenstein-Zierler (fPGZ) decoding algorithm. This improvement was presented in 1997 by exploiting the Hankel structure of the syndrom…
We present a simple syndrome-based fast Chase decoding algorithm for Reed--Solomon (RS) codes. Such an algorithm was initially presented by Wu (IEEE Trans. IT, Jan. 2012), building on properties of the Berlekamp--Massey (BM) algorithm. Wu devised a fast polynomial-update algorithm to construct the error-locator polynomial (ELP) as the solution of a certain linear-feedback shift register (LFSR) synthesis problem. This results in a conceptually complicated algorithm, divided into $8$ subtly different cases. Moreover, Wu's polynomial-update algorithm is not immediately suitable for working with v…
The order domains are a class of commutative rings introduced by Høholdt, van Lint, and Pellikaan to simplify the theory of error control codes using ideas from algebraic geometry. The definition is largely motivated by the structures utilized in the Berlekamp-Massey-Sakata (BMS) decoding algorithm, with Feng-Rao majority voting for unknown syndromes, applied to one-point geometric Goppa codes constructed from curves. However, order domains are much more general, and O'Sullivan has shown that the BMS algorithm can be applied to decode all codes constructed from order domains by a suitable gene…
