Algebraic Coding Theory

Berlekamp, Elwyn R.

doi:10.1142/9407

Cited by 1,101 publications

(394 citation statements)

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“…The Berlekamp/Massey algorithm ( [Berlekamp 1968;Massey 1969]) computes a minimal generator of a linearly generated scalar sequence. The algorithm is equivalent to the extended Euclidean algorithm [Dornstetter 1987].…”

Section: Introductionmentioning

confidence: 99%

A fraction free Matrix Berlekamp/Massey algorithm

Kaltofen

Yuhasz

2013

Linear Algebra and its Applications

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We describe a fraction free version of the Matrix Berlekamp/Massey algorithm. The algorithm computes a minimal matrix generator of linearly generated square matrix sequences over an integral domain. The algorithm performs all operations in the integral domain, so all divisions performed are exact. For scalar sequences, the matrix algorithm specializes to a more efficient algorithm than the algorithm currently in the literature. The proof of integrality of the matrix algorithm gives a new proof of integrality for the scalar specialization.

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Section: Introductionmentioning

confidence: 99%

A fraction free Matrix Berlekamp/Massey algorithm

Kaltofen

Yuhasz

2013

Linear Algebra and its Applications

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“…By taking the minimum value of the linear complexity for each number of errors, the results in the tree in Figure 1 give an incomplete approximate k-error linear complexity profile as being {(0, 8), (1,9), (2, 7), (3, 5), (5, 2)}. Applying Property 1 in Section 2 and using the fact that L w H (s) (s) = 0 the full approximate k-error linear complexity profile is found: {(0, 8), (1,8), (2,7), (3,5), (4,5), (5, 2), (6, 2), (7, 2), (8, 2), (9, 2), (10, 2), (11, 0)}.…”

Section: The Modified Berlekamp-massey Algorithmmentioning

confidence: 99%

“…Applying Property 1 in Section 2 and using the fact that L w H (s) (s) = 0 the full approximate k-error linear complexity profile is found: {(0, 8), (1,8), (2,7), (3,5), (4,5), (5, 2), (6, 2), (7, 2), (8, 2), (9, 2), (10, 2), (11, 0)}. The exact k-error linear complexity profile obtained by an exhaustive search algorithm is: (2,6), (3,4), (4, 2), (5, 1), (6, 1), (7,1), (8,1), (9, 1), (10, 1), (11, 0)}. : n ← 0 10: while sn = 0 and n < t − 1 do go over the initial zeros 11:…”

Section: The Modified Berlekamp-massey Algorithmmentioning

confidence: 99%

“…We propose adapting the Berlekamp-Massey Algorithm (Berlekamp [1], Massey [7]) which computes the linear complexity, in order to approximate the k-error linear complexity profile for a general sequence over a finite field. The main idea is to devise a heuristic algorithm which explores only some of all the possible error patterns.…”

Section: Introductionmentioning

confidence: 99%

“…The Berlekamp-Massey Algorithm ( [1], [7]) computes the characteristic polynomial and the linear complexity of a sequence over any field. It is iteratively processing each term of a finite sequence s 0 , s 1 , .…”

Section: Introductionmentioning

confidence: 99%

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Modified Berlekamp-Massey Algorithm for Approximating the k-Error Linear Complexity of Binary Sequences

Alecu

Sălăgean

Cryptography and Coding

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Abstract. Some cryptographical applications use pseudorandom sequences and require that the sequences are secure in the sense that they cannot be recovered by only knowing a small amount of consecutive terms. Such sequences should therefore have a large linear complexity and also a large k-error linear complexity. Efficient algorithms for computing the k-error linear complexity of a sequence only exist for sequences of period equal to a power of the characteristic of the field. It is therefore useful to find a general and efficient algorithm to compute a good approximation of the k-error linear complexity. We show that the Berlekamp-Massey Algorithm, which computes the linear complexity of a sequence, can be adapted to approximate the k-error linear complexity profile for a general sequence over a finite field. While the complexity of this algorithm is still exponential, it is considerably more efficient than the exhaustive search.

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A tutorial on Reed–Solomon coding for fault‐tolerance in RAID-like systems

Plank

1997

Softw: Pract. Exper.

203

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It is well-known that Reed-Solomon codes may be used to provide error correction for multiple failures in RAIDlike systems. The coding technique itself, however, is not as well-known. To the coding theorist, this technique is a straightforward extension to a basic coding paradigm and needs no special mention. However, to the systems programmer with no training in coding theory, the technique may be a mystery. Currently, there are no references that describe how to perform this coding that do not assume that the reader is already well-versed in algebra and coding theory. This paper is intended for the systems programmer. It presents a complete specification of the coding algorithm plus details on how it may be implemented. This specification assumes no prior knowledge of algebra or coding theory. The goal of this paper is for a systems programmer to be able to implement Reed-Solomon coding for reliability in RAID-like systems without needing to consult any external references. Problem Specification Let there be storage devices, ¡ £ ¢ ¥ ¤ ¦ ¡ § © ¤ ¤ ¦ ¡ , each of which holds bytes. These are called the "Data Devices." Let there be more storage devices ¢ ¥ ¤ § ¤ ¤ , each of which also holds bytes. These are called the "Checksum Devices." The contents of each checksum device will be calculated from the contents of the data devices. The goal is to define the calculation of each " ! such that if any of ¡ ¢¤ ¦ ¡ §¤ ¤ ¦ ¡ ¤ ¢¤ §¤ ¤ fail, then the contents of the failed devices can be reconstructed from the non-failed devices.

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Algebraic Coding Theory

Cited by 1,101 publications

References 0 publications

A fraction free Matrix Berlekamp/Massey algorithm

A fraction free Matrix Berlekamp/Massey algorithm

Modified Berlekamp-Massey Algorithm for Approximating the k-Error Linear Complexity of Binary Sequences

A tutorial on Reed–Solomon coding for fault‐tolerance in RAID-like systems

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