A class of array codes correcting multiple column erasures

In this paper, we present a new basis of polynomial over finite fields of characteristic two and then apply it to the encoding/decoding of Reed-Solomon erasure codes. The proposed polynomial basis allows that h-point polynomial evaluation can be computed in O(h log 2 (h)) finite field operations with small leading constant. As compared with the canonical polynomial basis, the proposed basis improves the arithmetic complexity of addition, multiplication, and the determination of polynomial degree from O(h log 2 (h) log 2 log 2 (h)) to O(h log 2 (h)). Based on this basis, we then develop the encoding and erasure decoding algorithms for the (n = 2 r , k) Reed-Solomon codes. Thanks to the efficiency of transform based on the polynomial basis, the encoding can be completed in O(n log 2 (k)) finite field operations, and the erasure decoding in O(n log 2 (n)) finite field operations. To the best of our knowledge, this is the first approach supporting Reed-Solomon erasure codes over characteristic-2 finite fields while achieving a complexity of O(n log 2 (n)), in both additive and multiplicative complexities. As the complexity leading factor is small, the algorithms are advantageous in practical applications.

2011 12th Canadian Workshop on Information Theory, 2011

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, i.e. we derive an adaptive erasing strategy which minimizes the residual codeword error probability after decoding. Our result is applicable to any error/erasure decoding algorithm as long as its decoding capabilities can be expressed by a decoder capability function. Examples are Bounded Minimum Distance decoding with the Berlekamp-Massey-or the Sugiyama algorithms and the Guruswami-Sudan list decoder.

2004

We present a new class of MDS array codes of size n 2 n (n a prime number) called X-code. The X-codes are of minimum column distance 3, namely, they can correct either one column error or two column erasures. The key novelty in X-code is that it has a simple geometrical construction which achieves encoding/update optimal complexity, i.e., a change of any single information bit affects exactly two parity bits. The key idea in our constructions is that all parity symbols are placed in rows rather than columns.

IEEE Transactions on Information Theory, 1986

IEEE Transactions on Information Theory, 1998

A novel construction method is presented for array codes correcting a single rectangular error cluster of size b 1 2 b 2 or less. Encoding is done in such a way that parity check symbols are calculated rowor columnwise which are a word of a one-dimensional code correcting a phased burst. These parity check symbols are not transmitted but can be calculated by the receiver. Corresponding decoding algorithms are given.

2007

We begin with the definition of Reed-Solomon codes. Definition 1.1 (Reed-Solomon code) . Let Fq be a finite field andFq[x] denote theFq-space of univariate polynomials where all the coefficients of x are fromFq. Pick {α1, α2, ...αn} distinct elements (also calledevaluation points ) of Fq and choosen and k such thatk ≤ n ≤ q. We define an encoding function for Reed-Solomon code as RS : Fq → F n q as follows. A message m = (m0, m1, ..., mk−1) with mi ∈ Fq is mapped to a degree k − 1 polynomial.

IEEE Transactions on Information Theory, 1991

New combinatorial and algebraic techniques are presented for systematically constructing different (d,k) block codes capable of detecting and correcting single bit-errors, single-peak shift-errors, double adjacent-errors and multiple adjacent erasures. Constructions utilizing channel side information, such as the magnetic recording ternary channel output string, or erasures, do not impose any restriction on the /c-constraint, while some of the other constructions require k = 2d. Due to the small and fixed number of redundant bits, the rates of both classes of constructions can be made to approach the capacity of the d constrained channel for long codeword lengths. All the codes can be encoded and decoded with simple, structured logic circuits.

2016

Tuan Anh Le DECODING OF NON-BINARY MULTIPLE INSERTION/DELETION ERROR CORRECTING CODES 2015-2016 Hieu D. Nguyen, Ph.D. Master of Arts in Mathematics Data that is either transmitted over a communication channel or stored in memory is not always completely error free. Many communication channels are subject to noise, and thus errors may occur during transmission from transmitter to receiver. For example, DRAM memory cell contents can change spuriously due to electromagnetic interference while magnetic flux can cause one or more bits to flip in magnetic storage devices. To combat these errors, codes capable of correcting insertion/deletion errors have been investigated. Levenshtein codes are the foundation of this thesis. His codes, first constructed by Varshamov-Tenengol’ts, are capable of correcting one insertion/deletion error. Helberg codes are based on Levenshtein codes and are able to correct multiple insertion/deletion errors. Even though there are codes with better rates, Helber...

2005

Reed-Solomon erasure codes provide efficient simple techni ques for re- dundantly encoding information so that the failure of a few disks in a disk array doesn't compromise the availability of data. This pap er presents a tech- nique for constructing a code that can correct up to three err ors with a sim- ple, regular encoding, which admits very efficient matrix in versions. It also presents new techniques for efficiently computing the sums o f streams of data elements in a finite field, such as is needed in recovering data after a disk failure.