LDPC codes from Singer cycles

The main goal of coding theory is to devise efficient systems to exploit the full capacity of a communication channel, thus achieving an arbitrarily small error probability. Low Density Parity Check (LDPC) codes are a family of block codes--characterised by admitting a sparse parity check matrix--with good correction capabilities. In the present paper the orbits of subspaces of a finite projective space under the action of a Singer cycle are investigated.Comment: 11 Page

Correction of single error bursts beyond the code correction capability using information sets

The most important method of ensuring data integrity is correcting errors that occur during information storage, processing or transmission. The error-correcting coding methods are used to correct errors. In real systems, noise processes are correlated. However, traditional coding and decoding methods use decorrelation, and it is known that this procedure reduces the maximum achievable characteristics of coding. Thus, constructing computationally efficient decoding methods that would correct grouped errors for a wide class of codes is an actual problem. In this paper the decoding by information sets is used to correct single bursts. This method has exponential complexity when correcting independent errors. The proposed approach uses a number of information sets linearly growing with code length, which provides polynomial decoding complexity. A further reduction of the number of information sets is possible with the proposed method of using dense information sets. It allows evaluating both the set of errors potentially corrected by the code and the characteristics of the decoder. An improvement of the decoding method using an error vector counter is proposed, which allows in some cases to increase the number of corrected error vectors. This method allows significantly reducing the number of information sets or increasing the number of corrected error vectors according to the minimum burst length criterion. The proposed decoders allow correction of single error bursts in polynomial time for arbitrary linear codes. The results of experiments based on standard array show that decoders not only correct all errors within the burst correcting capability of the code, but also a significant number of error vectors beyond of it. Possible directions of further research are the analysis of the proposed decoding algorithms for long codes where the method of analysis based on the standard array is not applicable; the development and analysis of decoding methods for multiple bursts and the joint correction of grouped and random errors

Generalization of the Ball-Collision Algorithm

In this paper we generalize the ball-collision algorithm by Bernstein, Lange, Peters from the binary field to a general finite field. We also provide a complexity analysis and compare the asymptotic complexity to other generalized information set decoding algorithms

DECODING OF MULTIPOINT ALGEBRAIC GEOMETRY CODES VIA LISTS

Algebraic geometry codes have been studied greatly since their introduction by Goppa . Early study had focused on algebraic geometry codes CL(D;G) where G was taken to be a multiple of a single point. However, it has been shown that if we allow G to be supported by more points, then the associated code may have better parameters. We call such a code a multipoint code and if G is supported by m points, then we call it an m-point code. In this dissertation, we wish to develop a decoding algorithm for multipoint codes. We show how we can embed a multipoint algebraic geometry code into a one-point supercode so that we can perform list decoding in the supercode. From the output list, we determine which of the elements is a codeword in the multipoint code. In this way we have unique decoding up to the minimum distance for multipoint algebraic geometry codes, provided the parameters of the list decoding algorithm are set appropriately

On the complexity of minimum distance decoding of long linear codes

We suggest a decoding algorithm of q-ary linear codes, which we call supercode decoding. It ensures the error probability that approaches the error probability of minimumdistance decoding as the length of the code grows. For ¡£¢¥¤ the algorithm has the maximum-likelihood performance. The asymptotic complexity of supercode decoding is exponentially smaller than the complexity of all other methods known. The algorithm develops the ideas of covering-set decoding and split syndrome decoding