Burst error-correcting code

Abstract: In computers and digital communication systems, information almost always is represented in a binary form as a sequence of bits each having the values 0 or 1. This sequence of bits is transmitted over a channel from a sender to a receiver. In some applications the channel is a storage medium like a DVD, where the information is written to the medium at a certain time and retrieved at a later time. Because of the physical limitations of the channel, some transmitted bits may be corrupted (the channel is noisy) and thus make it difficult for the receiver to reconstruct the information correctly. In algebraic coding theory, we are concerned mainly with developing methods to detect and correct errors that typically occur during transmission of information over a noisy channel. The basic technique to detect and correct errors is by introducing redundancy in the data that is to be transmitted. This article we describe the basic ideas, give some codes that are most important for applications, and explain their decoding algorithms. In particular, we describe the Reed–Solomon codes and give both the classic Berlekamp–Massery decoging algorithm and the recent Guruswami–Sudan decoding algorighm. Keywords: coding theory; decoding; Reed–Solomon codes

Abstract: An optimal family of array codes over GF(q) for correcting multiple phased burst errors and erasures, where each phased burst corresponds to an erroneous or erased column in a code array, is introduced. As for erasures, these array codes have an efficient decoding algorithm which avoids multiplications (or divisions) over extension fields, replacing these operations with cyclic shifts of vectors over GF(q). The erasure decoding algorithm can be adapted easily to handle single column errors as well. The codes are characterized geometrically by means of parity constraints along certain diagonal lines in each code array, thus generalizing a previously known construction for the special case of two erasures. Algebraically, they can be interpreted as Reed-Solomon codes. When q is primitive in GF(q), the resulting codes become (conventional) Reed-Solomon codes of length P over GF(q/sup p-1/), in which case the new erasure decoding technique can be incorporated into the Berlekamp-Massey algorithm, yielding a faster way to compute the values of any prescribed number of errors. >

Abstract: In the theory of cyclic codes, it is common practice to require that (n,q)=1, where n is the word length and F/sub q/ is the alphabet. It is shown that the even weight subcodes of the shortened binary Hamming codes form a sequence of repeated-root cyclic codes that are optimal. In nearly all other cases, one does not find good cyclic codes by dropping the usual restriction that n and q must be relatively prime. This statement is based on an analysis for lengths up to 100. A theorem shows why this was to be expected, but it also leads to low-complexity decoding methods. This is an advantage, especially for the codes that are not much worse than corresponding codes of odd length. It is demonstrated that a binary cyclic code of length 2n (n odd) can be obtained from two cyclic codes of length n by the well-known mod u mod u+v mod construction. This leads to an infinite sequence of optimal cyclic codes with distance 4. Furthermore, it is shown that low-complexity decoding methods can be used for these codes. The structure theorem generalizes to other characteristics and to other lengths. Some comparisons of the methods using earlier examples are given. >