@article{a5e9219b6c584dcea7028456492fffcb,
title = "Singleton-type bounds for blot-correcting codes",
abstract = "Consider the transmission of codewords over a channel which introduces dependent errors. Thinking of two-dimensional codewords, such errors can be viewed as blots of a particular shape on the codeword. For such blots of errors the combinatorial metric was introduced by Gabidulin and it was shown that a code with distance </ in combinatorial metric can correct d/2 blots. We propose an universal Singleton-type upper hound nn the rate P\ of a blot-correcting code with the distance </ in arbitrary combinatorial metric. The rate is bounded by R ≤ 1 -(d -1)/D, where D is the maximum possible distance between two words in this metric.",
keywords = "Blot-correcting codes, Combinatorial metric, Singleton-hound",
author = "Martin Bossert and Vladimir Sidorenko",
note = "Funding Information: Let us assume that one-or two-dimensional codewords (vectors or matrices) are transmitted over a channel with dependent errors. Thus the elements of the codeword are not corrupted independently but, one “error event” corrupts a number of elements in a region of given shape in the codeword. We will call this corrupted region of a codeword: “blot.” Examples of such channels are one-dimensional (usual) codewords with bursts of errors, two-dimensional codewords with two-dimensional bursts of rectangular (or another) shape, two-dimensional codewords with criss-cross or lattice-pattern errors, and so on. Generally, the Hamming metric for independent errors does not fit to such a channel with blots. However, a number of metrics were suggested for different channels with dependent errors; namely, the burst metric, the lattice pattern metric, the two-dimensional burst metric, and so on. All these metrics are special cases of the combinatorial metric introduced by Gabidulin [I]. This metric can be adjusted to any channel with blots. With such a metric, adapted to a given channel, codes can be constructed with minimal distance d in this metric. Such a code can correct up to d / 2 blots [l]. An important question is, what is the upper bound on the rate R of code with distance d in combinatorial metric? Singleton [2] suggested one way to obtain an upper bound for the Hamming metric. Recently, it was pointed out [3] that the bound can Manuscript received May 2, 1995; revised November 17, 1995. This work was supported by the DFG (Deutsche Forschungsgerneinschaft) and by ISF NKF000. The material in this correspondence was presented in part at the IEEE Workshop, Moscow, Russia, June 1994. M. Bossert is with the University of Ulm, Informationstechnik, D-89081 Ulm, Germany. V Sidorenko IS with the Institute for Problems of Information Transmission, Moscow 101447 GSP-4, Russia. Publisher Item Identifier S 001 8-9448(96)02934-3 be found in [4] which dates back to 1932. For each new proposed metric the Singleton-type bound was rederived. For the burst metric, the Reiger bound was obtained [SI, which is a Singleton-type bound for single-burst-correcting codes (d = 3). The Reiger bound was generalized for an arbitrary code distance d in (11, [6], and for two-dimensional burst metric in [7]. For linear codes with a lattice pattern metric the Singleton-type bound was obtained in [8] and for arbitrary codes in [SI.",
year = "1996",
doi = "10.1109/18.490569",
language = "English",
volume = "42",
pages = "1021--1023",
journal = "IEEE Transactions on Information Theory",
issn = "0018-9448",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
number = "3",
}