Coding Schemes for Crisscross Error Patterns Simon Plass German Aerospace Center (DLR), Institute of Communications and Navigation, 82234 Wessling, Germany Email: simon.plass@dlr.de Gerd Richter Ulm University, Department of TAIT, 89081 Ulm, Germany Email: gerd.richter@uni-ulm.de A. J. Han Vinck University of Essen, Institute for Experimental Mathematics, 45326 Essen, Germany Email: vinck@iem.uni-due.de Abstract. This paper addresses two coding schemes which can handle emerging errors with crisscross patterns. First, a code with maximum rank distance, so-called Rank-Codes, is described and a modiﬁed Berlekamp-Massey algorithm is provided. Secondly, a Permutation Code based coding scheme for crisscross error patterns is presented. The inﬂuence of diﬀerent types of noise are also discussed. Keywords: Rank Codes, Permutation Code, crisscross errors 1. Introduction In a number of applications, the following error protection problem occurs: The information symbols have to be stored in (N × n) arrays. Some of these symbols are transmitted erroneously in such a way that all corrupted symbols are conﬁned to a speciﬁed number of rows or columns (or both). We refer to such errors as crisscross errors. These crisscross errors can be found for example in memory chip arrays [1] or in magnetic tape recording [2]. Figure 1 shows a crisscross error pattern that is limited to two columns and three rows. Since the Hamming metric is not appropriate for these error pat- terns, Delsarte [3] introduced the rank of a matrix as a metric for error correction purpose. Gabidulin [4] and also Roth [5] introduced codes with maximum rank distance (Rank-Codes) that are capable of correcting a speciﬁed number of corrupted rows and columns. Rank- Codes cannot only correct erroneous rows and columns, they can even correct a certain number of rank errors. The number of rank errors is deﬁned as the rank of the error array. Furthermore, it is also possible to deﬁne a Permutation Code in which each codeword contains diﬀerent integers as symbols. This code c  2007 Kluwer Academic Publishers. Printed in the Netherlands. COST289_RankCode_Journal.tex; 31/05/2007; 9:44; p.1