Resynchronizing (d , k)-Constrained Sequences in the Presence of

Insertions and Deletions

Mario Blaum, Jehoshiia Rruck aad C. Michael Mclas IBM Research Division

Almaden Research Center 650 Harry Road

San Jose, CA 95120 USA

Henk C. A. van Tilborg Department oC Mathematics and Computer Scicii w

Eiridhoven University of Teclr~Idogy P.O. Box 513

5600 MB Eindhoven The Netherlands

A typical encoding configuration for a magnetic or optical record- ing channel consists of encoding the information bits with an error- correcting code, generally, a Reed Solomon code, followed by a (d , k) constrained code [I]. In general, Reed-Solomon codes can handle the most common type of errors: random errors and peak shifts A random error can bc of two types: a 0 becomes a 1, denoted O i l , or a 1 becomm a 0, denoted 1+0. Peak shifts are also of two typcs. 0 1-1 0 or 1 O+O 1. However, there are other types of errors that cause a catastrophic failure due to loss of synchronization. They are, deletion of a symbol (0 or 1) and insertion of a symbol (0 or 1). Although deletions and insertions are not as common as the other types of error, if we are able to determine how many insertions or deletions occurred in an interval, by inserting or deleting a proper amount of symbols we are going to have a burst error that will either be corrected by the outer error-correcting code or, if uncorrectablc, a t least it will have a limited length. Consider (1,7) sequences (the method can be generalized to any (d, k ) sequence). We make the following 1-1 mapping between a (1 ,7 ) sequence and symbols in 2, (i.e., set of integers modulo 7): to each run of O’s, we associate the number of zeros minus onc. If we denote by L the length of the binary string, by e the length of the 7-ary string and by S the sum of the symbols in the 7-ary string, these three parameters are related by L = S + 2P. At the 7-ary level, we encode the information using an [ T I , 7 1 - 21

block are redundant, while the middle 7% - 2 symbols carry the information. We require that in each block, the sum of the symbols

block code, where n 2 7 The first and thc last symbols in a

modulo 7 is 0. The last symbol in a block and the first symbol in the next block are chosen in such a way that their sum is equal to 6. Thus, we are inserting exactly 10 binary symbols between blocks in the binary sequence. It is important to have a fixed amount of redundancy while attempting to recover synchronization. Finally, we set the initial condition a,, = 0. At the receiving end, if a 7-ary sequence bo, b l , 02, has becn received, and errors have occurred, including possible insertions and/or deletions of symbols, we show how to recover synchroniza- tion with high probability under the following conditions (that are determined by the error statistics of the channel):

1. At most 3 errors in at most X consecutive 7-ary blocks of , 1 7 7 + 1 , length n have occurred, say in blocks m, vi + 1.

where 5 X - 1.

2. After the last block in error, say block 773 + T , there are at least s error-free blocks.

3. The length n of each block is a t least 7 (in general, il - d+1)

Under these conditions, we present a method that will allow us to determine how many symbols have been deleted, allowing for recovery of synchronization.

Re fer e nc e s [l] P. H. Siegel, “Recording Codes for Digital Magnetic Storagr,”

IEEE Trans. on Magnetics, Scpt. 1985, pp. 1344-1349.

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