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Zbl 0816.94023
Roth, Ron M.; Siegel, Paul H.
Lee-metric BCH codes and their application to constrained and partial- response channels. (English)
[J] IEEE Trans. Inf. Theory 40, No.4, 1083-1096 (1994). ISSN 0018-9448

Summary: We show that each code in a certain class of BCH codes over $GF(p)$, specified by a code length $n \le p\sp m-1$ and a runlength $r \le (p - 1)/2$ of consecutive roots in $GF(p\sp m)$, has minimum Lee distance $\ge 2r$. For the very high-rate range these codes approach the sphere-packing bound on the minimum Lee distance. Furthermore, for a given $r$, the length range of these codes is twice as large as that attainable by Berlekamp's extended negacyclic codes. We present an efficient decoding procedure, based on Euclid's algorithm, for correcting up to $r - 1$ errors and detecting $r$ errors, that is, up to the number of Lee errors guaranteed by the designed minimum Lee distance $2r$. Bounds on the minimum Lee distance for $r \ge (p + 1)/2$ are provided for the Reed- Solomon case, i.e., when the BCH code roots are in $GF(p)$. We present two applications. First, Lee-metric BCH codes can be used for protecting against bitshift errors and synchronization errors caused by insertion and/or deletion of zeros in $(d,k)$-constrained channels. Second, the code construction with its decoding algorithm can be formulated over the integer ring, providing an algebraic approach to correcting errors in partial-response channels where matched spectral-null codes are used.

MSC 2000:: *94B15 Cyclic codes
94A40 Channel models
94B35 Decoding

Keywords: constrained channels; BCH codes; Lee distance; decoding; Lee-metric; partial-response channels

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Zentralblatt MATH Berlin [Germany]