6 research outputs found
Systematic Codes for Rank Modulation
The goal of this paper is to construct systematic error-correcting codes for
permutations and multi-permutations in the Kendall's -metric. These codes
are important in new applications such as rank modulation for flash memories.
The construction is based on error-correcting codes for multi-permutations and
a partition of the set of permutations into error-correcting codes. For a given
large enough number of information symbols , and for any integer , we
present a construction for systematic -error-correcting codes,
for permutations from , with less redundancy symbols than the number
of redundancy symbols in the codes of the known constructions. In particular,
for a given and for sufficiently large we can obtain . The same
construction is also applied to obtain related systematic error-correcting
codes for multi-permutations.Comment: to be presented ISIT201
Limited-Magnitude Error-Correcting Gray Codes for Rank Modulation
We construct Gray codes over permutations for the rank-modulation scheme,
which are also capable of correcting errors under the infinity-metric. These
errors model limited-magnitude or spike errors, for which only
single-error-detecting Gray codes are currently known. Surprisingly, the
error-correcting codes we construct achieve a better asymptotic rate than that
of presently known constructions not having the Gray property, and exceed the
Gilbert-Varshamov bound. Additionally, we present efficient ranking and
unranking procedures, as well as a decoding procedure that runs in linear time.
Finally, we also apply our methods to solve an outstanding issue with
error-detecting rank-modulation Gray codes (snake-in-the-box codes) under a
different metric, the Kendall -metric, in the group of permutations over
an even number of elements , where we provide asymptotically optimal
codes.Comment: Revised version for journal submission. Additional results include
more tight auxiliary constructions, a decoding shcema, ranking/unranking
procedures, and application to snake-in-the-box codes under the Kendall
tau-metri
On the Labeling Problem of Permutation Group Codes Under the Infinity Metric
Abstract—We consider codes over permutations under the infinity norm. Given such a code, we show that a simple relabeling operation, which produces an isomorphic code, may drastically change the minimal distance of the code. Thus, we may choose a code structure for efficient encoding procedures, and then optimize the code’s minimal distance via relabeling. To establish that the relabeling problem is hard and is of interest, we formally define it and show that all codes may be relabeled to get a minimal distance at most 2. On the other hand, the decision problem of whether a code may be relabeled to distance 2 or more is shown to be NP-complete, and calculating the best achievable minimal distance after relabeling is proved to be hard to approximate up to a factor of 2. We then consider general bounds on the relabeling problem. We specifically construct the optimal relabeling for transitive cyclic groups. We conclude with the main result—a general probabilistic bound, whichwethenusetoshowboththe group and the dihedral group on elements may be relabeled to a minimal distance of. Index Terms—Error-correcting codes, group codes, permutations, rank modulation. I
On the Labeling Problem of Permutation Group Codes under the Infinity Metric
Abstract—Codes over permutations under the infinity norm have been recently suggested as a coding scheme for correcting limited-magnitude errors in the rank modulation scheme. Given such a code, we show that a simple relabeling operation, which producesanisomorphiccode,maydrasticallychangetheminimal distance of the code. Thus, we may choose a code structure for efficient encoding/decoding procedures, and then optimize the code’s minimal distance via relabeling. We formally define the relabeling problem, and show that all codes may be relabeled to get a minimal distance at most 2. The decision problem of whether a code may be relabeled to distance 1 is shown to be NP-complete, and calculating the best achievable minimal distance after relabeling is proved hard to approximate. Finally, we consider general bounds on the relabelingproblem. We specifically show the optimal relabeling distance of cyclic groups. A specific case of a general probabilistic argument is used to show AGL(p) may be relabeled to a minimal distance of p −O ( √ plnp). I