On Optimal Anticodes over Permutations with the Infinity Norm

Motivated by the set-antiset method for codes over permutations under the infinity norm, we study anticodes under this metric. For half of the parameter range we classify all the optimal anticodes, which is equivalent to finding the maximum permanent of certain $(0,1)$-matrices. For the rest of the cases we show constraints on the structure of optimal anticodes

Systematic Error-Correcting Codes for Rank Modulation

The rank-modulation scheme has been recently proposed for efficiently storing data in nonvolatile memories. Error-correcting codes are essential for rank modulation, however, existing results have been limited. In this work we explore a new approach, \emph{systematic error-correcting codes for rank modulation}. Systematic codes have the benefits of enabling efficient information retrieval and potentially supporting more efficient encoding and decoding procedures. We study systematic codes for rank modulation under Kendall's $\tau$-metric as well as under the $\ell_\infty$-metric. In Kendall's $\tau$-metric we present $[k+2,k,3]$-systematic codes for correcting one error, which have optimal rates, unless systematic perfect codes exist. We also study the design of multi-error-correcting codes, and provide two explicit constructions, one resulting in $[n+1,k+1,2t+2]$ systematic codes with redundancy at most $2t+1$. We use non-constructive arguments to show the existence of $[n,k,n-k]$-systematic codes for general parameters. Furthermore, we prove that for rank modulation, systematic codes achieve the same capacity as general error-correcting codes. Finally, in the $\ell_\infty$-metric we construct two $[n,k,d]$ systematic multi-error-correcting codes, the first for the case of $d=O(1)$, and the second for $d=\Theta(n)$. In the latter case, the codes have the same asymptotic rate as the best codes currently known in this metric

Systematic Error-Correcting Codes for Rank Modulation

The rank modulation scheme has been proposed recently for efficiently writing and storing data in nonvolatile memories. Error-correcting codes are very important for rank modulation, and they have attracted interest among researchers. In this work, we explore a new approach, systematic error-correcting codes for rank modulation. In an (n,k) systematic code, we use the permutation induced by the levels of n cells to store data, and the permutation induced by the first k cells (k < n) has a one-to-one mapping to information bits. Systematic codes have the benefits of enabling efficient information retrieval and potentially supporting more efficient encoding and decoding procedures. We study systematic codes for rank modulation equipped with the Kendall's τ-distance. We present (k + 2, k) systematic codes for correcting one error, which have optimal sizes unless perfect codes exist. We also study the design of multi-error-correcting codes, and prove that for any 2 ≤ k < n, there always exists an (n, k) systematic code of minimum distance n-k. Furthermore, we prove that for rank modulation, systematic codes achieve the same capacity as general error-correcting codes

Snake-in-the-Box Codes for Rank Modulation

Motivated by the rank-modulation scheme with applications to flash memory, we consider Gray codes capable of detecting a single error, also known as snake-in-the-box codes. We study two error metrics: Kendall's $\tau$-metric, which applies to charge-constrained errors, and the $\ell_\infty$-metric, which is useful in the case of limited magnitude errors. In both cases we construct snake-in-the-box codes with rate asymptotically tending to 1. We also provide efficient successor-calculation functions, as well as ranking and unranking functions. Finally, we also study bounds on the parameters of such codes

Data Representation for Efficient and Reliable Storage in Flash Memories

Recent years have witnessed a proliferation of flash memories as an emerging storage technology with wide applications in many important areas. Like magnetic recording and optimal recording, flash memories have their own distinct properties and usage environment, which introduce very interesting new challenges for data storage. They include accurate programming without overshooting, error correction, reliable writing data to flash memories under low-voltages and file recovery for flash memories. Solutions to these problems can significantly improve the longevity and performance of the storage systems based on flash memories. In this work, we explore several new data representation techniques for efficient and reliable data storage in flash memories. First, we present a new data representation scheme—rank modulation with multiplicity —to eliminate the overshooting and charge leakage problems for flash memories. Next, we study the Half-Wits — stochastic behavior of writing data to embedded flash memories at voltages lower than recommended by a microcontroller’s specifications—and propose three software- only algorithms that enable reliable storage at low voltages without modifying hard- ware, which can reduce energy consumption by 30%. Then, we address the file erasures recovery problem in flash memories. Instead of only using traditional error- correcting codes, we design a new content-assisted decoder (CAD) to recover text files. The new CAD can be combined with the existing error-correcting codes and the experiment results show CAD outperforms the traditional error-correcting codes