Constant-weight gray codes for local rank modulation

Eyal Gad, Michael Langberg, Moshe Schwartz, Jehoshua Bruck

Research output: Contribution to journalArticlepeer-review


We consider the local rank-modulation (LRM) scheme in which a sliding window going over a sequence of real-valued variables induces a sequence of permutations. LRM is a generalization of the rank-modulation scheme, which has been recently suggested as a way of storing information in flash memory. We study constant-weight Gray codes for the LRM scheme in order to simulate conventional multilevel flash cells while retaining the benefits of rank modulation. We present a practical construction of codes with asymptotically-optimal rate and weight asymptotically half the length, thus having an asymptotically-optimal charge difference between adjacent cells. Next, we turn to examine the existence of optimal codes by specifically studying codes of weight 2 and 3. In the former case, we upper bound the code efficiency, proving that there are no such asymptotically-optimal cyclic codes. In contrast, for the latter case we construct codes which are asymptotically- optimal. We conclude by providing necessary conditions for the existence of cyclic and cyclic optimal Gray codes.

Original languageEnglish
Article number5959206
Pages (from-to)7431-7442
Number of pages12
JournalIEEE Transactions on Information Theory
Issue number11
StatePublished - Nov 2011

Bibliographical note

Funding Information:
Manuscript received December 29, 2010; revised May 13, 2011; accepted July 07, 2011. Date of current version November 11, 2011. This work was supported in part by the ISF under Grant 134/10 and Grant 480/08, by the Open University of Israel’s research fund under Grant 46114, and by the NSF under Grant NSF-0801795, and under an NSF-NRI award.


  • Flash memory
  • gray code
  • local rank modulation
  • permutations
  • rank modulation


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