Constructions of Rank Modulation Codes

Rank modulation is a way of encoding information to correct errors in flash memory devices as well as impulse noise in transmission lines. Modeling rank modulation involves construction of packings of the space of permutations equipped with the Kendall tau distance. We present several general constructions of codes in permutations that cover a broad range of code parameters. In particular, we show a number of ways in which conventional error-correcting codes can be modified to correct errors in the Kendall space. Codes that we construct afford simple encoding and decoding algorithms of essentially the same complexity as required to correct errors in the Hamming metric. For instance, from binary BCH codes we obtain codes correcting $t$ Kendall errors in $n$ memory cells that support the order of $n!/(\log_2n!)^t$ messages, for any constant $t= 1,2,...$ We also construct families of codes that correct a number of errors that grows with $n$ at varying rates, from $\Theta(n)$ to $\Theta(n^{2})$. One of our constructions gives rise to a family of rank modulation codes for which the trade-off between the number of messages and the number of correctable Kendall errors approaches the optimal scaling rate. Finally, we list a number of possibilities for constructing codes of finite length, and give examples of rank modulation codes with specific parameters.Comment: Submitted to IEEE Transactions on Information Theor

On generic erasure correcting sets and related problems

Motivated by iterative decoding techniques for the binary erasure channel Hollmann and Tolhuizen introduced and studied the notion of generic erasure correcting sets for linear codes. A generic $(r,s)$--erasure correcting set generates for all codes of codimension $r$ a parity check matrix that allows iterative decoding of all correctable erasure patterns of size $s$ or less. The problem is to derive bounds on the minimum size $F(r,s)$ of generic erasure correcting sets and to find constructions for such sets. In this paper we continue the study of these sets. We derive better lower and upper bounds. Hollmann and Tolhuizen also introduced the stronger notion of $(r,s)$--sets and derived bounds for their minimum size $G(r,s)$. Here also we improve these bounds. We observe that these two conceps are closely related to so called $s$--wise intersecting codes, an area, in which $G(r,s)$ has been studied primarily with respect to ratewise performance. We derive connections. Finally, we observed that hypergraph covering can be used for both problems to derive good upper bounds.Comment: 9 pages, to appear in IEEE Transactions on Information Theor

Pragmatic Languages with Universal Grammars

This paper shows the existence of an equilibrium pragmatic Language with a universal grammar as a coordination device under communication misunderstandings. Such a language plays a key role in achieving efficient outcomes in those Sender-Receiver games where there may exist noisy information transmission. The Language is pragmatic in the sense that the Receiver’ best response depends on the context, i.e, on the payoffs and on the initial probability distribution of the states of nature of the underlying game. The Language has a universal grammar because the coding rule does not depend on such specific parameters and can then be applied to any Sender-Receiver game with noisy communication.grammar, pragmatic language, prototypes, separating equilibria