Interleaving schemes for multidimensional cluster errors

We present two-dimensional and three-dimensional interleaving techniques for correcting two- and three-dimensional bursts (or clusters) of errors, where a cluster of errors is characterized by its area or volume. Correction of multidimensional error clusters is required in holographic storage, an emerging application of considerable importance. Our main contribution is the construction of efficient two-dimensional and three-dimensional interleaving schemes. The proposed schemes are based on t-interleaved arrays of integers, defined by the property that every connected component of area or volume t consists of distinct integers. In the two-dimensional case, our constructions are optimal: they have the lowest possible interleaving degree. That is, the resulting t-interleaved arrays contain the smallest possible number of distinct integers, hence minimizing the number of codewords required in an interleaving scheme. In general, we observe that the interleaving problem can be interpreted as a graph-coloring problem, and introduce the useful special class of lattice interleavers. We employ a result of Minkowski, dating back to 1904, to establish both upper and lower bounds on the interleaving degree of lattice interleavers in three dimensions. For the case t≡0 mod 6, the upper and lower bounds coincide, and the Minkowski lattice directly yields an optimal lattice interleaver. For t≠0 mod 6, we construct efficient lattice interleavers using approximations of the Minkowski lattice

Space-time coding techniques with bit-interleaved coded modulations for MIMO block-fading channels

The space-time bit-interleaved coded modulation (ST-BICM) is an efficient technique to obtain high diversity and coding gain on a block-fading MIMO channel. Its maximum-likelihood (ML) performance is computed under ideal interleaving conditions, which enables a global optimization taking into account channel coding. Thanks to a diversity upperbound derived from the Singleton bound, an appropriate choice of the time dimension of the space-time coding is possible, which maximizes diversity while minimizing complexity. Based on the analysis, an optimized interleaver and a set of linear precoders, called dispersive nucleo algebraic (DNA) precoders are proposed. The proposed precoders have good performance with respect to the state of the art and exist for any number of transmit antennas and any time dimension. With turbo codes, they exhibit a frame error rate which does not increase with frame length.Comment: Submitted to IEEE Trans. on Information Theory, Submission: January 2006 - First review: June 200

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

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

Multilevel Coded Modulation for Unequal Error Protection and Multistage Decoding—Part II: Asymmetric Constellations

In this paper, multilevel coded asymmetric modulation with multistage decoding and unequal error protection (UEP) is discussed. These results further emphasize the fact that unconventional signal set partitionings are more promising than traditional (Ungerboeck-type) partitionings, to achieve UEP capabilities with multilevel coding and multistage decoding. Three types of unconventional partitionings are analyzed for asymmetric 8-PSK and 16-QAM constellations over the additive white Gaussian noise channel to introduce design guidelines. Generalizations to other PSK and QAM type constellations follow the same lines. Upper bounds on the bit-error probability based on union bound arguments are first derived. In some cases, these bounds become loose due to the large overlappings of decision regions associated with asymmetric constellations and unconventional partitionings. To overcome this problem, simpler and tighter approximated bounds are derived. Based on these bounds, it is shown that additional refinements can be achieved in the construction of multilevel UEP codes, by introducing asymmetries in PSK and QAM signal constellations

A coding approach for detection of tampering in write-once optical disks

We present coding methods for protecting against tampering of write-once optical disks, which turns them into a secure digital medium for applications where critical information must be stored in a way that prevents or allows detection of an attempt at falsification. Our method involves adding a small amount of redundancy to a modulated sector of data. This extra redundancy is not used for normal operation, but can be used for determining, say, as a testimony in court, that a disk has not been tampered with

Iterative decoding for MIMO channels via modified sphere decoding

In recent years, soft iterative decoding techniques have been shown to greatly improve the bit error rate performance of various communication systems. For multiantenna systems employing space-time codes, however, it is not clear what is the best way to obtain the soft information required of the iterative scheme with low complexity. In this paper, we propose a modification of the Fincke-Pohst (sphere decoding) algorithm to estimate the maximum a posteriori probability of the received symbol sequence. The new algorithm solves a nonlinear integer least squares problem and, over a wide range of rates and signal-to-noise ratios, has polynomial-time complexity. Performance of the algorithm, combined with convolutional, turbo, and low-density parity check codes, is demonstrated on several multiantenna channels. The results for systems that employ space-time modulation schemes seem to indicate that the best performing schemes are those that support the highest mutual information between the transmitted and received signals, rather than the best diversity gain

Correcting Charge-Constrained Errors in the Rank-Modulation Scheme

We investigate error-correcting codes for a the rank-modulation scheme with an application to flash memory devices. In this scheme, a set of n cells stores information in the permutation induced by the different charge levels of the individual cells. The resulting scheme eliminates the need for discrete cell levels, overcomes overshoot errors when programming cells (a serious problem that reduces the writing speed), and mitigates the problem of asymmetric errors. In this paper, we study the properties of error-correcting codes for charge-constrained errors in the rank-modulation scheme. In this error model the number of errors corresponds to the minimal number of adjacent transpositions required to change a given stored permutation to another erroneous one—a distance measure known as Kendall’s τ-distance.We show bounds on the size of such codes, and use metric-embedding techniques to give constructions which translate a wealth of knowledge of codes in the Lee metric to codes over permutations in Kendall’s τ-metric. Specifically, the one-error-correcting codes we construct are at least half the ball-packing upper bound