Vitaly Skachek

Recursive Code Construction for Random Networks

A modification of Ko¿tter-Kschischang codes for random networks is presented (these codes were also studied by Wang in the context of authentication problems). The new codes have higher information rate, while maintaining the same error-correcting capabilities. An efficient error-correcting algorithm is proposed for these codes.

Linear-Programming Decoding of Nonbinary Linear Codes

A framework for linear-programming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates LP-based reception for coded modulation systems which use direct modulation mapping of coded symbols. It is proved that the resulting LP decoder has the ldquomaximum-likelihood (ML) certificaterdquo property. It is also shown that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. It is also proved that if the modulator-channel combination satisfies a particular symmetry condition, the codeword error rate performance is independent of the transmitted codeword. Two alternative polytopes for use with LP decoding are studied, and it is shown that for many classes of codes these polytopes yield a complexity advantage for decoding. These polytope representations lead to polynomial-time decoders for a wide variety of classical nonbinary linear codes. LP decoding performance is illustrated for ternary Golay code with ternary phase-shift keying (PSK) modulation over additive white Gaussian noise (AWGN), and in this case it is shown that the performance of the LP decoder is comparable to codeword-error-rate-optimum hard-decision-based decoding. LP decoding is also simulated for medium-length ternary and quaternary low-density parity-check (LDPC) codes with corresponding PSK modulations over AWGN.

Error-Correction in Flash Memories via Codes in the Ulam Metric

We consider rank modulation codes for flash memories that allow for handling arbitrary charge-drop errors. Unlike classical rank modulation codes used for correcting errors that manifest themselves as swaps of two adjacently ranked elements, the proposed translocation rank codes account for more general forms of errors that arise in storage systems. Translocations represent a natural extension of the notion of adjacent transpositions and as such may be analyzed using related concepts in combinatorics and rank modulation coding. Our results include derivation of the asymptotic capacity of translocation rank codes, construction techniques for asymptotically good codes, as well as simple decoding methods for one class of constructed codes. As part of our exposition, we also highlight the close connections between the new code family and permutations with short common subsequences, deletion and insertion error-correcting codes for permutations, and permutation codes in the Hamming distance.

Error Correction for Index Coding With Side Information

A problem of index coding with side information was first considered by Birk and Kol in 1998. In this study, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Error-correcting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols δ , what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortest length error-correcting code with a prescribed error-correcting capability in the classical coding theory. The Singleton bound and two other bounds, referred to as the α-bound and the κ -bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with a maximum distance separable classical code is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another inexplicit bound on the length of an optimal linear ECIC. Further, the problem of error-correcting decoding by a linear ECIC is studied. It is shown that in order to decode correctly the desired symbol, the decoder is required to find one of the vectors, belonging to an affine space containing the actual error vector. The syndrome decoding is shown to produce the correct output if the weight of the error pattern is less or equal to the error-correcting capability of the corresponding ECIC. Finally, the notion of static ECIC, which is suitable for use with a family of instances of an index coding problem, is introduced. Several bounds on the length of static ECICs are derived, and constructions for static ECICs are discussed. Connections of these codes to weakly resilient Boolean functions are established.

Improved Nearly-MDS Expander Codes

A construction of expander codes is presented with the following three properties: i) the codes lie close to the Singleton bound, ii) they can be encoded in time complexity that is linear in their code length, and iii) they have a linear-time bounded-distance decoder. By using a version of the decoder that corrects also erasures, the codes can replace maximum-distance separable (MDS) outer codes in concatenated constructions, thus resulting in linear-time encodable and decodable codes that approach the Zyablov bound or the capacity of memoryless channels. The presented construction improves on an earlier result by Guruswami and Indyk in that any rate and relative minimum distance that lies below the Singleton bound is attainable for a significantly smaller alphabet size

On the Security of Index Coding With Side Information

Security aspects of the index coding with side information (ICSI) problem are investigated. Building on the results of Bar-Yossef (2006), the properties of linear index codes are further explored. The notion of weak security, considered by Bhattad and Narayanan (2005) in the context of network coding, is generalized to block security. It is shown that the linear index code based on a matrix <i>L</i>, whose column space code <i>C</i>(<i>L</i>) has length <i>n</i>, minimum distance <i>d</i> , and dual distance <i>d</i>^⊥ , is (<i>d</i>-1-<i>t</i>) -block secure (and hence also weakly secure) if the adversary knows in advance <i>t</i> ≤ <i>d</i>-2 messages, and is completely insecure if the adversary knows in advance more than <i>n</i> - <i>d</i>^⊥ messages. Strong security is examined under the conditions that the adversary: 1) possesses <i>t</i> messages in advance; 2) eavesdrops at most μ transmissions; 3) corrupts at most δ transmissions. We prove that for sufficiently large <i>q</i> , an optimal linear index code which is strongly secure against such an adversary has length κ_q+μ+2δ . Here, κ_q is a generalization of the min-rank over F_q of the side information graph for the ICSI problem in its original formulation in the work of Bar-Yossef et al.

Generalized minimum distance iterative decoding of expander codes

Recently, G. Zemor (see IEEE Trans. Inf. Theory, vol.47, p.835-7, 2001) proposed an improvement on the Sipser-Spielman analysis of expander codes (Sipser, M. and Spielman, D.A., IEEE Trans. Inf. Theory, vol.42 , p.1710-22, 1996) and presented a linear-time iterative decoder that can correct a number of errors up to approximately 1/4 the known lower bound on the minimum distance of the code. We propose an improvement on Zemors decoder for F=GF(2), with the number of correctable errors becoming close to half the lower bound on the minimum distance. The improvement is obtained by inserting into the decoding algorithm features akin to generalized minimum distance decoding of concatenated codes.

Probabilistic algorithm for finding roots of linearized polynomials

AbstractA probabilistic algorithm is presented for finding a basis of the root space of a linearized polynomial

Linear Batch Codes

L(x) = \sum_{i=0}^t L_i x^{q^i}