Navin Kashyap

Shortened Array Codes of Large Girth

One approach to designing structured low-density parity-check (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity-check matrix contain all the variables involved in short cycles. This approach is especially effective if the parity-check matrix of a code is a matrix composed of blocks of circulant permutation matrices, as is the case for the class of codes known as array codes. We show how to shorten array codes by deleting certain columns of their parity-check matrices so as to increase their girth. The shortening approach is based on the observation that for array codes, and in fact for a slightly more general class of LDPC codes, the cycles in the corresponding Tanner graph are governed by certain homogeneous linear equations with integer coefficients. Consequently, we can selectively eliminate cycles from an array code by only retaining those columns from the parity-check matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles. We provide Ramsey-theoretic estimates for the maximum number of columns that can be retained from the original parity-check matrix with the property that the sequence of their indices avoid solutions to various types of cycle-governing equations. This translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles. Simulation results show that for the codes considered, shortening them to increase the girth can lead to significant gains in signal-to-noise ratio (SNR) in the case of communication over an additive white Gaussian noise (AWGN) channel

Stopping sets in codes from designs

The size of the smallest stopping set in LDPC codes helps in analyzing their performance under iterative decoding, just a minimum distance helps in analyzing the performance under maximum likelihood decoding. We study stopping sets in LDPC codes arising from 2-designs, in particular LDPC codes derived from projective and Euclidean geometries. We derive upper and lower bounds on the size of the smallest stopping set in such codes, and provide examples of codes that achieve these bounds.

A Decomposition Theory for Binary Linear Codes

The decomposition theory of matroids initiated by Paul Seymour in the 1980s has had an enormous impact on research in matroid theory. This theory, when applied to matrices over the binary field, yields a powerful decomposition theory for binary linear codes. In this paper, we give an overview of this code decomposition theory, and discuss some of its implications in the context of the recently discovered formulation of maximum-likelihood (ML) decoding of a binary linear code over a binary-input discrete memoryless channel as a linear programming problem. We translate matroid-theoretic results of Grotschel and Truemper from the combinatorial optimization literature to give examples of nontrivial families of codes for which the ML decoding problem can be solved in time polynomial in the length of the code. One such family is that consisting of codes for which the codeword polytope is identical to the Koetter-Vontobel fundamental polytope derived from the entire dual code Cperp. However, we also show that such families of codes are not good in a coding-theoretic sense-either their dimension or their minimum distance must grow sublinearly with code length. As a consequence, we have that decoding by linear programming, when applied to good codes, cannot avoid failing occasionally due to the presence of pseudocode words.

Coding for High-Density Recording on a 1-D Granular Magnetic Medium

In terabit-density magnetic recording, several bits of data can be replaced by the values of their neighbors in the storage medium. As a result, errors in the medium are dependent on each other and also on the data written. We consider a simple 1-D combinatorial model of this medium. In our model, we assume a setting where binary data is sequentially written on the medium and a bit can erroneously change to the immediately preceding value. We derive several properties of codes that correct this type of errors, focusing on bounds on their cardinality. We also define a probabilistic finite-state channel model of the storage medium, and derive lower and upper estimates of its capacity. A lower bound is derived by evaluating the symmetric capacity of the channel, i.e., the maximum transmission rate under the assumption of the uniform input distribution of the channel. An upper bound is found by showing that the original channel is a stochastic degradation of another, related channel model whose capacity we can compute explicitly.

Matroid Pathwidth and Code Trellis Complexity

We relate the notion of matroid pathwidth to the minimum trellis state-complexity (which we term trellis-width) of a linear code and to the pathwidth of a graph. By reducing from the problem of computing the pathwidth of a graph, we show that the problem of determining the pathwidth of a representable matroid is NP-hard. Consequently, the problem of computing the trellis-width of a linear code is also NP-hard. For a finite field

Secure Compute-and-Forward in a Bidirectional Relay

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Achieving SK capacity in the source model: When must all terminals talk?

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On Minimal Tree Realizations of Linear Codes

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Data synchronization with timing

-representable matroids of pathwidth at most

The Feedback Capacity of the Binary Erasure Channel With a No-Consecutive-Ones Input Constraint

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