Analysis of the Second Moment of the LT Decoder

We analyze the second moment of the ripple size during the LT decoding process and prove that the standard deviation of the ripple size for an LT-code with length $k$ is of the order of $\sqrt k.$ Together with a result by Karp et. al stating that the expectation of the ripple size is of the order of $k$ [3], this gives bounds on the error probability of the LT decoder. We also give an analytic expression for the variance of the ripple size up to terms of constant order, and refine the expression in [3] for the expectation of the ripple size up to terms of the order of $1/k$, thus providing a first step towards an analytic finite-length analysis of LT decoding.Comment: 5 pages, 1 figure; submitted to ISIT 200

Good Ensembles of Goppa Codes

It is well-known that random error-correcting codes achieve the Gilbert-Varshamov bound with high probability. In [2], the authors describe a construction which can be used to yield a polynomially large family of codes of which a large fraction achieve the Gilbert-Varshamov bound. In this project, we investigate ways to obtain codes known to achieve this bound, given such a family of codes. Since computing the minimum distance of a code is NP-hard, we work with a class of Goppa codes described in [1] whose minimum distance is known. We know that there exist Goppa codes which achieve the Gilbert-Varshamov bound, but we do not know if there are codes in this class which achieve it. We investigate various approaches to determining the rate of a code and try to apply them to this class of codes in order to determine if they achieve the Gilbert-Varshamov bound. These approaches include investigating upper bounds on the covering radius of a code and refining an existing lower bound on the code dimension. We also implemented the described class of Goppa codes using the PARI/GP computer algebra system [5], in order to obtain numerical values which would allow us to detect patterns and formulate conjectures regarding those codes

Untrusting network coding

In networks that perform linear network coding, an intermediate network node may receive a much larger number of linear equations of the source symbols than the number of messages it needs to send. For networks constructed by untrusted nodes, we propose a relaxed measure of security: we want to be untrusting, and allow each intermediate node to only learn as much information as the number of independent messages it needs to send. In this paper we formulate this problem and provide sufficient and necessary conditions for classes of combination networks. ©2012 IEEE

Irregular Product Codes

We introduce irregular product codes, a class of codes where each codeword is represented by a matrix and the entries in each row (column) of the matrix come from a component row (column) code. As opposed to standard product codes, we do not require that all component row codes nor all component column codes be the same. Relaxing this requirement can provide some additional attractive features such as allowing some regions of the codeword to be more error-resilient, providing a more refined spectrum of rates for finite lengths, and improved performance for some of these rates. We study these codes over erasure channels and prove that for any 0 < ε < 1, for many rate distributions on component row codes, there is a matching rate distribution on component column codes such that an irregular product code based on MDS codes with those rate distributions on the component codes has asymptotic rate 1 - ε and can decode on erasure channels having erasure probability <; ε (and having alphabet size equal to the alphabet size of the component MDS codes)

Symmetric LDPC codes are not necessarily locally testable

Locally testable codes, i.e., codes where membership in the code is testable with a constant number of queries, have played a central role in complexity theory. It is well known that a code must be a “low-density parity check ” (LDPC) code for it to be locally testable, but few LDPC codes are known to the locally testable, and even fewer classes of LDPC codes are known not to be locally testable. Indeed, most previous examples of codes that are not locally testable were also not LDPC. The only exception was in the work of Ben-Sasson et al. [2005] who showed that random LDPC codes are not locally testable. Random codes lack “structure ” and in particular “symmetries ” motivating the possibility that “symmetric LDPC ” codes are locally testable, a question raised in the work of Alon et al. [2005]. If true such a result would capture many of the basic ingredients of known locally testable codes. In this work we rule out such a possibility by giving a highly symmetric (“2-transitive”) family of LDPC codes that are not testable with constant number of queries. We do so by continuing the exploration of “affine-invariant codes ” — codes where the coordinates of the words are associated with a finite field, and the code is invariant under affine transformation

Good Ensembles of Goppa Codes

It is well-known that random error-correcting codes achieve the Gilbert-Varshamov bound with high probability. In [2], the authors describe a construction which can be used to yield a polynomially large family of codes of which a large fraction achieve the Gilbert Varshamov bound. In this project, we investigate ways to obtain codes known to achieve this bound, given such a family of codes. Since computing the minimum distance of a code is NP-hard, we work with a class of Goppa codes described in [1] whose minimum distance is known. We know that there exist Goppa codes which achieve the Gilbert-Varshamov bound, but we do not know if there are codes in this class which achieve it. We investigate various approaches to determining the rate of a code and try to apply them to this class of codes in order to determine if they achieve the Gilbert-Varshamov bound. These approaches include investigating upper bounds on th