Improved error bounds for the erasure/list scheme: the binary and spherical cases

We derive improved bounds on the error and erasure rate for spherical codes and for binary linear codes under Forney's erasure/list decoding scheme and prove some related results.Comment: 18 pages, 3 figures. Submitted to IEEE Transactions on Informatin Theory in May 2001, will appear in Oct. 2004 (tentative

On the Number of Errors Correctable with Codes on Graphs

We study ensembles of codes on graphs (generalized low-density parity-check, or LDPC codes) constructed from random graphs and fixed local constrained codes, and their extension to codes on hypergraphs. It is known that the average minimum distance of codes in these ensembles grows linearly with the code length. We show that these codes can correct a linearly growing number of errors under simple iterative decoding algorithms. In particular, we show that this property extends to codes constructed by parallel concatenation of Hamming codes and other codes with small minimum distance. Previously known results that proved this property for graph codes relied on graph expansion and required the choice of local codes with large distance relative to their length.Comment: Published in the Ralf Koetter Memorial Issue of IEEE Transactions on Information Theor

Polar Codes for Distributed Hierarchical Source Coding

We show that polar codes can be used to achieve the rate-distortion functions in the problem of hierarchical source coding also known as the successive refinement problem. We also analyze the distributed version of this problem, constructing a polar coding scheme that achieves the rate distortion functions for successive refinement with side information.Comment: 14 page

New bounds for equiangular lines

A set of lines in $\mathbb{R}^n$ is called equiangular if the angle between each pair of lines is the same. We address the question of determining the maximum size of equiangular line sets in $\mathbb{R}^n$, using semidefinite programming to improve the upper bounds on this quantity. Improvements are obtained in dimensions $24 \leq n \leq 136$. In particular, we show that the maximum number of equiangular lines in $\mathbb{R}^n$ is $276$ for all $24 \leq n \leq 41$ and is 344 for $n=43.$ This provides a partial resolution of the conjecture set forth by Lemmens and Seidel (1973).Comment: Minor corrections; added one new reference. To appear in "Discrete Geometry and Algebraic Combinatorics," A. Barg and O. R. Musin, Editors, Providence: RI, AMS (2014). AMS Contemporary Mathematics serie